q-gaussian densities and behaviors of solutions to related...
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Information Geometry on q-Gaussian Densities
and Behaviors of Solutions to
Related Diffusion Equations*
Atsumi Ohara Osaka UniversityJanuary 24 2009 at Osaka City University
(Updated April 23 2009)
*Joint work with Tatsuaki
Wada (Ibaraki University)
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Introduction (1)
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Porous medium equation (PME)
Example: Diffusion coefficient depending on the power of u
Percolation in porous medium, intensive thermal wave, …
Slow diffusion (anomalous diffusion): Finite propagation speedm=1 (normal diffusion): Infinite propagation speed
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Solution of the PME for 1D case (initial function with bounded support)
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m =1.5
propagation speed is finite
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Introduction (2)
Nonlinear Fokker-Planck equation (NFPE)
Corresponding physical phenomena Slow diffusion + drift force (by quadratic potential)
equilibrium density existsNonlinear transformation between the PME and the NFPE
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Previous work
[Aronson], [Vazquez], [Toscani] and many others…
Existence, uniqueness & mass conservationW.l.o.g. we consider probability densities
Special solution: self-similar solutionConvergence rate to the self-similar solutionLyapunov functional (free energy) technique
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Introduction (3)
The purpose of the presentation:Behavioral analysis of the PME type diffusion eq. focusing on a stable invariant manifold
the family of q-Gaussian densities
A new point of viewTechnique and concepts from Information Geometry can be applied
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Outline1.Generalized entropy and exponential family2.Information geometry on the q-Gaussian family and analytical tools3.Behavioral analysis of the PME and NFPE
Invariant manifoldThe second moments, m-projection, geodesicPeculiar phenomena to slow diffusionConvergence rate to the q-Gaussian family
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1.
Generalized entropy and exp family (1) [Naudts
02 & 04], [Eguchi04]
:strictly increasing and positive on generalized logarithmic function
generalized exponential function: the inverse of
convex function
- Strictly inc.- Concave-
to define entropy
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Generalized entropy and exp family (2)
Bregman divergence
Generalized entropy
Generalized exponential model
: canonical paramtr., : normalizing const: vector of stochastic variables (Hamiltonian)
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Remark [Naudts 02, 04]
Requirements to the generalized entropy:1. For a certain , the entropy should be of the form:
2. -Gaussian is an ME equilibrium for Then in the previous slide is determined.
is called the deduced log func of
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Another representation of
Conjugate function of
U-divergence [Eguchi 04]
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Example (1): used later
Generalized log q-logarithm q: real
Generalized exp q-exponential
PMEGeneralized entropy
(2-q)-Tsallis entropy
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Example (2): used later
Bregman divergence
Gen. exp family q-Gaussian family
When q goes to 1, all of them recover to the standard ones. 13
q-Gaussian density
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2. Information geometry [Amari,Nagaoka00]
on q-Gaussian family
:finite dimensional manifold in Potential function on
:Legendre transform of Legendre structure on compatible with statistical physics
Riemannian metric, covariant derivatives, geodesics and so on.
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Important tools from IG (1)
1. dual coordinates (Expectation parameters)
Expectation of each (= the 1st and 2nd moments for q-Gaussian)
2. m-geodesica curve on represented as a straight line in the -coordinates
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Important tools from IG (2)
3. m-projection of
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(-1)
m-projection
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Useful properties of the m-projection
Rem: The property i) claims that the 1st and 2nd moments are conserved.
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3. Behavioral analysis of PME and NFPE
PME:
NFPE:
Relation between u and p [Vazquez 03]
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Key preliminary result
Assumption: 1<m=2-q<2Proposition
The q-Gaussian family is a stable invariant manifold of the PME and NFPE.
Idea of the proof)Show the R.H.S. of the PME is tangent to when is on .
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Trajectories of m-projections (PME)
The 1st and 2nd moments of u(t)
where
ThmThe m-projection of the solution to the PME
evolves following an m-geodesic curve, i.e., its expectation coordinate is a straight line.
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Properties of the m-projection and behavioral analysis
m-geodesic
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Idea of the proof
Time derivatives of the moments:
straight line in the -coordinates
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Implication of the theorem (1)
The theorem implies the existence of nontrivial N-1 constants of motions. N=dim
A solution of the PME on the invariant manifold is possibly solvable by quadratures. 23
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Implication of the theorem (2)
Corollary: Let and be solutions of the PME.
If at , then
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Implication of the theorem (3)
Idea of the proofThe formula of the 2nd moments + the property i) of the m-projection
The corollary shows that the evolutional speed of each solution depends on the Bregman divergence from .(=the difference of the entropies)When m=1 (normal diffusion), such a phenomenon does not occur.
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Difference of the second moments
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Difference of the evolutional speed
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Convergence analysis for the NFPE and its application to the PME
Generalized free energy
It works as a Lyapunov functional for the NFPE:
The equilibrium density is a q-Gaussian:
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Convergence analysis for the NFPE and its application to the PME
Difference of the free energy from the equilibrium density:
Thus, is monotone decreasing.
Interpreted as a generalized H-theorem
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Convergence analysis for the NFPE and its application to the PME
1. The property ii) of the m-projection:
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Convergence analysis for the NFPE and its application to the PME
2. The known convergence result [Toscani05]
3. The property of the transformation between the PME and the NFPE
If is transform of , then is an m-projection of u
is an m-projection of p
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Convergence analysis for the NFPE and its application to the PME
Using 1, 2 and 3, we have the following:
Convergence rate to the q-Gaussian familyL1-norm convergence rate is derived from this result via the Csiszar-Kullback inequality.
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Convergence analysis for the NFPE and its application to the PME
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Convergence analysis for the NFPE and its application to the PME
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Remark: L1-norm convergence rate
Csiszar-Kullback inequality [Carrillo & Toscani 00]
The proposition implies that L1 convergence rate to is
faster than L1 convergence rate to the self-similar solution[Toscani 05]
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Self-similar solution
PropositionSelf-similar solution is an m- and e-geodesic
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Conclusions
Behavioral analysis of the solutions to the PME and NFPE focusing on the q-Gaussian family.
Constants of motions, evolutional speeds, convergence rate to .Generalized concepts of statistical physics
Future work Relation with Otto’s resultThe other parameter range: m<1 (fast diffusion), 2<m, or the other type of diffusion equation 37
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ReferencesS. Eguchi, Sugaku Expositions, Vol. 19, No. 2, 197-216 (2006), (originally S¯ugaku, 56, No.4, 380-399 (2004) in Japanese.)J. Naudts, Physica A 316, 323-334 (2002).J. Naudts, Reviews in Mathematical Physics, 16, 6, 809-822 (2004).A. Ohara and T. Wada, arXiv: 0810.0624v1 (2008).F. Otto, Comm. Partial Differential Equations 26, 101-174 (2001).G. Toscani, J. Evol. Equ. 5:185-203 (2005).J. L. Vazquez, J. Evol. Equ. 3, 67-118 (2003).