fluids and particles: doodads and kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics...
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![Page 1: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/1.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Fluids and Particles: Doodads and Kinetics
Peter Constantin
Department of MathematicsThe University of Chicago
CIME, Cetraro September 2010
![Page 2: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/2.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Outline:
1 Equilibrium: Onsager Equation on Metric Spaces
2 Kinetics: Nonlinear Fokker-Planck Equation
3 Gradient System in Metric Spaces
![Page 3: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/3.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Outline:
1 Equilibrium: Onsager Equation on Metric Spaces
2 Kinetics: Nonlinear Fokker-Planck Equation
3 Gradient System in Metric Spaces
![Page 4: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/4.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Outline:
1 Equilibrium: Onsager Equation on Metric Spaces
2 Kinetics: Nonlinear Fokker-Planck Equation
3 Gradient System in Metric Spaces
![Page 5: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/5.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Major Problems
1 Modeling of interactions in the appropriate phase space
2 Relaxation mechanisms in the appropriate phase space
3 Derivation of Micro-Macro coupling
4 PDE existence theory for coupled system
5 Dimension reduction
![Page 6: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/6.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Major Problems
1 Modeling of interactions in the appropriate phase space
2 Relaxation mechanisms in the appropriate phase space
3 Derivation of Micro-Macro coupling
4 PDE existence theory for coupled system
5 Dimension reduction
![Page 7: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/7.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Major Problems
1 Modeling of interactions in the appropriate phase space
2 Relaxation mechanisms in the appropriate phase space
3 Derivation of Micro-Macro coupling
4 PDE existence theory for coupled system
5 Dimension reduction
![Page 8: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/8.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Major Problems
1 Modeling of interactions in the appropriate phase space
2 Relaxation mechanisms in the appropriate phase space
3 Derivation of Micro-Macro coupling
4 PDE existence theory for coupled system
5 Dimension reduction
![Page 9: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/9.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Major Problems
1 Modeling of interactions in the appropriate phase space
2 Relaxation mechanisms in the appropriate phase space
3 Derivation of Micro-Macro coupling
4 PDE existence theory for coupled system
5 Dimension reduction
![Page 10: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/10.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
• Configuration space: M = (locally) compact, separable,metric space, d distance. m ∈ M = corpus = doodad.
• Reference measure: dµ – Borel Probability on M.• Corpora state f (m)dµ(m) – Probabililty, AC w.r. dµ.• Interaction kernel k : M ×M → R+,• symmetric: k(m, p) = k(p,m)• uniformly bi-Lipschitz:
|k(m, n)− k(p, n)| ≤ Ld(m, p)
• Potential U[f ](m) =∫M k(m, p)f (p)dµ(p)
• Potential U = micro-micro interaction• Free Energy
E [f ] =
∫M
f log fdµ +1
2
∫M
U[f ]fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1e−U[f ].
![Page 11: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/11.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
• Configuration space: M = (locally) compact, separable,metric space, d distance. m ∈ M = corpus = doodad.
• Reference measure: dµ – Borel Probability on M.
• Corpora state f (m)dµ(m) – Probabililty, AC w.r. dµ.• Interaction kernel k : M ×M → R+,• symmetric: k(m, p) = k(p,m)• uniformly bi-Lipschitz:
|k(m, n)− k(p, n)| ≤ Ld(m, p)
• Potential U[f ](m) =∫M k(m, p)f (p)dµ(p)
• Potential U = micro-micro interaction• Free Energy
E [f ] =
∫M
f log fdµ +1
2
∫M
U[f ]fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1e−U[f ].
![Page 12: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/12.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
• Configuration space: M = (locally) compact, separable,metric space, d distance. m ∈ M = corpus = doodad.
• Reference measure: dµ – Borel Probability on M.• Corpora state f (m)dµ(m) – Probabililty, AC w.r. dµ.
• Interaction kernel k : M ×M → R+,• symmetric: k(m, p) = k(p,m)• uniformly bi-Lipschitz:
|k(m, n)− k(p, n)| ≤ Ld(m, p)
• Potential U[f ](m) =∫M k(m, p)f (p)dµ(p)
• Potential U = micro-micro interaction• Free Energy
E [f ] =
∫M
f log fdµ +1
2
∫M
U[f ]fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1e−U[f ].
![Page 13: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/13.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
• Configuration space: M = (locally) compact, separable,metric space, d distance. m ∈ M = corpus = doodad.
• Reference measure: dµ – Borel Probability on M.• Corpora state f (m)dµ(m) – Probabililty, AC w.r. dµ.• Interaction kernel k : M ×M → R+,
• symmetric: k(m, p) = k(p,m)• uniformly bi-Lipschitz:
|k(m, n)− k(p, n)| ≤ Ld(m, p)
• Potential U[f ](m) =∫M k(m, p)f (p)dµ(p)
• Potential U = micro-micro interaction• Free Energy
E [f ] =
∫M
f log fdµ +1
2
∫M
U[f ]fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1e−U[f ].
![Page 14: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/14.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
• Configuration space: M = (locally) compact, separable,metric space, d distance. m ∈ M = corpus = doodad.
• Reference measure: dµ – Borel Probability on M.• Corpora state f (m)dµ(m) – Probabililty, AC w.r. dµ.• Interaction kernel k : M ×M → R+,• symmetric: k(m, p) = k(p,m)
• uniformly bi-Lipschitz:
|k(m, n)− k(p, n)| ≤ Ld(m, p)
• Potential U[f ](m) =∫M k(m, p)f (p)dµ(p)
• Potential U = micro-micro interaction• Free Energy
E [f ] =
∫M
f log fdµ +1
2
∫M
U[f ]fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1e−U[f ].
![Page 15: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/15.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
• Configuration space: M = (locally) compact, separable,metric space, d distance. m ∈ M = corpus = doodad.
• Reference measure: dµ – Borel Probability on M.• Corpora state f (m)dµ(m) – Probabililty, AC w.r. dµ.• Interaction kernel k : M ×M → R+,• symmetric: k(m, p) = k(p,m)• uniformly bi-Lipschitz:
|k(m, n)− k(p, n)| ≤ Ld(m, p)
• Potential U[f ](m) =∫M k(m, p)f (p)dµ(p)
• Potential U = micro-micro interaction• Free Energy
E [f ] =
∫M
f log fdµ +1
2
∫M
U[f ]fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1e−U[f ].
![Page 16: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/16.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
• Configuration space: M = (locally) compact, separable,metric space, d distance. m ∈ M = corpus = doodad.
• Reference measure: dµ – Borel Probability on M.• Corpora state f (m)dµ(m) – Probabililty, AC w.r. dµ.• Interaction kernel k : M ×M → R+,• symmetric: k(m, p) = k(p,m)• uniformly bi-Lipschitz:
|k(m, n)− k(p, n)| ≤ Ld(m, p)
• Potential U[f ](m) =∫M k(m, p)f (p)dµ(p)
• Potential U = micro-micro interaction• Free Energy
E [f ] =
∫M
f log fdµ +1
2
∫M
U[f ]fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1e−U[f ].
![Page 17: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/17.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
• Configuration space: M = (locally) compact, separable,metric space, d distance. m ∈ M = corpus = doodad.
• Reference measure: dµ – Borel Probability on M.• Corpora state f (m)dµ(m) – Probabililty, AC w.r. dµ.• Interaction kernel k : M ×M → R+,• symmetric: k(m, p) = k(p,m)• uniformly bi-Lipschitz:
|k(m, n)− k(p, n)| ≤ Ld(m, p)
• Potential U[f ](m) =∫M k(m, p)f (p)dµ(p)
• Potential U = micro-micro interaction
• Free Energy
E [f ] =
∫M
f log fdµ +1
2
∫M
U[f ]fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1e−U[f ].
![Page 18: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/18.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
• Configuration space: M = (locally) compact, separable,metric space, d distance. m ∈ M = corpus = doodad.
• Reference measure: dµ – Borel Probability on M.• Corpora state f (m)dµ(m) – Probabililty, AC w.r. dµ.• Interaction kernel k : M ×M → R+,• symmetric: k(m, p) = k(p,m)• uniformly bi-Lipschitz:
|k(m, n)− k(p, n)| ≤ Ld(m, p)
• Potential U[f ](m) =∫M k(m, p)f (p)dµ(p)
• Potential U = micro-micro interaction• Free Energy
E [f ] =
∫M
f log fdµ +1
2
∫M
U[f ]fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1e−U[f ].
![Page 19: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/19.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
• Configuration space: M = (locally) compact, separable,metric space, d distance. m ∈ M = corpus = doodad.
• Reference measure: dµ – Borel Probability on M.• Corpora state f (m)dµ(m) – Probabililty, AC w.r. dµ.• Interaction kernel k : M ×M → R+,• symmetric: k(m, p) = k(p,m)• uniformly bi-Lipschitz:
|k(m, n)− k(p, n)| ≤ Ld(m, p)
• Potential U[f ](m) =∫M k(m, p)f (p)dµ(p)
• Potential U = micro-micro interaction• Free Energy
E [f ] =
∫M
f log fdµ +1
2
∫M
U[f ]fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1e−U[f ].
![Page 20: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/20.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Examples: packing of soft spheres, low energy many particlequantum systems, ensembles of stick-and-ball models ofmolecules.
m is the system not the particle.
Goals:
1 Existence theory for solutions of Onsager’s equation
2 Classification of high intensity limits
3 Selection mechanisms for high intensity limits
4 Gradient system relaxation
![Page 21: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/21.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Examples: packing of soft spheres, low energy many particlequantum systems, ensembles of stick-and-ball models ofmolecules. m is the system not the particle.
Goals:
1 Existence theory for solutions of Onsager’s equation
2 Classification of high intensity limits
3 Selection mechanisms for high intensity limits
4 Gradient system relaxation
![Page 22: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/22.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Examples: packing of soft spheres, low energy many particlequantum systems, ensembles of stick-and-ball models ofmolecules. m is the system not the particle.
Goals:
1 Existence theory for solutions of Onsager’s equation
2 Classification of high intensity limits
3 Selection mechanisms for high intensity limits
4 Gradient system relaxation
![Page 23: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/23.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Examples: packing of soft spheres, low energy many particlequantum systems, ensembles of stick-and-ball models ofmolecules. m is the system not the particle.
Goals:
1 Existence theory for solutions of Onsager’s equation
2 Classification of high intensity limits
3 Selection mechanisms for high intensity limits
4 Gradient system relaxation
![Page 24: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/24.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Example: Rods, Maier-Saupe potential
M = Sn−1, dµ = area.
U[f ](p) = −b
∫Sn−1
((p · q)2 − 1
n
)f (q)dµ
b = intensity, inverse temperature.
![Page 25: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/25.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Dimension Reduction, Maier-Saupe
n × n symmetric, traceless matrix S :
S 7→ Z (S)
Z (S) =
∫Sn−1
eb(S ijmimj )dµ.
fS(m) = (Z (S))−1eb(S ijmimj )
σ(S)ij =
∫Sn−1
(mimj −
δij
n
)fS(m)dµ.
TheoremOnsager’s equation with Maier-Saupe potential is equivalent to
σ(S) = S .
![Page 26: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/26.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Dimension Reduction, Maier-Saupe
n × n symmetric, traceless matrix S :
S 7→ Z (S)
Z (S) =
∫Sn−1
eb(S ijmimj )dµ.
fS(m) = (Z (S))−1eb(S ijmimj )
σ(S)ij =
∫Sn−1
(mimj −
δij
n
)fS(m)dµ.
TheoremOnsager’s equation with Maier-Saupe potential is equivalent to
σ(S) = S .
![Page 27: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/27.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Dimension Reduction, Maier-Saupe
n × n symmetric, traceless matrix S :
S 7→ Z (S)
Z (S) =
∫Sn−1
eb(S ijmimj )dµ.
fS(m) = (Z (S))−1eb(S ijmimj )
σ(S)ij =
∫Sn−1
(mimj −
δij
n
)fS(m)dµ.
TheoremOnsager’s equation with Maier-Saupe potential is equivalent to
σ(S) = S .
![Page 28: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/28.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Dimension Reduction, Maier-Saupe
n × n symmetric, traceless matrix S :
S 7→ Z (S)
Z (S) =
∫Sn−1
eb(S ijmimj )dµ.
fS(m) = (Z (S))−1eb(S ijmimj )
σ(S)ij =
∫Sn−1
(mimj −
δij
n
)fS(m)dµ.
TheoremOnsager’s equation with Maier-Saupe potential is equivalent to
σ(S) = S .
![Page 29: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/29.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Onsager Equation, Maier-Saupe n = 3.
S ij = λiδij
λi ∈ [−13 , 2
3 ],λ1 + λ2 + λ3 = 0.
Let
v1 =1
2(λ1 + λ2), v2 =
1
2(λ1 − λ2).
y1(p) = 1− 3p2
y2(p, t) = (1− p2) cos t
for (p, t) ∈ K = [−1, 1]× [0, 2π].
y = y(p, t) = (y1(p), y2(p, t)), v = (v1, v2).
![Page 30: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/30.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Onsager Equation, Maier-Saupe n = 3.
S ij = λiδij
λi ∈ [−13 , 2
3 ],λ1 + λ2 + λ3 = 0.
Let
v1 =1
2(λ1 + λ2), v2 =
1
2(λ1 − λ2).
y1(p) = 1− 3p2
y2(p, t) = (1− p2) cos t
for (p, t) ∈ K = [−1, 1]× [0, 2π].
y = y(p, t) = (y1(p), y2(p, t)), v = (v1, v2).
![Page 31: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/31.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Onsager Equation, Maier-Saupe n = 3.
S ij = λiδij
λi ∈ [−13 , 2
3 ],λ1 + λ2 + λ3 = 0.
Let
v1 =1
2(λ1 + λ2), v2 =
1
2(λ1 − λ2).
y1(p) = 1− 3p2
y2(p, t) = (1− p2) cos t
for (p, t) ∈ K = [−1, 1]× [0, 2π].
y = y(p, t) = (y1(p), y2(p, t)), v = (v1, v2).
![Page 32: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/32.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
TheoremLet
Z2(v) =
∫K
ebv ·y(p,t)dpdt
F(v) = log(Z2(v))− b(3v2
1 + v22
).
Onsager’s equation: critical points of F , v ∈ [−13 , 2
3 ]× [0, 12 ],
i.e.: 6v1 = [y1](v)2v2 = [y2](v)
where, for any φ : K → R,
[φ](v) = (Z2(v))−1
∫K
φ(p, t)ebv ·y(p,t)dpdt
![Page 33: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/33.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Limit b →∞
[φ] =
∫S2
φ(m)f (m)dµ.
Isotropic:
limb→∞
[φ] =1
4π
∫S2
φ(p)dµ
Oblate:
limb→∞
[φ] =1
2π
∫ 2π
0φ(cos ϕ, sin ϕ, 0)dϕ
Prolate:lim
b→∞[φ] = φ(m), m ∈ S2.
![Page 34: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/34.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Limit b →∞
[φ] =
∫S2
φ(m)f (m)dµ.
Isotropic:
limb→∞
[φ] =1
4π
∫S2
φ(p)dµ
Oblate:
limb→∞
[φ] =1
2π
∫ 2π
0φ(cos ϕ, sin ϕ, 0)dϕ
Prolate:lim
b→∞[φ] = φ(m), m ∈ S2.
![Page 35: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/35.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Limit b →∞
[φ] =
∫S2
φ(m)f (m)dµ.
Isotropic:
limb→∞
[φ] =1
4π
∫S2
φ(p)dµ
Oblate:
limb→∞
[φ] =1
2π
∫ 2π
0φ(cos ϕ, sin ϕ, 0)dϕ
Prolate:lim
b→∞[φ] = φ(m), m ∈ S2.
![Page 36: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/36.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Limit b →∞
[φ] =
∫S2
φ(m)f (m)dµ.
Isotropic:
limb→∞
[φ] =1
4π
∫S2
φ(p)dµ
Oblate:
limb→∞
[φ] =1
2π
∫ 2π
0φ(cos ϕ, sin ϕ, 0)dϕ
Prolate:lim
b→∞[φ] = φ(m), m ∈ S2.
![Page 37: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/37.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
u(p1, q1, p2, q2, . . . ) =∑
i
ui (pi , qi )
U[f ] =N∑
i=1
Ui [f ], with
Ui [f ](pi ) =
∫eM ui (pi , qi )f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1e−eUef
Z = ΠNj=1Zj , with Zj =
∫Mj
e−Uj [fj ]dµj , fj = (Zj)−1e−Uj [fj ]
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 38: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/38.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
u(p1, q1, p2, q2, . . . ) =∑
i
ui (pi , qi )
U[f ] =N∑
i=1
Ui [f ], with
Ui [f ](pi ) =
∫eM ui (pi , qi )f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1e−eUef
Z = ΠNj=1Zj , with Zj =
∫Mj
e−Uj [fj ]dµj , fj = (Zj)−1e−Uj [fj ]
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 39: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/39.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
u(p1, q1, p2, q2, . . . ) =∑
i
ui (pi , qi )
U[f ] =N∑
i=1
Ui [f ],
with
Ui [f ](pi ) =
∫eM ui (pi , qi )f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1e−eUef
Z = ΠNj=1Zj , with Zj =
∫Mj
e−Uj [fj ]dµj , fj = (Zj)−1e−Uj [fj ]
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 40: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/40.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
u(p1, q1, p2, q2, . . . ) =∑
i
ui (pi , qi )
U[f ] =N∑
i=1
Ui [f ], with
Ui [f ](pi ) =
∫eM ui (pi , qi )f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1e−eUef
Z = ΠNj=1Zj , with Zj =
∫Mj
e−Uj [fj ]dµj , fj = (Zj)−1e−Uj [fj ]
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 41: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/41.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
u(p1, q1, p2, q2, . . . ) =∑
i
ui (pi , qi )
U[f ] =N∑
i=1
Ui [f ], with
Ui [f ](pi ) =
∫eM ui (pi , qi )f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1e−eUef
Z = ΠNj=1Zj , with Zj =
∫Mj
e−Uj [fj ]dµj , fj = (Zj)−1e−Uj [fj ]
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 42: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/42.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
u(p1, q1, p2, q2, . . . ) =∑
i
ui (pi , qi )
U[f ] =N∑
i=1
Ui [f ], with
Ui [f ](pi ) =
∫eM ui (pi , qi )f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1e−eUef
Z = ΠNj=1Zj , with Zj =
∫Mj
e−Uj [fj ]dµj , fj = (Zj)−1e−Uj [fj ]
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 43: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/43.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
u(p1, q1, p2, q2, . . . ) =∑
i
ui (pi , qi )
U[f ] =N∑
i=1
Ui [f ], with
Ui [f ](pi ) =
∫eM ui (pi , qi )f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1e−eUef
Z = ΠNj=1Zj , with Zj =
∫Mj
e−Uj [fj ]dµj , fj = (Zj)−1e−Uj [fj ]
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 44: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/44.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Example of Interacting Corpora
M = S1, M = S1 × S1.
U[f ](p1, p2) =b
∫T2 ‖e(p1) ∧ e(p2)− e(q1) ∧ e(q2)‖2f (q1, q2)dq1dq2
with e(p) = (cos p, sin p) if p ∈ [0, 2π].
‖e(p1)∧e(p2)−e(q1)∧e(q2)‖2 = (sin(p1 − p2)− sin(q1 − q2))2
Dimension reduction: Onsager’s equation f = Z−1e−U[f ]
reduces toa = [sin θ](a)
with [φ](a) =
∫ 2π0 φ(θ)g(θ)dθ
g(θ) = Z−1e−b(sin(θ)−a)2
Z =∫ 2π0 e−b(sin(θ)−a)2dθ
![Page 45: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/45.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Example of Interacting Corpora
M = S1, M = S1 × S1.
U[f ](p1, p2) =b
∫T2 ‖e(p1) ∧ e(p2)− e(q1) ∧ e(q2)‖2f (q1, q2)dq1dq2
with e(p) = (cos p, sin p) if p ∈ [0, 2π].
‖e(p1)∧e(p2)−e(q1)∧e(q2)‖2 = (sin(p1 − p2)− sin(q1 − q2))2
Dimension reduction: Onsager’s equation f = Z−1e−U[f ]
reduces toa = [sin θ](a)
with [φ](a) =
∫ 2π0 φ(θ)g(θ)dθ
g(θ) = Z−1e−b(sin(θ)−a)2
Z =∫ 2π0 e−b(sin(θ)−a)2dθ
![Page 46: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/46.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Example of Interacting Corpora
M = S1, M = S1 × S1.
U[f ](p1, p2) =b
∫T2 ‖e(p1) ∧ e(p2)− e(q1) ∧ e(q2)‖2f (q1, q2)dq1dq2
with e(p) = (cos p, sin p) if p ∈ [0, 2π].
‖e(p1)∧e(p2)−e(q1)∧e(q2)‖2 = (sin(p1 − p2)− sin(q1 − q2))2
Dimension reduction: Onsager’s equation f = Z−1e−U[f ]
reduces toa = [sin θ](a)
with [φ](a) =
∫ 2π0 φ(θ)g(θ)dθ
g(θ) = Z−1e−b(sin(θ)−a)2
Z =∫ 2π0 e−b(sin(θ)−a)2dθ
![Page 47: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/47.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Example of Interacting Corpora
M = S1, M = S1 × S1.
U[f ](p1, p2) =b
∫T2 ‖e(p1) ∧ e(p2)− e(q1) ∧ e(q2)‖2f (q1, q2)dq1dq2
with e(p) = (cos p, sin p) if p ∈ [0, 2π].
‖e(p1)∧e(p2)−e(q1)∧e(q2)‖2 = (sin(p1 − p2)− sin(q1 − q2))2
Dimension reduction: Onsager’s equation f = Z−1e−U[f ]
reduces toa = [sin θ](a)
with [φ](a) =
∫ 2π0 φ(θ)g(θ)dθ
g(θ) = Z−1e−b(sin(θ)−a)2
Z =∫ 2π0 e−b(sin(θ)−a)2dθ
![Page 48: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/48.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Example of Interacting Corpora
M = S1, M = S1 × S1.
U[f ](p1, p2) =b
∫T2 ‖e(p1) ∧ e(p2)− e(q1) ∧ e(q2)‖2f (q1, q2)dq1dq2
with e(p) = (cos p, sin p) if p ∈ [0, 2π].
‖e(p1)∧e(p2)−e(q1)∧e(q2)‖2 = (sin(p1 − p2)− sin(q1 − q2))2
Dimension reduction: Onsager’s equation f = Z−1e−U[f ]
reduces toa = [sin θ](a)
with [φ](a) =
∫ 2π0 φ(θ)g(θ)dθ
g(θ) = Z−1e−b(sin(θ)−a)2
Z =∫ 2π0 e−b(sin(θ)−a)2dθ
![Page 49: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/49.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Example of Interacting Corpora
M = S1, M = S1 × S1.
U[f ](p1, p2) =b
∫T2 ‖e(p1) ∧ e(p2)− e(q1) ∧ e(q2)‖2f (q1, q2)dq1dq2
with e(p) = (cos p, sin p) if p ∈ [0, 2π].
‖e(p1)∧e(p2)−e(q1)∧e(q2)‖2 = (sin(p1 − p2)− sin(q1 − q2))2
Dimension reduction: Onsager’s equation f = Z−1e−U[f ]
reduces toa = [sin θ](a)
with [φ](a) =
∫ 2π0 φ(θ)g(θ)dθ
g(θ) = Z−1e−b(sin(θ)−a)2
Z =∫ 2π0 e−b(sin(θ)−a)2dθ
![Page 50: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/50.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The solution is f (θ1, θ2) = g(θ1 − θ2).
Let
φ(θ, a) = sin θ − a,
and let
[φ](b, a) =
∫ 2π0 φ(θ, a)e−bφ2(θ,a)dθ∫ 2π
0 e−bφ2(θ,a)dθ.
The Onsager equation is equivalent to
[φ](b, a) = 0.
This determines a, which in turn determines g , f .a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 51: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/51.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The solution is f (θ1, θ2) = g(θ1 − θ2). Let
φ(θ, a) = sin θ − a,
and let
[φ](b, a) =
∫ 2π0 φ(θ, a)e−bφ2(θ,a)dθ∫ 2π
0 e−bφ2(θ,a)dθ.
The Onsager equation is equivalent to
[φ](b, a) = 0.
This determines a, which in turn determines g , f .a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 52: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/52.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The solution is f (θ1, θ2) = g(θ1 − θ2). Let
φ(θ, a) = sin θ − a,
and let
[φ](b, a) =
∫ 2π0 φ(θ, a)e−bφ2(θ,a)dθ∫ 2π
0 e−bφ2(θ,a)dθ.
The Onsager equation is equivalent to
[φ](b, a) = 0.
This determines a, which in turn determines g , f .a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 53: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/53.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The solution is f (θ1, θ2) = g(θ1 − θ2). Let
φ(θ, a) = sin θ − a,
and let
[φ](b, a) =
∫ 2π0 φ(θ, a)e−bφ2(θ,a)dθ∫ 2π
0 e−bφ2(θ,a)dθ.
The Onsager equation is equivalent to
[φ](b, a) = 0.
This determines a, which in turn determines g , f .a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 54: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/54.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The solution is f (θ1, θ2) = g(θ1 − θ2). Let
φ(θ, a) = sin θ − a,
and let
[φ](b, a) =
∫ 2π0 φ(θ, a)e−bφ2(θ,a)dθ∫ 2π
0 e−bφ2(θ,a)dθ.
The Onsager equation is equivalent to
[φ](b, a) = 0.
This determines a, which in turn determines g , f .
a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 55: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/55.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The solution is f (θ1, θ2) = g(θ1 − θ2). Let
φ(θ, a) = sin θ − a,
and let
[φ](b, a) =
∫ 2π0 φ(θ, a)e−bφ2(θ,a)dθ∫ 2π
0 e−bφ2(θ,a)dθ.
The Onsager equation is equivalent to
[φ](b, a) = 0.
This determines a, which in turn determines g , f .a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 56: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/56.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The solution is f (θ1, θ2) = g(θ1 − θ2). Let
φ(θ, a) = sin θ − a,
and let
[φ](b, a) =
∫ 2π0 φ(θ, a)e−bφ2(θ,a)dθ∫ 2π
0 e−bφ2(θ,a)dθ.
The Onsager equation is equivalent to
[φ](b, a) = 0.
This determines a, which in turn determines g , f .a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 57: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/57.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Consider
λ(a, τ) = b12
∫ 2π
0e−b(sin θ−a)2dθ
with τ = b−1.
Note
[φ] =1
2b
∂aλ
λ
and
∂τλ =1
4∂2
aλ
limτ→0
λ(a, τ) = 2√
π1√
1− a2, 0 < a < 1.
Increasing. But things are subtle, ∂λ∂a (1, τ) < 0.
![Page 58: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/58.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Consider
λ(a, τ) = b12
∫ 2π
0e−b(sin θ−a)2dθ
with τ = b−1.Note
[φ] =1
2b
∂aλ
λ
and
∂τλ =1
4∂2
aλ
limτ→0
λ(a, τ) = 2√
π1√
1− a2, 0 < a < 1.
Increasing. But things are subtle, ∂λ∂a (1, τ) < 0.
![Page 59: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/59.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Consider
λ(a, τ) = b12
∫ 2π
0e−b(sin θ−a)2dθ
with τ = b−1.Note
[φ] =1
2b
∂aλ
λ
and
∂τλ =1
4∂2
aλ
limτ→0
λ(a, τ) = 2√
π1√
1− a2, 0 < a < 1.
Increasing. But things are subtle, ∂λ∂a (1, τ) < 0.
![Page 60: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/60.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Consider
λ(a, τ) = b12
∫ 2π
0e−b(sin θ−a)2dθ
with τ = b−1.Note
[φ] =1
2b
∂aλ
λ
and
∂τλ =1
4∂2
aλ
limτ→0
λ(a, τ) = 2√
π1√
1− a2, 0 < a < 1.
Increasing.
But things are subtle, ∂λ∂a (1, τ) < 0.
![Page 61: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/61.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Consider
λ(a, τ) = b12
∫ 2π
0e−b(sin θ−a)2dθ
with τ = b−1.Note
[φ] =1
2b
∂aλ
λ
and
∂τλ =1
4∂2
aλ
limτ→0
λ(a, τ) = 2√
π1√
1− a2, 0 < a < 1.
Increasing. But things are subtle, ∂λ∂a (1, τ) < 0.
![Page 62: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/62.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
In fact, phase transition at positive τ
∂aλ((a(τ), τ) = 0
and limit limτ→0 a(τ) = 1.
Consider
hb(a) =
∫ 2π
0ue−bu2
dθ
with u(θ, a) = sin θ − a. We seek zeros of hb(a). Clearly,hb(1) < 0. ∃ 0 < a < 1 such that hb(a) > 0: Changingvariables:
hb(a) = 2
∫ 1−a
−1−ae−bu2 udu√
1− (u + a)2
hb(a) = 4∫ 1−a0 e−bu2
u
1√
1−(u+a)2− 1√
1−(a−u)2
du
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2
= 16a∫ 1−a0
e−bu2u2du√
(1−(u+a)2)(1−(u−a)2)“√
1−(u−a)2+√
1−(u+a)2”
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2.
![Page 63: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/63.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
In fact, phase transition at positive τ
∂aλ((a(τ), τ) = 0
and limit limτ→0 a(τ) = 1. Consider
hb(a) =
∫ 2π
0ue−bu2
dθ
with u(θ, a) = sin θ − a.
We seek zeros of hb(a). Clearly,hb(1) < 0. ∃ 0 < a < 1 such that hb(a) > 0: Changingvariables:
hb(a) = 2
∫ 1−a
−1−ae−bu2 udu√
1− (u + a)2
hb(a) = 4∫ 1−a0 e−bu2
u
1√
1−(u+a)2− 1√
1−(a−u)2
du
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2
= 16a∫ 1−a0
e−bu2u2du√
(1−(u+a)2)(1−(u−a)2)“√
1−(u−a)2+√
1−(u+a)2”
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2.
![Page 64: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/64.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
In fact, phase transition at positive τ
∂aλ((a(τ), τ) = 0
and limit limτ→0 a(τ) = 1. Consider
hb(a) =
∫ 2π
0ue−bu2
dθ
with u(θ, a) = sin θ − a. We seek zeros of hb(a).
Clearly,hb(1) < 0. ∃ 0 < a < 1 such that hb(a) > 0: Changingvariables:
hb(a) = 2
∫ 1−a
−1−ae−bu2 udu√
1− (u + a)2
hb(a) = 4∫ 1−a0 e−bu2
u
1√
1−(u+a)2− 1√
1−(a−u)2
du
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2
= 16a∫ 1−a0
e−bu2u2du√
(1−(u+a)2)(1−(u−a)2)“√
1−(u−a)2+√
1−(u+a)2”
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2.
![Page 65: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/65.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
In fact, phase transition at positive τ
∂aλ((a(τ), τ) = 0
and limit limτ→0 a(τ) = 1. Consider
hb(a) =
∫ 2π
0ue−bu2
dθ
with u(θ, a) = sin θ − a. We seek zeros of hb(a). Clearly,hb(1) < 0.
∃ 0 < a < 1 such that hb(a) > 0: Changingvariables:
hb(a) = 2
∫ 1−a
−1−ae−bu2 udu√
1− (u + a)2
hb(a) = 4∫ 1−a0 e−bu2
u
1√
1−(u+a)2− 1√
1−(a−u)2
du
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2
= 16a∫ 1−a0
e−bu2u2du√
(1−(u+a)2)(1−(u−a)2)“√
1−(u−a)2+√
1−(u+a)2”
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2.
![Page 66: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/66.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
In fact, phase transition at positive τ
∂aλ((a(τ), τ) = 0
and limit limτ→0 a(τ) = 1. Consider
hb(a) =
∫ 2π
0ue−bu2
dθ
with u(θ, a) = sin θ − a. We seek zeros of hb(a). Clearly,hb(1) < 0. ∃ 0 < a < 1 such that hb(a) > 0:
Changingvariables:
hb(a) = 2
∫ 1−a
−1−ae−bu2 udu√
1− (u + a)2
hb(a) = 4∫ 1−a0 e−bu2
u
1√
1−(u+a)2− 1√
1−(a−u)2
du
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2
= 16a∫ 1−a0
e−bu2u2du√
(1−(u+a)2)(1−(u−a)2)“√
1−(u−a)2+√
1−(u+a)2”
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2.
![Page 67: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/67.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
In fact, phase transition at positive τ
∂aλ((a(τ), τ) = 0
and limit limτ→0 a(τ) = 1. Consider
hb(a) =
∫ 2π
0ue−bu2
dθ
with u(θ, a) = sin θ − a. We seek zeros of hb(a). Clearly,hb(1) < 0. ∃ 0 < a < 1 such that hb(a) > 0: Changingvariables:
hb(a) = 2
∫ 1−a
−1−ae−bu2 udu√
1− (u + a)2
hb(a) = 4∫ 1−a0 e−bu2
u
1√
1−(u+a)2− 1√
1−(a−u)2
du
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2
= 16a∫ 1−a0
e−bu2u2du√
(1−(u+a)2)(1−(u−a)2)“√
1−(u−a)2+√
1−(u+a)2”
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2.
![Page 68: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/68.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
In fact, phase transition at positive τ
∂aλ((a(τ), τ) = 0
and limit limτ→0 a(τ) = 1. Consider
hb(a) =
∫ 2π
0ue−bu2
dθ
with u(θ, a) = sin θ − a. We seek zeros of hb(a). Clearly,hb(1) < 0. ∃ 0 < a < 1 such that hb(a) > 0: Changingvariables:
hb(a) = 2
∫ 1−a
−1−ae−bu2 udu√
1− (u + a)2
hb(a) = 4∫ 1−a0 e−bu2
u
1√
1−(u+a)2− 1√
1−(a−u)2
du
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2
= 16a∫ 1−a0
e−bu2u2du√
(1−(u+a)2)(1−(u−a)2)“√
1−(u−a)2+√
1−(u+a)2”
−2∫ 1+a1−a e−bu2 udu√
1−(a−u)2.
![Page 69: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/69.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The last integral is negative but exponentially small as b →∞.
The positive integral is larger than a fixed multiple of∫ δ0 u2e−bu2
du for small fixed δ (depending on a). This integral
asymptotically equals Cb−3/2 with positive C , as b →∞.Thus, for any fixed 0 < a < 1 and b large enough, we havehb(a) > 0. This proves for all large enough b the existence ofa(b) > 0 such that hb(a(b)) = 0. Moreover, because 0 < a < 1was arbitrary, this also proves limb→∞ a(b) = 1 for any sucha(b) > 0.
limb→∞
f (p1 − p2) = δ((
p1 − p2 −π
2
)modπ
)
![Page 70: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/70.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The last integral is negative but exponentially small as b →∞.The positive integral is larger than a fixed multiple of∫ δ0 u2e−bu2
du for small fixed δ (depending on a).
This integral
asymptotically equals Cb−3/2 with positive C , as b →∞.Thus, for any fixed 0 < a < 1 and b large enough, we havehb(a) > 0. This proves for all large enough b the existence ofa(b) > 0 such that hb(a(b)) = 0. Moreover, because 0 < a < 1was arbitrary, this also proves limb→∞ a(b) = 1 for any sucha(b) > 0.
limb→∞
f (p1 − p2) = δ((
p1 − p2 −π
2
)modπ
)
![Page 71: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/71.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The last integral is negative but exponentially small as b →∞.The positive integral is larger than a fixed multiple of∫ δ0 u2e−bu2
du for small fixed δ (depending on a). This integral
asymptotically equals Cb−3/2 with positive C , as b →∞.
Thus, for any fixed 0 < a < 1 and b large enough, we havehb(a) > 0. This proves for all large enough b the existence ofa(b) > 0 such that hb(a(b)) = 0. Moreover, because 0 < a < 1was arbitrary, this also proves limb→∞ a(b) = 1 for any sucha(b) > 0.
limb→∞
f (p1 − p2) = δ((
p1 − p2 −π
2
)modπ
)
![Page 72: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/72.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The last integral is negative but exponentially small as b →∞.The positive integral is larger than a fixed multiple of∫ δ0 u2e−bu2
du for small fixed δ (depending on a). This integral
asymptotically equals Cb−3/2 with positive C , as b →∞.Thus, for any fixed 0 < a < 1 and b large enough, we havehb(a) > 0.
This proves for all large enough b the existence ofa(b) > 0 such that hb(a(b)) = 0. Moreover, because 0 < a < 1was arbitrary, this also proves limb→∞ a(b) = 1 for any sucha(b) > 0.
limb→∞
f (p1 − p2) = δ((
p1 − p2 −π
2
)modπ
)
![Page 73: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/73.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The last integral is negative but exponentially small as b →∞.The positive integral is larger than a fixed multiple of∫ δ0 u2e−bu2
du for small fixed δ (depending on a). This integral
asymptotically equals Cb−3/2 with positive C , as b →∞.Thus, for any fixed 0 < a < 1 and b large enough, we havehb(a) > 0. This proves for all large enough b the existence ofa(b) > 0 such that hb(a(b)) = 0. Moreover, because 0 < a < 1was arbitrary, this also proves limb→∞ a(b) = 1 for any sucha(b) > 0.
limb→∞
f (p1 − p2) = δ((
p1 − p2 −π
2
)modπ
)
![Page 74: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/74.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The last integral is negative but exponentially small as b →∞.The positive integral is larger than a fixed multiple of∫ δ0 u2e−bu2
du for small fixed δ (depending on a). This integral
asymptotically equals Cb−3/2 with positive C , as b →∞.Thus, for any fixed 0 < a < 1 and b large enough, we havehb(a) > 0. This proves for all large enough b the existence ofa(b) > 0 such that hb(a(b)) = 0. Moreover, because 0 < a < 1was arbitrary, this also proves limb→∞ a(b) = 1 for any sucha(b) > 0.
limb→∞
f (p1 − p2) = δ((
p1 − p2 −π
2
)modπ
)
![Page 75: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/75.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
More degrees of freedom
M = [0, L]× [0, L]× [0, π], dµ = 1πL2 dx1dx2dθ.
U[f ](x1, x2, θ) =
∫M
(x1x2 sin(θ)− y1y2 sin(φ))2f (y1, y2, φ)dµ
The solutions of Onsager’s equation are of the form
g(x1, x2, θ) = Z−1e−b(x1x2 sin θ−a)2
with Z determined by the requirement of normalization∫M gdµ = 1, a determined by
a =
∫M
(x1x2 sin θ)g(x1, x2, θ)dµ
![Page 76: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/76.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
More degrees of freedom
M = [0, L]× [0, L]× [0, π], dµ = 1πL2 dx1dx2dθ.
U[f ](x1, x2, θ) =
∫M
(x1x2 sin(θ)− y1y2 sin(φ))2f (y1, y2, φ)dµ
The solutions of Onsager’s equation are of the form
g(x1, x2, θ) = Z−1e−b(x1x2 sin θ−a)2
with Z determined by the requirement of normalization∫M gdµ = 1, a determined by
a =
∫M
(x1x2 sin θ)g(x1, x2, θ)dµ
![Page 77: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/77.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
More degrees of freedom
M = [0, L]× [0, L]× [0, π], dµ = 1πL2 dx1dx2dθ.
U[f ](x1, x2, θ) =
∫M
(x1x2 sin(θ)− y1y2 sin(φ))2f (y1, y2, φ)dµ
The solutions of Onsager’s equation are of the form
g(x1, x2, θ) = Z−1e−b(x1x2 sin θ−a)2
with Z determined by the requirement of normalization∫M gdµ = 1, a determined by
a =
∫M
(x1x2 sin θ)g(x1, x2, θ)dµ
![Page 78: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/78.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
More degrees of freedom
M = [0, L]× [0, L]× [0, π], dµ = 1πL2 dx1dx2dθ.
U[f ](x1, x2, θ) =
∫M
(x1x2 sin(θ)− y1y2 sin(φ))2f (y1, y2, φ)dµ
The solutions of Onsager’s equation are of the form
g(x1, x2, θ) = Z−1e−b(x1x2 sin θ−a)2
with Z determined by the requirement of normalization∫M gdµ = 1,
a determined by
a =
∫M
(x1x2 sin θ)g(x1, x2, θ)dµ
![Page 79: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/79.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
More degrees of freedom
M = [0, L]× [0, L]× [0, π], dµ = 1πL2 dx1dx2dθ.
U[f ](x1, x2, θ) =
∫M
(x1x2 sin(θ)− y1y2 sin(φ))2f (y1, y2, φ)dµ
The solutions of Onsager’s equation are of the form
g(x1, x2, θ) = Z−1e−b(x1x2 sin θ−a)2
with Z determined by the requirement of normalization∫M gdµ = 1, a determined by
a =
∫M
(x1x2 sin θ)g(x1, x2, θ)dµ
![Page 80: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/80.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Letφ(x1, x2, θ, a) = x1x2 sin θ − a
[φ] =
∫M
φgdµ
a is determined by [φ] = 0.
λ(a, τ) = τ−1/2
∫M
e−φ2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[φ] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 81: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/81.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Letφ(x1, x2, θ, a) = x1x2 sin θ − a
[φ] =
∫M
φgdµ
a is determined by [φ] = 0.
λ(a, τ) = τ−1/2
∫M
e−φ2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[φ] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 82: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/82.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Letφ(x1, x2, θ, a) = x1x2 sin θ − a
[φ] =
∫M
φgdµ
a is determined by [φ] = 0.
λ(a, τ) = τ−1/2
∫M
e−φ2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[φ] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 83: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/83.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Letφ(x1, x2, θ, a) = x1x2 sin θ − a
[φ] =
∫M
φgdµ
a is determined by [φ] = 0.
λ(a, τ) = τ−1/2
∫M
e−φ2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[φ] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 84: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/84.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Letφ(x1, x2, θ, a) = x1x2 sin θ − a
[φ] =
∫M
φgdµ
a is determined by [φ] = 0.
λ(a, τ) = τ−1/2
∫M
e−φ2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[φ] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 85: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/85.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Letφ(x1, x2, θ, a) = x1x2 sin θ − a
[φ] =
∫M
φgdµ
a is determined by [φ] = 0.
λ(a, τ) = τ−1/2
∫M
e−φ2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[φ] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 86: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/86.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Letφ(x1, x2, θ, a) = x1x2 sin θ − a
[φ] =
∫M
φgdµ
a is determined by [φ] = 0.
λ(a, τ) = τ−1/2
∫M
e−φ2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[φ] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 87: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/87.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Even More Degrees of Freedom...
V (r) nonnegative, nonincreasing, compactly supported.p = (x1, . . . xN), xi ∈ Ω ⊂ Rn.
Packing energy:
F (p) =∑i<j
V (|xi − xj |).
M = Ω× · · · × Ω ∩ F ≤ F0.
U[f ](p) =
∫eM
[|F (p)− F (q)|2 + d2(p, q)
]f (q)dq
Connection to the example of freely articulated 2n doodads,jamming, perhaps...
![Page 88: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/88.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Even More Degrees of Freedom...
V (r) nonnegative, nonincreasing, compactly supported.p = (x1, . . . xN), xi ∈ Ω ⊂ Rn. Packing energy:
F (p) =∑i<j
V (|xi − xj |).
M = Ω× · · · × Ω ∩ F ≤ F0.
U[f ](p) =
∫eM
[|F (p)− F (q)|2 + d2(p, q)
]f (q)dq
Connection to the example of freely articulated 2n doodads,jamming, perhaps...
![Page 89: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/89.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Even More Degrees of Freedom...
V (r) nonnegative, nonincreasing, compactly supported.p = (x1, . . . xN), xi ∈ Ω ⊂ Rn. Packing energy:
F (p) =∑i<j
V (|xi − xj |).
M = Ω× · · · × Ω ∩ F ≤ F0.
U[f ](p) =
∫eM
[|F (p)− F (q)|2 + d2(p, q)
]f (q)dq
Connection to the example of freely articulated 2n doodads,jamming, perhaps...
![Page 90: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/90.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Even More Degrees of Freedom...
V (r) nonnegative, nonincreasing, compactly supported.p = (x1, . . . xN), xi ∈ Ω ⊂ Rn. Packing energy:
F (p) =∑i<j
V (|xi − xj |).
M = Ω× · · · × Ω ∩ F ≤ F0.
U[f ](p) =
∫eM
[|F (p)− F (q)|2 + d2(p, q)
]f (q)dq
Connection to the example of freely articulated 2n doodads,jamming, perhaps...
![Page 91: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/91.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Even More Degrees of Freedom...
V (r) nonnegative, nonincreasing, compactly supported.p = (x1, . . . xN), xi ∈ Ω ⊂ Rn. Packing energy:
F (p) =∑i<j
V (|xi − xj |).
M = Ω× · · · × Ω ∩ F ≤ F0.
U[f ](p) =
∫eM
[|F (p)− F (q)|2 + d2(p, q)
]f (q)dq
Connection to the example of freely articulated 2n doodads,jamming, perhaps...
![Page 92: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/92.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
TheoremFor any b > 0 there exists a solution g that minimizes theenergy:
E [g ] = minf≥0,
RM fdµ=1
E [f ]
The function g solves the Onsager equation
g(x) = (Z (b))−1e−bU(x)
with
Z (b) =
∫M
e−bU(x)dµ(x)
and
U(x) =
∫M
k(x , y)g(y)dµ(y).
The function g is normalized∫
gdµ = 1, strictly positive andLipschitz continuous.
![Page 93: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/93.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
TheoremFor any b > 0 there exists a solution g that minimizes theenergy:
E [g ] = minf≥0,
RM fdµ=1
E [f ]
The function g solves the Onsager equation
g(x) = (Z (b))−1e−bU(x)
with
Z (b) =
∫M
e−bU(x)dµ(x)
and
U(x) =
∫M
k(x , y)g(y)dµ(y).
The function g is normalized∫
gdµ = 1, strictly positive andLipschitz continuous.
![Page 94: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/94.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
TheoremFor any b > 0 there exists a solution g that minimizes theenergy:
E [g ] = minf≥0,
RM fdµ=1
E [f ]
The function g solves the Onsager equation
g(x) = (Z (b))−1e−bU(x)
with
Z (b) =
∫M
e−bU(x)dµ(x)
and
U(x) =
∫M
k(x , y)g(y)dµ(y).
The function g is normalized∫
gdµ = 1, strictly positive andLipschitz continuous.
![Page 95: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/95.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
TheoremFor any b > 0 there exists a solution g that minimizes theenergy:
E [g ] = minf≥0,
RM fdµ=1
E [f ]
The function g solves the Onsager equation
g(x) = (Z (b))−1e−bU(x)
with
Z (b) =
∫M
e−bU(x)dµ(x)
and
U(x) =
∫M
k(x , y)g(y)dµ(y).
The function g is normalized∫
gdµ = 1, strictly positive andLipschitz continuous.
![Page 96: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/96.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
TheoremFor any b > 0 there exists a solution g that minimizes theenergy:
E [g ] = minf≥0,
RM fdµ=1
E [f ]
The function g solves the Onsager equation
g(x) = (Z (b))−1e−bU(x)
with
Z (b) =
∫M
e−bU(x)dµ(x)
and
U(x) =
∫M
k(x , y)g(y)dµ(y).
The function g is normalized∫
gdµ = 1, strictly positive andLipschitz continuous.
![Page 97: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/97.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
TheoremFor any b > 0 there exists a solution g that minimizes theenergy:
E [g ] = minf≥0,
RM fdµ=1
E [f ]
The function g solves the Onsager equation
g(x) = (Z (b))−1e−bU(x)
with
Z (b) =
∫M
e−bU(x)dµ(x)
and
U(x) =
∫M
k(x , y)g(y)dµ(y).
The function g is normalized∫
gdµ = 1, strictly positive andLipschitz continuous.
![Page 98: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/98.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Idea of Proof0 ≤ U[f ](p) ≤ ‖u‖∞
|U[f ](p)− U[f ](q)| ≤ Ld(p, q)
Jensen, µ(M) = 1 ⇒ ∫M
f log fdµ ≥ 0.
Consequently:Eb[f ] ≥ 0.
Minimizing sequence fj ,
a = inff≥0,
RM fdµ=1
Eb[f ] = limj→∞
Eb[fj ].
WLOG fjdµ converge weakly to a measure dν. Uj = U[fj ]converge uniformly to a non-negative Lipschitz continuousfunction U. Then it follows that
U(p) =
∫M
k(p, q)dν(q)
![Page 99: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/99.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Idea of Proof0 ≤ U[f ](p) ≤ ‖u‖∞
|U[f ](p)− U[f ](q)| ≤ Ld(p, q)
Jensen, µ(M) = 1 ⇒ ∫M
f log fdµ ≥ 0.
Consequently:Eb[f ] ≥ 0.
Minimizing sequence fj ,
a = inff≥0,
RM fdµ=1
Eb[f ] = limj→∞
Eb[fj ].
WLOG fjdµ converge weakly to a measure dν. Uj = U[fj ]converge uniformly to a non-negative Lipschitz continuousfunction U. Then it follows that
U(p) =
∫M
k(p, q)dν(q)
![Page 100: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/100.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Idea of Proof0 ≤ U[f ](p) ≤ ‖u‖∞
|U[f ](p)− U[f ](q)| ≤ Ld(p, q)
Jensen, µ(M) = 1 ⇒ ∫M
f log fdµ ≥ 0.
Consequently:Eb[f ] ≥ 0.
Minimizing sequence fj ,
a = inff≥0,
RM fdµ=1
Eb[f ] = limj→∞
Eb[fj ].
WLOG fjdµ converge weakly to a measure dν. Uj = U[fj ]converge uniformly to a non-negative Lipschitz continuousfunction U. Then it follows that
U(p) =
∫M
k(p, q)dν(q)
![Page 101: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/101.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Idea of Proof0 ≤ U[f ](p) ≤ ‖u‖∞
|U[f ](p)− U[f ](q)| ≤ Ld(p, q)
Jensen, µ(M) = 1 ⇒ ∫M
f log fdµ ≥ 0.
Consequently:Eb[f ] ≥ 0.
Minimizing sequence fj ,
a = inff≥0,
RM fdµ=1
Eb[f ] = limj→∞
Eb[fj ].
WLOG fjdµ converge weakly to a measure dν. Uj = U[fj ]converge uniformly to a non-negative Lipschitz continuousfunction U. Then it follows that
U(p) =
∫M
k(p, q)dν(q)
![Page 102: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/102.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Idea of Proof0 ≤ U[f ](p) ≤ ‖u‖∞
|U[f ](p)− U[f ](q)| ≤ Ld(p, q)
Jensen, µ(M) = 1 ⇒ ∫M
f log fdµ ≥ 0.
Consequently:Eb[f ] ≥ 0.
Minimizing sequence fj ,
a = inff≥0,
RM fdµ=1
Eb[f ] = limj→∞
Eb[fj ].
WLOG fjdµ converge weakly to a measure dν.
Uj = U[fj ]converge uniformly to a non-negative Lipschitz continuousfunction U. Then it follows that
U(p) =
∫M
k(p, q)dν(q)
![Page 103: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/103.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Idea of Proof0 ≤ U[f ](p) ≤ ‖u‖∞
|U[f ](p)− U[f ](q)| ≤ Ld(p, q)
Jensen, µ(M) = 1 ⇒ ∫M
f log fdµ ≥ 0.
Consequently:Eb[f ] ≥ 0.
Minimizing sequence fj ,
a = inff≥0,
RM fdµ=1
Eb[f ] = limj→∞
Eb[fj ].
WLOG fjdµ converge weakly to a measure dν. Uj = U[fj ]converge uniformly to a non-negative Lipschitz continuousfunction U. Then it follows that
U(p) =
∫M
k(p, q)dν(q)
![Page 104: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/104.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
We claim dν AC wr dµ, i.e. dν = gdµ with g ≥ 0 andg ∈ L1(dµ).
Indeed, this is because the sequence fjdµ isuniformly absolutely continuous. The latter is proved usingthe convexity of the function y log y and Jensen’s inequality
µ(A)−1
∫A
fj log fjdµ ≥ m log m,
where m = m(A, j) = µ(A)−1∫A fjdµ. Thus for all A, j ,
m log m ≤ C
µ(A)
with a fixed C ≥ 1. Let us choose R = R(A) ≥ 1 so thatR log R = C/µ(A). Then m ≤ R and so∫A fjdµ ≤ µ(A)R = C/ log R. This means∫
Afjdµ ≤ δ(µ(A))
with limx→0 δ(x) = 0 and δ(·) independent of A and j . Itfollows that ν is absolutely continuous with respect to dµ.
![Page 105: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/105.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
We claim dν AC wr dµ, i.e. dν = gdµ with g ≥ 0 andg ∈ L1(dµ). Indeed, this is because the sequence fjdµ isuniformly absolutely continuous.
The latter is proved usingthe convexity of the function y log y and Jensen’s inequality
µ(A)−1
∫A
fj log fjdµ ≥ m log m,
where m = m(A, j) = µ(A)−1∫A fjdµ. Thus for all A, j ,
m log m ≤ C
µ(A)
with a fixed C ≥ 1. Let us choose R = R(A) ≥ 1 so thatR log R = C/µ(A). Then m ≤ R and so∫A fjdµ ≤ µ(A)R = C/ log R. This means∫
Afjdµ ≤ δ(µ(A))
with limx→0 δ(x) = 0 and δ(·) independent of A and j . Itfollows that ν is absolutely continuous with respect to dµ.
![Page 106: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/106.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
We claim dν AC wr dµ, i.e. dν = gdµ with g ≥ 0 andg ∈ L1(dµ). Indeed, this is because the sequence fjdµ isuniformly absolutely continuous. The latter is proved usingthe convexity of the function y log y and Jensen’s inequality
µ(A)−1
∫A
fj log fjdµ ≥ m log m,
where m = m(A, j) = µ(A)−1∫A fjdµ.
Thus for all A, j ,
m log m ≤ C
µ(A)
with a fixed C ≥ 1. Let us choose R = R(A) ≥ 1 so thatR log R = C/µ(A). Then m ≤ R and so∫A fjdµ ≤ µ(A)R = C/ log R. This means∫
Afjdµ ≤ δ(µ(A))
with limx→0 δ(x) = 0 and δ(·) independent of A and j . Itfollows that ν is absolutely continuous with respect to dµ.
![Page 107: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/107.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
We claim dν AC wr dµ, i.e. dν = gdµ with g ≥ 0 andg ∈ L1(dµ). Indeed, this is because the sequence fjdµ isuniformly absolutely continuous. The latter is proved usingthe convexity of the function y log y and Jensen’s inequality
µ(A)−1
∫A
fj log fjdµ ≥ m log m,
where m = m(A, j) = µ(A)−1∫A fjdµ. Thus for all A, j ,
m log m ≤ C
µ(A)
with a fixed C ≥ 1.
Let us choose R = R(A) ≥ 1 so thatR log R = C/µ(A). Then m ≤ R and so∫A fjdµ ≤ µ(A)R = C/ log R. This means∫
Afjdµ ≤ δ(µ(A))
with limx→0 δ(x) = 0 and δ(·) independent of A and j . Itfollows that ν is absolutely continuous with respect to dµ.
![Page 108: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/108.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
We claim dν AC wr dµ, i.e. dν = gdµ with g ≥ 0 andg ∈ L1(dµ). Indeed, this is because the sequence fjdµ isuniformly absolutely continuous. The latter is proved usingthe convexity of the function y log y and Jensen’s inequality
µ(A)−1
∫A
fj log fjdµ ≥ m log m,
where m = m(A, j) = µ(A)−1∫A fjdµ. Thus for all A, j ,
m log m ≤ C
µ(A)
with a fixed C ≥ 1. Let us choose R = R(A) ≥ 1 so thatR log R = C/µ(A). Then m ≤ R and so∫A fjdµ ≤ µ(A)R = C/ log R. This means∫
Afjdµ ≤ δ(µ(A))
with limx→0 δ(x) = 0 and δ(·) independent of A and j . Itfollows that ν is absolutely continuous with respect to dµ.
![Page 109: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/109.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
In general, weak convergence of measures is not enough toshow lower semicontinuity of nonlinear integrals or almosteverywhere convergence.
We claim however that, in fact, theconvergence fn → g takes place strongly in L1(dµ):
limn→∞
∫M|fn(p)− g(p)|dµ(p) = 0.
In order to prove this, we prove that fn is a Cauchy sequencein L1(dµ). We take ε > 0 and choose N large enough so that
supp∈M
∣∣Un(p)− U(p)∣∣ ≤ ε2
16b,
and
Eb[fn] ≤ a +ε2
16
holds for n ≥ N.
![Page 110: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/110.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
In general, weak convergence of measures is not enough toshow lower semicontinuity of nonlinear integrals or almosteverywhere convergence. We claim however that, in fact, theconvergence fn → g takes place strongly in L1(dµ):
limn→∞
∫M|fn(p)− g(p)|dµ(p) = 0.
In order to prove this, we prove that fn is a Cauchy sequencein L1(dµ). We take ε > 0 and choose N large enough so that
supp∈M
∣∣Un(p)− U(p)∣∣ ≤ ε2
16b,
and
Eb[fn] ≤ a +ε2
16
holds for n ≥ N.
![Page 111: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/111.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
In general, weak convergence of measures is not enough toshow lower semicontinuity of nonlinear integrals or almosteverywhere convergence. We claim however that, in fact, theconvergence fn → g takes place strongly in L1(dµ):
limn→∞
∫M|fn(p)− g(p)|dµ(p) = 0.
In order to prove this, we prove that fn is a Cauchy sequencein L1(dµ).
We take ε > 0 and choose N large enough so that
supp∈M
∣∣Un(p)− U(p)∣∣ ≤ ε2
16b,
and
Eb[fn] ≤ a +ε2
16
holds for n ≥ N.
![Page 112: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/112.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
In general, weak convergence of measures is not enough toshow lower semicontinuity of nonlinear integrals or almosteverywhere convergence. We claim however that, in fact, theconvergence fn → g takes place strongly in L1(dµ):
limn→∞
∫M|fn(p)− g(p)|dµ(p) = 0.
In order to prove this, we prove that fn is a Cauchy sequencein L1(dµ). We take ε > 0 and choose N large enough so that
supp∈M
∣∣Un(p)− U(p)∣∣ ≤ ε2
16b,
and
Eb[fn] ≤ a +ε2
16
holds for n ≥ N.
![Page 113: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/113.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
In general, weak convergence of measures is not enough toshow lower semicontinuity of nonlinear integrals or almosteverywhere convergence. We claim however that, in fact, theconvergence fn → g takes place strongly in L1(dµ):
limn→∞
∫M|fn(p)− g(p)|dµ(p) = 0.
In order to prove this, we prove that fn is a Cauchy sequencein L1(dµ). We take ε > 0 and choose N large enough so that
supp∈M
∣∣Un(p)− U(p)∣∣ ≤ ε2
16b,
and
Eb[fn] ≤ a +ε2
16
holds for n ≥ N.
![Page 114: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/114.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Let s(p) = 12(fn(p) + fm(p)) with n,m ≥ N. Then
∫M sdµ = 1,
s ≥ 0, soa ≤ Eb[s].
Therefore1
2Eb[fn] + Eb[fm] − Eb[s] ≤
ε2
16.
On the other hand,∫M
12 (fn log fn + fm log fm)− s log s
dµ
≤ 12 Eb[fn] + Eb[fm] − Eb[s] + ε2
16
using U[s] = 12(U[fn] + U[fm]) so,∫
M
1
2(fn log fn + fm log fm)− s log s
dµ ≤ ε2
8.
![Page 115: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/115.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Let s(p) = 12(fn(p) + fm(p)) with n,m ≥ N. Then
∫M sdµ = 1,
s ≥ 0, soa ≤ Eb[s].
Therefore1
2Eb[fn] + Eb[fm] − Eb[s] ≤
ε2
16.
On the other hand,∫M
12 (fn log fn + fm log fm)− s log s
dµ
≤ 12 Eb[fn] + Eb[fm] − Eb[s] + ε2
16
using U[s] = 12(U[fn] + U[fm]) so,∫
M
1
2(fn log fn + fm log fm)− s log s
dµ ≤ ε2
8.
![Page 116: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/116.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Let s(p) = 12(fn(p) + fm(p)) with n,m ≥ N. Then
∫M sdµ = 1,
s ≥ 0, soa ≤ Eb[s].
Therefore1
2Eb[fn] + Eb[fm] − Eb[s] ≤
ε2
16.
On the other hand,∫M
12 (fn log fn + fm log fm)− s log s
dµ
≤ 12 Eb[fn] + Eb[fm] − Eb[s] + ε2
16
using U[s] = 12(U[fn] + U[fm])
so,∫M
1
2(fn log fn + fm log fm)− s log s
dµ ≤ ε2
8.
![Page 117: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/117.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Let s(p) = 12(fn(p) + fm(p)) with n,m ≥ N. Then
∫M sdµ = 1,
s ≥ 0, soa ≤ Eb[s].
Therefore1
2Eb[fn] + Eb[fm] − Eb[s] ≤
ε2
16.
On the other hand,∫M
12 (fn log fn + fm log fm)− s log s
dµ
≤ 12 Eb[fn] + Eb[fm] − Eb[s] + ε2
16
using U[s] = 12(U[fn] + U[fm]) so,∫
M
1
2(fn log fn + fm log fm)− s log s
dµ ≤ ε2
8.
![Page 118: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/118.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Denote χ = fn−fmfn+fm
and note that −1 ≤ χ ≤ 1 holds µ - a.e.Also, elementary calculations show that
1
2(fn log fn + fm log fm)− s log s
=
s
2G (χ)
holds with
G (χ) = log(1− χ2) + χ log
(1 + χ
1− χ
).
G is even on (−1, 1), G ′(χ) = log(
1+χ1−χ
), G (0) = G ′(0) = 0
and G ′′(χ) = 21−χ2 ≥ 2 on (−1, 1). Consequently,
0 ≤ χ2 ≤ G (χ)
holds for −1 ≤ χ ≤ 1.
![Page 119: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/119.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
It follows that we have∫M
(fn − fm)2
fn + fmdµ ≤ ε2
2
Writing |fn − fm| =√
fn + fm|fn−fm|√
fn+fmand using the Schwartz
inequality we deduce ∫M|fn − fm|dµ ≤ ε.
Therefore the sequence fn is Cauchy in L1(dµ). This provesthat the weak limit fndµ → gdµ is actually strong fn → g inL1(dµ). By passing to a subsequence if necessary, we mayassume that fn → g holds also µ- a.e. Then from Fatou’sLemma, ∫
Mg log gdµ ≤ lim
j→∞
∫M
fj log fjdµ.
and thus g is a minimizer of Eb with Eb[g ] = a. It also followsthat g ≥ δ where δ > 0 is such that (x log x)′ < −3‖u‖∞ forall x ≤ δ.
![Page 120: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/120.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The dad-doodad
Let M be a compact metrizable space and let k(x , y) besymmetric, bi-Lipschitz and bounded below.
In addition,assume:
k(x , x) = 0.
Theorem(C-Zlatos) Let ν be any weak limit of a sequence fndµ ofminima of the free energy E corresponding to bn →∞. Thenthere exists m ∈ M such that ν is concentrated on the level setΣ(m) = p | k(m, p) = 0.Explains the selection of prolate states.Pattern recognition example: Rhombi centered at the origin,with k the area of the symmetric difference. The dad-rhombusis the square.
![Page 121: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/121.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The dad-doodad
Let M be a compact metrizable space and let k(x , y) besymmetric, bi-Lipschitz and bounded below. In addition,assume:
k(x , x) = 0.
Theorem(C-Zlatos) Let ν be any weak limit of a sequence fndµ ofminima of the free energy E corresponding to bn →∞. Thenthere exists m ∈ M such that ν is concentrated on the level setΣ(m) = p | k(m, p) = 0.Explains the selection of prolate states.Pattern recognition example: Rhombi centered at the origin,with k the area of the symmetric difference. The dad-rhombusis the square.
![Page 122: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/122.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The dad-doodad
Let M be a compact metrizable space and let k(x , y) besymmetric, bi-Lipschitz and bounded below. In addition,assume:
k(x , x) = 0.
Theorem(C-Zlatos) Let ν be any weak limit of a sequence fndµ ofminima of the free energy E corresponding to bn →∞. Thenthere exists m ∈ M such that ν is concentrated on the level setΣ(m) = p | k(m, p) = 0.
Explains the selection of prolate states.Pattern recognition example: Rhombi centered at the origin,with k the area of the symmetric difference. The dad-rhombusis the square.
![Page 123: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/123.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The dad-doodad
Let M be a compact metrizable space and let k(x , y) besymmetric, bi-Lipschitz and bounded below. In addition,assume:
k(x , x) = 0.
Theorem(C-Zlatos) Let ν be any weak limit of a sequence fndµ ofminima of the free energy E corresponding to bn →∞. Thenthere exists m ∈ M such that ν is concentrated on the level setΣ(m) = p | k(m, p) = 0.Explains the selection of prolate states.
Pattern recognition example: Rhombi centered at the origin,with k the area of the symmetric difference. The dad-rhombusis the square.
![Page 124: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/124.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The dad-doodad
Let M be a compact metrizable space and let k(x , y) besymmetric, bi-Lipschitz and bounded below. In addition,assume:
k(x , x) = 0.
Theorem(C-Zlatos) Let ν be any weak limit of a sequence fndµ ofminima of the free energy E corresponding to bn →∞. Thenthere exists m ∈ M such that ν is concentrated on the level setΣ(m) = p | k(m, p) = 0.Explains the selection of prolate states.Pattern recognition example: Rhombi centered at the origin,with k the area of the symmetric difference.
The dad-rhombusis the square.
![Page 125: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/125.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
The dad-doodad
Let M be a compact metrizable space and let k(x , y) besymmetric, bi-Lipschitz and bounded below. In addition,assume:
k(x , x) = 0.
Theorem(C-Zlatos) Let ν be any weak limit of a sequence fndµ ofminima of the free energy E corresponding to bn →∞. Thenthere exists m ∈ M such that ν is concentrated on the level setΣ(m) = p | k(m, p) = 0.Explains the selection of prolate states.Pattern recognition example: Rhombi centered at the origin,with k the area of the symmetric difference. The dad-rhombusis the square.
![Page 126: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/126.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Idea of proof: First establish
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
then
ε
∫ ∫k(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0, and
Q(p, ε) = q|k(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn. Pass to subsequencepn → p.
![Page 127: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/127.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Idea of proof: First establish
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
then
ε
∫ ∫k(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0, and
Q(p, ε) = q|k(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn. Pass to subsequencepn → p.
![Page 128: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/128.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Idea of proof: First establish
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
then
ε
∫ ∫k(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0,
and
Q(p, ε) = q|k(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn. Pass to subsequencepn → p.
![Page 129: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/129.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Idea of proof: First establish
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
then
ε
∫ ∫k(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0, and
Q(p, ε) = q|k(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn. Pass to subsequencepn → p.
![Page 130: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/130.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Idea of proof: First establish
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
then
ε
∫ ∫k(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0, and
Q(p, ε) = q|k(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn. Pass to subsequencepn → p.
![Page 131: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/131.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Idea of proof: First establish
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
then
ε
∫ ∫k(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0, and
Q(p, ε) = q|k(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn.
Pass to subsequencepn → p.
![Page 132: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/132.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Idea of proof: First establish
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
then
ε
∫ ∫k(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0, and
Q(p, ε) = q|k(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn. Pass to subsequencepn → p.
![Page 133: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/133.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Principle: if a µ measure-preserving transformation Texists such that locally around p = p0,k(Tp, Tq) ≤ ck(p, q) with c < 1, then p0 cannot be adad-doodad.
If a local k-preserving transformation around p = p0 hasthe property that µ(T (B)) ≥ Cµ(B) for small balls aroundp0, with C > 1, then p0 cannot be a dad-doodad.
Explains a number of examples with multiple states.
![Page 134: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/134.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Principle: if a µ measure-preserving transformation Texists such that locally around p = p0,k(Tp, Tq) ≤ ck(p, q) with c < 1, then p0 cannot be adad-doodad.If a local k-preserving transformation around p = p0 hasthe property that µ(T (B)) ≥ Cµ(B) for small balls aroundp0, with C > 1, then p0 cannot be a dad-doodad.
Explains a number of examples with multiple states.
![Page 135: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/135.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Principle: if a µ measure-preserving transformation Texists such that locally around p = p0,k(Tp, Tq) ≤ ck(p, q) with c < 1, then p0 cannot be adad-doodad.If a local k-preserving transformation around p = p0 hasthe property that µ(T (B)) ≥ Cµ(B) for small balls aroundp0, with C > 1, then p0 cannot be a dad-doodad.
Explains a number of examples with multiple states.
![Page 136: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/136.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Theorem(Zlatos-C) Let A0,A1 ⊆ M be compacts such that k(p, q) = 0for any p, q ∈ Aj ( j = 0, 1) and letBj(ε) = p ∈ M | d(p,Aj) < ε. Assume that for some εj > 0there is a 1-1 map T : B1(ε1) → B0(ε0)
such that T ,T−1 aremeasurable and there is c > 1 such that
∀p, q ∈ B1(ε1) : k(T (p),T (q)) ≤ k(p, q)
∀B ⊆ B1(ε1) measurable : µ(T (B)) ≥ cµ(B).
Assume also that for each p ∈ A1, q ∈ M \ B1(ε1) we havek(p, q) > 0.Then ν(A1) < 1 for each measure ν that is a zero temperatureweak limit of minimizers.
![Page 137: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/137.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Theorem(Zlatos-C) Let A0,A1 ⊆ M be compacts such that k(p, q) = 0for any p, q ∈ Aj ( j = 0, 1) and letBj(ε) = p ∈ M | d(p,Aj) < ε. Assume that for some εj > 0there is a 1-1 map T : B1(ε1) → B0(ε0) such that T ,T−1 aremeasurable
and there is c > 1 such that
∀p, q ∈ B1(ε1) : k(T (p),T (q)) ≤ k(p, q)
∀B ⊆ B1(ε1) measurable : µ(T (B)) ≥ cµ(B).
Assume also that for each p ∈ A1, q ∈ M \ B1(ε1) we havek(p, q) > 0.Then ν(A1) < 1 for each measure ν that is a zero temperatureweak limit of minimizers.
![Page 138: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/138.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Theorem(Zlatos-C) Let A0,A1 ⊆ M be compacts such that k(p, q) = 0for any p, q ∈ Aj ( j = 0, 1) and letBj(ε) = p ∈ M | d(p,Aj) < ε. Assume that for some εj > 0there is a 1-1 map T : B1(ε1) → B0(ε0) such that T ,T−1 aremeasurable and there is c > 1 such that
∀p, q ∈ B1(ε1) : k(T (p),T (q)) ≤ k(p, q)
∀B ⊆ B1(ε1) measurable : µ(T (B)) ≥ cµ(B).
Assume also that for each p ∈ A1, q ∈ M \ B1(ε1) we havek(p, q) > 0.Then ν(A1) < 1 for each measure ν that is a zero temperatureweak limit of minimizers.
![Page 139: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/139.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Theorem(Zlatos-C) Let A0,A1 ⊆ M be compacts such that k(p, q) = 0for any p, q ∈ Aj ( j = 0, 1) and letBj(ε) = p ∈ M | d(p,Aj) < ε. Assume that for some εj > 0there is a 1-1 map T : B1(ε1) → B0(ε0) such that T ,T−1 aremeasurable and there is c > 1 such that
∀p, q ∈ B1(ε1) : k(T (p),T (q)) ≤ k(p, q)
∀B ⊆ B1(ε1) measurable : µ(T (B)) ≥ cµ(B).
Assume also that for each p ∈ A1, q ∈ M \ B1(ε1) we havek(p, q) > 0.
Then ν(A1) < 1 for each measure ν that is a zero temperatureweak limit of minimizers.
![Page 140: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/140.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Theorem(Zlatos-C) Let A0,A1 ⊆ M be compacts such that k(p, q) = 0for any p, q ∈ Aj ( j = 0, 1) and letBj(ε) = p ∈ M | d(p,Aj) < ε. Assume that for some εj > 0there is a 1-1 map T : B1(ε1) → B0(ε0) such that T ,T−1 aremeasurable and there is c > 1 such that
∀p, q ∈ B1(ε1) : k(T (p),T (q)) ≤ k(p, q)
∀B ⊆ B1(ε1) measurable : µ(T (B)) ≥ cµ(B).
Assume also that for each p ∈ A1, q ∈ M \ B1(ε1) we havek(p, q) > 0.Then ν(A1) < 1 for each measure ν that is a zero temperatureweak limit of minimizers.
![Page 141: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/141.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Kinetics in Riemannian Setting
M compact, connected Riemannian manifold of dimension d
with metric gαβ , g = det(gαβ), (gαβ) = (gαβ)−1, volumeelement dµ =
√gdφ in local coordinates φ. Generalized
Doi-Smoluchowski equation
∂t f = ∆g f + divg (f∇gU)
∆g Laplace-Beltrami,
divg (f∇gU) =1√
g∂α
(√ggαβf ∂βU
).
U[f ](p) =
∫M
k(p, q)f (q)dµ(q)
![Page 142: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/142.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Kinetics in Riemannian Setting
M compact, connected Riemannian manifold of dimension dwith metric gαβ ,
g = det(gαβ), (gαβ) = (gαβ)−1, volumeelement dµ =
√gdφ in local coordinates φ. Generalized
Doi-Smoluchowski equation
∂t f = ∆g f + divg (f∇gU)
∆g Laplace-Beltrami,
divg (f∇gU) =1√
g∂α
(√ggαβf ∂βU
).
U[f ](p) =
∫M
k(p, q)f (q)dµ(q)
![Page 143: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/143.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Kinetics in Riemannian Setting
M compact, connected Riemannian manifold of dimension dwith metric gαβ , g = det(gαβ),
(gαβ) = (gαβ)−1, volumeelement dµ =
√gdφ in local coordinates φ. Generalized
Doi-Smoluchowski equation
∂t f = ∆g f + divg (f∇gU)
∆g Laplace-Beltrami,
divg (f∇gU) =1√
g∂α
(√ggαβf ∂βU
).
U[f ](p) =
∫M
k(p, q)f (q)dµ(q)
![Page 144: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/144.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Kinetics in Riemannian Setting
M compact, connected Riemannian manifold of dimension dwith metric gαβ , g = det(gαβ), (gαβ) = (gαβ)−1,
volumeelement dµ =
√gdφ in local coordinates φ. Generalized
Doi-Smoluchowski equation
∂t f = ∆g f + divg (f∇gU)
∆g Laplace-Beltrami,
divg (f∇gU) =1√
g∂α
(√ggαβf ∂βU
).
U[f ](p) =
∫M
k(p, q)f (q)dµ(q)
![Page 145: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/145.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Kinetics in Riemannian Setting
M compact, connected Riemannian manifold of dimension dwith metric gαβ , g = det(gαβ), (gαβ) = (gαβ)−1, volumeelement dµ =
√gdφ in local coordinates φ.
GeneralizedDoi-Smoluchowski equation
∂t f = ∆g f + divg (f∇gU)
∆g Laplace-Beltrami,
divg (f∇gU) =1√
g∂α
(√ggαβf ∂βU
).
U[f ](p) =
∫M
k(p, q)f (q)dµ(q)
![Page 146: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/146.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Kinetics in Riemannian Setting
M compact, connected Riemannian manifold of dimension dwith metric gαβ , g = det(gαβ), (gαβ) = (gαβ)−1, volumeelement dµ =
√gdφ in local coordinates φ. Generalized
Doi-Smoluchowski equation
∂t f = ∆g f + divg (f∇gU)
∆g Laplace-Beltrami,
divg (f∇gU) =1√
g∂α
(√ggαβf ∂βU
).
U[f ](p) =
∫M
k(p, q)f (q)dµ(q)
![Page 147: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/147.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Kinetics in Riemannian Setting
M compact, connected Riemannian manifold of dimension dwith metric gαβ , g = det(gαβ), (gαβ) = (gαβ)−1, volumeelement dµ =
√gdφ in local coordinates φ. Generalized
Doi-Smoluchowski equation
∂t f = ∆g f + divg (f∇gU)
∆g Laplace-Beltrami,
divg (f∇gU) =1√
g∂α
(√ggαβf ∂βU
).
U[f ](p) =
∫M
k(p, q)f (q)dµ(q)
![Page 148: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/148.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Kinetics in Riemannian Setting
M compact, connected Riemannian manifold of dimension dwith metric gαβ , g = det(gαβ), (gαβ) = (gαβ)−1, volumeelement dµ =
√gdφ in local coordinates φ. Generalized
Doi-Smoluchowski equation
∂t f = ∆g f + divg (f∇gU)
∆g Laplace-Beltrami,
divg (f∇gU) =1√
g∂α
(√ggαβf ∂βU
).
U[f ](p) =
∫M
k(p, q)f (q)dµ(q)
![Page 149: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/149.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Kinetics in Riemannian Setting
M compact, connected Riemannian manifold of dimension dwith metric gαβ , g = det(gαβ), (gαβ) = (gαβ)−1, volumeelement dµ =
√gdφ in local coordinates φ. Generalized
Doi-Smoluchowski equation
∂t f = ∆g f + divg (f∇gU)
∆g Laplace-Beltrami,
divg (f∇gU) =1√
g∂α
(√ggαβf ∂βU
).
U[f ](p) =
∫M
k(p, q)f (q)dµ(q)
![Page 150: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/150.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Kinetics in Riemannian Setting
M compact, connected Riemannian manifold of dimension dwith metric gαβ , g = det(gαβ), (gαβ) = (gαβ)−1, volumeelement dµ =
√gdφ in local coordinates φ. Generalized
Doi-Smoluchowski equation
∂t f = ∆g f + divg (f∇gU)
∆g Laplace-Beltrami,
divg (f∇gU) =1√
g∂α
(√ggαβf ∂βU
).
U[f ](p) =
∫M
k(p, q)f (q)dµ(q)
![Page 151: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/151.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Dissipative Structure
Free Energy:
E [f ] =
∫M
log f +
1
2U[f ]
fdµ
NLFP:
∂t f = divg ·(
f∇g
(δE [f ]
δf
))Lyapunov functional:
d
dtE [f ] = −
∫M
f
∣∣∣∣∇g
(δE [f ]
δf
)∣∣∣∣2 dµ
![Page 152: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/152.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Dissipative Structure
Free Energy:
E [f ] =
∫M
log f +
1
2U[f ]
fdµ
NLFP:
∂t f = divg ·(
f∇g
(δE [f ]
δf
))Lyapunov functional:
d
dtE [f ] = −
∫M
f
∣∣∣∣∇g
(δE [f ]
δf
)∣∣∣∣2 dµ
![Page 153: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/153.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Dissipative Structure
Free Energy:
E [f ] =
∫M
log f +
1
2U[f ]
fdµ
NLFP:
∂t f = divg ·(
f∇g
(δE [f ]
δf
))Lyapunov functional:
d
dtE [f ] = −
∫M
f
∣∣∣∣∇g
(δE [f ]
δf
)∣∣∣∣2 dµ
![Page 154: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/154.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Dissipative Structure
Free Energy:
E [f ] =
∫M
log f +
1
2U[f ]
fdµ
NLFP:
∂t f = divg ·(
f∇g
(δE [f ]
δf
))
Lyapunov functional:
d
dtE [f ] = −
∫M
f
∣∣∣∣∇g
(δE [f ]
δf
)∣∣∣∣2 dµ
![Page 155: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/155.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Dissipative Structure
Free Energy:
E [f ] =
∫M
log f +
1
2U[f ]
fdµ
NLFP:
∂t f = divg ·(
f∇g
(δE [f ]
δf
))Lyapunov functional:
d
dtE [f ] = −
∫M
f
∣∣∣∣∇g
(δE [f ]
δf
)∣∣∣∣2 dµ
![Page 156: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/156.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Connection to Onsager’s equation
For Doi-Smoluchowski
δE [f ]
δf= log f + U[f ]
Time independent solutions: Onsager equation:
f = Z−1e−U[f ]
Dynamics: nontrivial. Multiple steady states, gradient system,finite dimensional attractor. Inertial Manifolds: Vukadinovic(2008-9).
![Page 157: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/157.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Connection to Onsager’s equation
For Doi-Smoluchowski
δE [f ]
δf= log f + U[f ]
Time independent solutions: Onsager equation:
f = Z−1e−U[f ]
Dynamics: nontrivial. Multiple steady states, gradient system,finite dimensional attractor. Inertial Manifolds: Vukadinovic(2008-9).
![Page 158: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/158.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Connection to Onsager’s equation
For Doi-Smoluchowski
δE [f ]
δf= log f + U[f ]
Time independent solutions:
Onsager equation:
f = Z−1e−U[f ]
Dynamics: nontrivial. Multiple steady states, gradient system,finite dimensional attractor. Inertial Manifolds: Vukadinovic(2008-9).
![Page 159: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/159.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Connection to Onsager’s equation
For Doi-Smoluchowski
δE [f ]
δf= log f + U[f ]
Time independent solutions: Onsager equation:
f = Z−1e−U[f ]
Dynamics: nontrivial. Multiple steady states, gradient system,finite dimensional attractor. Inertial Manifolds: Vukadinovic(2008-9).
![Page 160: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/160.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Connection to Onsager’s equation
For Doi-Smoluchowski
δE [f ]
δf= log f + U[f ]
Time independent solutions: Onsager equation:
f = Z−1e−U[f ]
Dynamics: nontrivial. Multiple steady states, gradient system,finite dimensional attractor.
Inertial Manifolds: Vukadinovic(2008-9).
![Page 161: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/161.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Connection to Onsager’s equation
For Doi-Smoluchowski
δE [f ]
δf= log f + U[f ]
Time independent solutions: Onsager equation:
f = Z−1e−U[f ]
Dynamics: nontrivial. Multiple steady states, gradient system,finite dimensional attractor. Inertial Manifolds: Vukadinovic(2008-9).
![Page 162: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/162.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Gradient System in Metric Space
M metric. Length space:∀m, n ∈ M ∃p ∈ M, d(p,m) = d(p, n) = 1
2d(m, n).P2(M), probability space with the Wasserstein 2 distance (it isa length space if M is).
Problem: define
∂t f = −grad(E [f ])
DeGiorgi, Ambrosio et al: Energy Dissipation Identity (EDI):
d
dtE [f ] ≤ −1
2|f ′(t)|2 − 1
2|∂E [f ]|2
|f ′(t)| = lims→0
d2(f (t + s), f (t))
|s|and descending slope
|∂E [f ]| = lim supg→f
(E [f ]− E [g ])+d2(f , g)
![Page 163: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/163.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Gradient System in Metric Space
M metric. Length space:∀m, n ∈ M ∃p ∈ M, d(p,m) = d(p, n) = 1
2d(m, n).P2(M), probability space with the Wasserstein 2 distance (it isa length space if M is). Problem: define
∂t f = −grad(E [f ])
DeGiorgi, Ambrosio et al: Energy Dissipation Identity (EDI):
d
dtE [f ] ≤ −1
2|f ′(t)|2 − 1
2|∂E [f ]|2
|f ′(t)| = lims→0
d2(f (t + s), f (t))
|s|and descending slope
|∂E [f ]| = lim supg→f
(E [f ]− E [g ])+d2(f , g)
![Page 164: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/164.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Gradient System in Metric Space
M metric. Length space:∀m, n ∈ M ∃p ∈ M, d(p,m) = d(p, n) = 1
2d(m, n).P2(M), probability space with the Wasserstein 2 distance (it isa length space if M is). Problem: define
∂t f = −grad(E [f ])
DeGiorgi, Ambrosio et al: Energy Dissipation Identity (EDI):
d
dtE [f ] ≤ −1
2|f ′(t)|2 − 1
2|∂E [f ]|2
|f ′(t)| = lims→0
d2(f (t + s), f (t))
|s|and descending slope
|∂E [f ]| = lim supg→f
(E [f ]− E [g ])+d2(f , g)
![Page 165: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/165.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Gradient System in Metric Space
M metric. Length space:∀m, n ∈ M ∃p ∈ M, d(p,m) = d(p, n) = 1
2d(m, n).P2(M), probability space with the Wasserstein 2 distance (it isa length space if M is). Problem: define
∂t f = −grad(E [f ])
DeGiorgi, Ambrosio et al: Energy Dissipation Identity (EDI):
d
dtE [f ] ≤ −1
2|f ′(t)|2 − 1
2|∂E [f ]|2
|f ′(t)| = lims→0
d2(f (t + s), f (t))
|s|and descending slope
|∂E [f ]| = lim supg→f
(E [f ]− E [g ])+d2(f , g)
![Page 166: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/166.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Gradient System in Metric Space
M metric. Length space:∀m, n ∈ M ∃p ∈ M, d(p,m) = d(p, n) = 1
2d(m, n).P2(M), probability space with the Wasserstein 2 distance (it isa length space if M is). Problem: define
∂t f = −grad(E [f ])
DeGiorgi, Ambrosio et al: Energy Dissipation Identity (EDI):
d
dtE [f ] ≤ −1
2|f ′(t)|2 − 1
2|∂E [f ]|2
|f ′(t)| = lims→0
d2(f (t + s), f (t))
|s|and descending slope
|∂E [f ]| = lim supg→f
(E [f ]− E [g ])+d2(f , g)
![Page 167: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/167.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Gradient System in Metric Space
M metric. Length space:∀m, n ∈ M ∃p ∈ M, d(p,m) = d(p, n) = 1
2d(m, n).P2(M), probability space with the Wasserstein 2 distance (it isa length space if M is). Problem: define
∂t f = −grad(E [f ])
DeGiorgi, Ambrosio et al: Energy Dissipation Identity (EDI):
d
dtE [f ] ≤ −1
2|f ′(t)|2 − 1
2|∂E [f ]|2
|f ′(t)| = lims→0
d2(f (t + s), f (t))
|s|
and descending slope
|∂E [f ]| = lim supg→f
(E [f ]− E [g ])+d2(f , g)
![Page 168: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/168.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Gradient System in Metric Space
M metric. Length space:∀m, n ∈ M ∃p ∈ M, d(p,m) = d(p, n) = 1
2d(m, n).P2(M), probability space with the Wasserstein 2 distance (it isa length space if M is). Problem: define
∂t f = −grad(E [f ])
DeGiorgi, Ambrosio et al: Energy Dissipation Identity (EDI):
d
dtE [f ] ≤ −1
2|f ′(t)|2 − 1
2|∂E [f ]|2
|f ′(t)| = lims→0
d2(f (t + s), f (t))
|s|and descending slope
|∂E [f ]| = lim supg→f
(E [f ]− E [g ])+d2(f , g)
![Page 169: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/169.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Gradient System in Metric Space
M metric. Length space:∀m, n ∈ M ∃p ∈ M, d(p,m) = d(p, n) = 1
2d(m, n).P2(M), probability space with the Wasserstein 2 distance (it isa length space if M is). Problem: define
∂t f = −grad(E [f ])
DeGiorgi, Ambrosio et al: Energy Dissipation Identity (EDI):
d
dtE [f ] ≤ −1
2|f ′(t)|2 − 1
2|∂E [f ]|2
|f ′(t)| = lims→0
d2(f (t + s), f (t))
|s|and descending slope
|∂E [f ]| = lim supg→f
(E [f ]− E [g ])+d2(f , g)
![Page 170: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/170.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Implicit Euler
Piece-wise constant fτ (t) = fk
for t ∈ (kτ, (k + 1)τ ]
fk+1 := argmin
(E [g ] +
1
2τd22 (g , fk)
)
Limits as τ → 0 exist in general, and givegradient flows in the sense of the energy dis-sipation identity EDI.
Motivation for definition:
In metric space, kinetic equation = EDI ofE in P2(M).
![Page 171: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/171.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Implicit Euler
Piece-wise constant fτ (t) = fk for t ∈ (kτ, (k + 1)τ ]
fk+1 := argmin
(E [g ] +
1
2τd22 (g , fk)
)
Limits as τ → 0 exist in general, and givegradient flows in the sense of the energy dis-sipation identity EDI.
Motivation for definition:
In metric space, kinetic equation = EDI ofE in P2(M).
![Page 172: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/172.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Implicit Euler
Piece-wise constant fτ (t) = fk for t ∈ (kτ, (k + 1)τ ]
fk+1 := argmin
(E [g ] +
1
2τd22 (g , fk)
)
Limits as τ → 0 exist in general, and givegradient flows in the sense of the energy dis-sipation identity EDI.
Motivation for definition:
In metric space, kinetic equation = EDI ofE in P2(M).
![Page 173: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/173.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Implicit Euler
Piece-wise constant fτ (t) = fk for t ∈ (kτ, (k + 1)τ ]
fk+1 := argmin
(E [g ] +
1
2τd22 (g , fk)
)
Limits as τ → 0 exist in general, and givegradient flows in the sense of the energy dis-sipation identity EDI.
Motivation for definition:
In metric space, kinetic equation = EDI ofE in P2(M).
![Page 174: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/174.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Implicit Euler
Piece-wise constant fτ (t) = fk for t ∈ (kτ, (k + 1)τ ]
fk+1 := argmin
(E [g ] +
1
2τd22 (g , fk)
)
Limits as τ → 0 exist in general, and givegradient flows in the sense of the energy dis-sipation identity EDI.
Motivation for definition:
In metric space, kinetic equation = EDI ofE in P2(M).
![Page 175: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/175.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Implicit Euler
Piece-wise constant fτ (t) = fk for t ∈ (kτ, (k + 1)τ ]
fk+1 := argmin
(E [g ] +
1
2τd22 (g , fk)
)
Limits as τ → 0 exist in general, and givegradient flows in the sense of the energy dis-sipation identity EDI.
Motivation for definition:
In metric space, kinetic equation = EDI ofE in P2(M).
![Page 176: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/176.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Evolutional Variational Inequalities
In the convex energy, Hilbert space case: well understood.
Assume E [f ] = E [f ] + Q[f ] with E convex andQ[f ] = 1
2(Lf , f ), with L bounded, selfadjoint in Hilbert spaceH.Definition: Evolutional Variational Inequality (EVI)
1
2
d
dt‖f (t)−g‖2 ≤ E [g ]−E [f (t)]−Q[f (t)−g ], ∀g
recover classical definition for Q = 0. Interesting for L notpositive. Uniqueness of solutions of initial value problem canbe obtained in the non-convex, Hilbert case.
• Open: Define Evolutional Variational Inequality EVI fornon-convex energies in length spaces.
• Open: EDI implies uniqueness of solutions of initial valueproblem in the nonconvex case in length spaces.
• Open: Time asymptotics to Onsager solution, in P2(M).
![Page 177: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/177.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Evolutional Variational Inequalities
In the convex energy, Hilbert space case: well understood.Assume E [f ] = E [f ] + Q[f ]
with E convex andQ[f ] = 1
2(Lf , f ), with L bounded, selfadjoint in Hilbert spaceH.Definition: Evolutional Variational Inequality (EVI)
1
2
d
dt‖f (t)−g‖2 ≤ E [g ]−E [f (t)]−Q[f (t)−g ], ∀g
recover classical definition for Q = 0. Interesting for L notpositive. Uniqueness of solutions of initial value problem canbe obtained in the non-convex, Hilbert case.
• Open: Define Evolutional Variational Inequality EVI fornon-convex energies in length spaces.
• Open: EDI implies uniqueness of solutions of initial valueproblem in the nonconvex case in length spaces.
• Open: Time asymptotics to Onsager solution, in P2(M).
![Page 178: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/178.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Evolutional Variational Inequalities
In the convex energy, Hilbert space case: well understood.Assume E [f ] = E [f ] + Q[f ] with E convex and
Q[f ] = 12(Lf , f ), with L bounded, selfadjoint in Hilbert space
H.Definition: Evolutional Variational Inequality (EVI)
1
2
d
dt‖f (t)−g‖2 ≤ E [g ]−E [f (t)]−Q[f (t)−g ], ∀g
recover classical definition for Q = 0. Interesting for L notpositive. Uniqueness of solutions of initial value problem canbe obtained in the non-convex, Hilbert case.
• Open: Define Evolutional Variational Inequality EVI fornon-convex energies in length spaces.
• Open: EDI implies uniqueness of solutions of initial valueproblem in the nonconvex case in length spaces.
• Open: Time asymptotics to Onsager solution, in P2(M).
![Page 179: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/179.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Evolutional Variational Inequalities
In the convex energy, Hilbert space case: well understood.Assume E [f ] = E [f ] + Q[f ] with E convex andQ[f ] = 1
2(Lf , f ),
with L bounded, selfadjoint in Hilbert spaceH.Definition: Evolutional Variational Inequality (EVI)
1
2
d
dt‖f (t)−g‖2 ≤ E [g ]−E [f (t)]−Q[f (t)−g ], ∀g
recover classical definition for Q = 0. Interesting for L notpositive. Uniqueness of solutions of initial value problem canbe obtained in the non-convex, Hilbert case.
• Open: Define Evolutional Variational Inequality EVI fornon-convex energies in length spaces.
• Open: EDI implies uniqueness of solutions of initial valueproblem in the nonconvex case in length spaces.
• Open: Time asymptotics to Onsager solution, in P2(M).
![Page 180: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/180.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Evolutional Variational Inequalities
In the convex energy, Hilbert space case: well understood.Assume E [f ] = E [f ] + Q[f ] with E convex andQ[f ] = 1
2(Lf , f ), with L bounded, selfadjoint
in Hilbert spaceH.Definition: Evolutional Variational Inequality (EVI)
1
2
d
dt‖f (t)−g‖2 ≤ E [g ]−E [f (t)]−Q[f (t)−g ], ∀g
recover classical definition for Q = 0. Interesting for L notpositive. Uniqueness of solutions of initial value problem canbe obtained in the non-convex, Hilbert case.
• Open: Define Evolutional Variational Inequality EVI fornon-convex energies in length spaces.
• Open: EDI implies uniqueness of solutions of initial valueproblem in the nonconvex case in length spaces.
• Open: Time asymptotics to Onsager solution, in P2(M).
![Page 181: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/181.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Evolutional Variational Inequalities
In the convex energy, Hilbert space case: well understood.Assume E [f ] = E [f ] + Q[f ] with E convex andQ[f ] = 1
2(Lf , f ), with L bounded, selfadjoint in Hilbert spaceH.
Definition: Evolutional Variational Inequality (EVI)
1
2
d
dt‖f (t)−g‖2 ≤ E [g ]−E [f (t)]−Q[f (t)−g ], ∀g
recover classical definition for Q = 0. Interesting for L notpositive. Uniqueness of solutions of initial value problem canbe obtained in the non-convex, Hilbert case.
• Open: Define Evolutional Variational Inequality EVI fornon-convex energies in length spaces.
• Open: EDI implies uniqueness of solutions of initial valueproblem in the nonconvex case in length spaces.
• Open: Time asymptotics to Onsager solution, in P2(M).
![Page 182: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/182.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Evolutional Variational Inequalities
In the convex energy, Hilbert space case: well understood.Assume E [f ] = E [f ] + Q[f ] with E convex andQ[f ] = 1
2(Lf , f ), with L bounded, selfadjoint in Hilbert spaceH.Definition: Evolutional Variational Inequality (EVI)
1
2
d
dt‖f (t)−g‖2 ≤ E [g ]−E [f (t)]−Q[f (t)−g ], ∀g
recover classical definition for Q = 0. Interesting for L notpositive. Uniqueness of solutions of initial value problem canbe obtained in the non-convex, Hilbert case.
• Open: Define Evolutional Variational Inequality EVI fornon-convex energies in length spaces.
• Open: EDI implies uniqueness of solutions of initial valueproblem in the nonconvex case in length spaces.
• Open: Time asymptotics to Onsager solution, in P2(M).
![Page 183: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/183.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Evolutional Variational Inequalities
In the convex energy, Hilbert space case: well understood.Assume E [f ] = E [f ] + Q[f ] with E convex andQ[f ] = 1
2(Lf , f ), with L bounded, selfadjoint in Hilbert spaceH.Definition: Evolutional Variational Inequality (EVI)
1
2
d
dt‖f (t)−g‖2 ≤ E [g ]−E [f (t)]−Q[f (t)−g ], ∀g
recover classical definition for Q = 0.
Interesting for L notpositive. Uniqueness of solutions of initial value problem canbe obtained in the non-convex, Hilbert case.
• Open: Define Evolutional Variational Inequality EVI fornon-convex energies in length spaces.
• Open: EDI implies uniqueness of solutions of initial valueproblem in the nonconvex case in length spaces.
• Open: Time asymptotics to Onsager solution, in P2(M).
![Page 184: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/184.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Evolutional Variational Inequalities
In the convex energy, Hilbert space case: well understood.Assume E [f ] = E [f ] + Q[f ] with E convex andQ[f ] = 1
2(Lf , f ), with L bounded, selfadjoint in Hilbert spaceH.Definition: Evolutional Variational Inequality (EVI)
1
2
d
dt‖f (t)−g‖2 ≤ E [g ]−E [f (t)]−Q[f (t)−g ], ∀g
recover classical definition for Q = 0. Interesting for L notpositive.
Uniqueness of solutions of initial value problem canbe obtained in the non-convex, Hilbert case.
• Open: Define Evolutional Variational Inequality EVI fornon-convex energies in length spaces.
• Open: EDI implies uniqueness of solutions of initial valueproblem in the nonconvex case in length spaces.
• Open: Time asymptotics to Onsager solution, in P2(M).
![Page 185: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/185.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Evolutional Variational Inequalities
In the convex energy, Hilbert space case: well understood.Assume E [f ] = E [f ] + Q[f ] with E convex andQ[f ] = 1
2(Lf , f ), with L bounded, selfadjoint in Hilbert spaceH.Definition: Evolutional Variational Inequality (EVI)
1
2
d
dt‖f (t)−g‖2 ≤ E [g ]−E [f (t)]−Q[f (t)−g ], ∀g
recover classical definition for Q = 0. Interesting for L notpositive. Uniqueness of solutions of initial value problem canbe obtained in the non-convex, Hilbert case.
• Open: Define Evolutional Variational Inequality EVI fornon-convex energies in length spaces.
• Open: EDI implies uniqueness of solutions of initial valueproblem in the nonconvex case in length spaces.
• Open: Time asymptotics to Onsager solution, in P2(M).
![Page 186: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/186.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Evolutional Variational Inequalities
In the convex energy, Hilbert space case: well understood.Assume E [f ] = E [f ] + Q[f ] with E convex andQ[f ] = 1
2(Lf , f ), with L bounded, selfadjoint in Hilbert spaceH.Definition: Evolutional Variational Inequality (EVI)
1
2
d
dt‖f (t)−g‖2 ≤ E [g ]−E [f (t)]−Q[f (t)−g ], ∀g
recover classical definition for Q = 0. Interesting for L notpositive. Uniqueness of solutions of initial value problem canbe obtained in the non-convex, Hilbert case.
• Open: Define Evolutional Variational Inequality EVI fornon-convex energies in length spaces.
• Open: EDI implies uniqueness of solutions of initial valueproblem in the nonconvex case in length spaces.
• Open: Time asymptotics to Onsager solution, in P2(M).
![Page 187: Fluids and Particles: Doodads and Kineticspeople.cs.uchicago.edu/~const/cime101talk.pdfkinetics Outline: 1 Equilibrium: Onsager Equation on Metric Spaces 2 Kinetics: Nonlinear Fokker-Planck](https://reader036.vdocuments.site/reader036/viewer/2022081623/613f4c89a7a58608c268d587/html5/thumbnails/187.jpg)
Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Evolutional Variational Inequalities
In the convex energy, Hilbert space case: well understood.Assume E [f ] = E [f ] + Q[f ] with E convex andQ[f ] = 1
2(Lf , f ), with L bounded, selfadjoint in Hilbert spaceH.Definition: Evolutional Variational Inequality (EVI)
1
2
d
dt‖f (t)−g‖2 ≤ E [g ]−E [f (t)]−Q[f (t)−g ], ∀g
recover classical definition for Q = 0. Interesting for L notpositive. Uniqueness of solutions of initial value problem canbe obtained in the non-convex, Hilbert case.
• Open: Define Evolutional Variational Inequality EVI fornon-convex energies in length spaces.
• Open: EDI implies uniqueness of solutions of initial valueproblem in the nonconvex case in length spaces.
• Open: Time asymptotics to Onsager solution, in P2(M).
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Fluids andParticles:
Doodads andKinetics
PeterConstantin
Introduction
ComplexMicroscopicSystems
Examples
OnsagerEquation
Relaxationkinetics
Evolutional Variational Inequalities
In the convex energy, Hilbert space case: well understood.Assume E [f ] = E [f ] + Q[f ] with E convex andQ[f ] = 1
2(Lf , f ), with L bounded, selfadjoint in Hilbert spaceH.Definition: Evolutional Variational Inequality (EVI)
1
2
d
dt‖f (t)−g‖2 ≤ E [g ]−E [f (t)]−Q[f (t)−g ], ∀g
recover classical definition for Q = 0. Interesting for L notpositive. Uniqueness of solutions of initial value problem canbe obtained in the non-convex, Hilbert case.
• Open: Define Evolutional Variational Inequality EVI fornon-convex energies in length spaces.
• Open: EDI implies uniqueness of solutions of initial valueproblem in the nonconvex case in length spaces.
• Open: Time asymptotics to Onsager solution, in P2(M).