the john r. klein wayne state...

StochasticProcesses

Concrete FieldTheories

TrajectorySpaces

TheProbabilityMeasure

The Current

Quantization

Topological Stochastics

John R. KleinWayne State University

Topology in Dubrovnik

June 24, 2014

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Collaborators

Vladimir Chernyak (Dept. of Chemistry, Wayne State)

Mike Catanzaro (PhD Student, Wayne State)

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Outline

..1 Stochastic Processes

..2 Concrete Field Theories

..3 Trajectory Spaces

..4 The Probability Measure

..5 The Current

..6 Quantization

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Definition

Let Y be a space.

A stochastic process with values in Y is a one parameterfamily of random variables Xt , i.e., a map

X : [0, τ ]× Ω → Y ,

where,

• Ω is a probability measure space, and

• Xt(σ) := X (t, σ) .

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Using the exponential law, view this as a map

X : Ω → trajectories in Y .

Let ΩY denote the space of trajectories in Y .

Then X induces a probability measure PY on ΩY bypushforward:

PY := PΩ X−1 .

This is called the law of the process.

Conversely, a probability measure on ΩY defines a stochasticprocess by setting Ω = ΩY .

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In a Nutshell...

The theory of stochastic processes amounts to the study ofprobability measures on trajectory spaces.

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Context

Classically, Y is the real line R or, more generally, a smoothmanifold.

However, for this talk:

Y := a finite connected CW complex of dimension d ≥ 1.

(We will give a precise definition of ΩY below.)

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..5 The Current

..6 Quantization

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Observables

An observable on ΩY is a measurable function

B : ΩY → R .

Think of this as the result of taking a measurement.

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The Path Integral

If we are given an observable

B : ΩY → R

then we can form its expectation

⟨B⟩ :=∫σ∈ΩY

dP(σ)B(σ) .

This is a path integral.

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The observables we will consider arise from the real-valuedcellular d-cocycles on Y .

We will construct a linear transformation

J :Zd(Y ;R) −→ F (ΩY ,R) = observables .

called the current density.

Path integration composed with J results in a “flux” oraverage current:

⟨J⟩ :Zd(Y ;R) J−→ F (ΩY ,R)path integral−−−−−−−→ R .

Call this set-up a combinatorial field theory.

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A combinatorial field theory is homological if

⟨J⟩ :Zd(Y ;R) → R

vanishes on coboundaries, i.e., it factors over Hd(Y ;R) to givea linear functional

⟨J⟩∗ :Hd(Y ;R) → R .

If this occurs we obtain a real homology class

q ∈ Hd(Y ;R) .

which pairs perfectly with ⟨J⟩∗.

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Summary

In favorable circumstances:

stochastic process homology class

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To Do List:

• Define the space of trajectories ΩY ;

• Construct the probability measure PY ;

• Construct the combinatorial observable;

• Determine whether the combinatorial observable ishomological;

• If not, make repairs.

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..5 The Current

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The Graph Case

SupposeY := Γ

is a finite connected CW complex of dimension d = 1.

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Definition of a Trajectory

A trajectory in Γ of time duration τ is a pair

(ψ, t) ,

in which ψ is a path in Γ:

i1α1−→ i2

α2−→ · · · αk−→ ik+1 ,

i.e.,

• ij is a vertex;

• αj is an edge connecting ij and ij+1;

and t := (t1, · · · , tk+1) denotes a sequence of jump times:

0 ≤ t1 ≤ t2 ≤ · · · ≤ tk ≤ tk+1 = τ ,

The .. formula for the probability of (ψ, t) appears below.

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The collection of all trajectories of fixed time duration τ isdenoted

ΩτΓ ,

but when τ is understood we simply denote it by

ΩΓ .

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Context

We will be describing a Markov process on Γ where at thestate ij , we wait until time tj and then instantaneously jumpto the state ij+1 with a certain transition probability.

Markov process = stochastic process without memory.

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Example

Figure: a Markov process.

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Higher Dimensions

Now letY := X

be a finite connected CW complex of dimension d . We willassociate to X a locally finite graph ΓX .

The desired stochastic process is then defined on thetrajectories of ΓX .

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The Cycle Incidence Graph

We fix throughout a non-trivial homology class

x ∈ Hd−1(X ;Z) .

The vertices of ΓX are the set of integer-valued (cellular)(d − 1)-cycles in X representing x .

An oriented edge from z0 to z1 is specified by a pair

(e, f )

such that

• e is a d-cell, f is a (d − 1)-cell;

• ⟨z0, f ⟩ = 0 and ⟨z1, f ⟩ = 0;

• z0 + n∂e = z1 for some integer n (⇒ ⟨∂e, f ⟩ = 0).

(note: n is unique).

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The Cycle Incidence Graph

We setΩX := ΩΓX .

Hence, the state space consists of integer (d − 1)-cycles andtrajectories are formed by jumping across d-cells which areincident to the (d − 1)-cycles.

(To avoid clutter, we do not include the given homology class xin the notation.)

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Example

Figure: A trajectory on a polyhedral torus with a meridian as theinitial state.

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Remark

There is a canonical chain map

r∗ : C∗(ΓX ;Z) → C∗+d−1(X ;Z)

which is given when ∗ = 0 by

z 7→ z

and when ∗ = 1 by(e, f ) 7→ ne .

So we have a map on cohomology

r∗ :Hd(X ;R) → H1(ΓX ;R) ,

and a field theory for ΓX can be pulled-back to one for X .

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The Case d = 1

Consider the case dimX = 1 and X is a simple graph. Takex ∈ H0(X ) = Z to be the generator.

Then the evident inclusion

i :X⊂−→ ΓX ,

is a connected component. Furthermore, i∗r∗ is the identity, sor∗ :H1(X ;R) → H1(ΓX ;R) is injective.

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..5 The Current

..6 Quantization

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Overview

The probability measure on our sample space arises mostnaturally by considering how the process evolves according to acertain differential equation, called the Master Equation.1

However, because of time constraints, I will give a more directconstruction using the biased divergence operator.

For simplicity, I’ll give the construction when d = 1. Theconstruction easily extends to higher dimensions using the cycleincidence graph.

1A combinatorial version of the Kolmogorov Equation.28 / 63

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Graph Energies

We choose:

• for each vertex j a real number Ej , and

• for each edge α a real number Wα.

Setγ := (E•,W•) .

Then γ assigns a real number to each vertex and each edge.Call these graph energies.

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Time Dependence

To get something other than a diffusion process, we’ll allowγ = (E•,W•) to be time dependent and periodic.

I.e.,γ(t) = (E•(t),W•(t))

is a smooth one parameter family of graph energies such that

γ(0) = γ(τ) .

This is known as periodic driving.

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Basis Rescaling

Choose a real number β > 0 (inverse temperature).

Letκ :C0(Γ;R) → C0(Γ;R)

be the diagonal matrix with entries

κjj = eβEj

and letg :C1(Γ;R) → C1(Γ;R)

be the diagonal matrix with entries

gαα = eβWα

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Biased Divergence

The biased divergence operator is

g−1∂∗κ :C0(Γ;R) → C1(Γ;R) ,

where ∂∗ :C0(Γ;R) → C1(Γ;R) is the formal adjoint to ∂.

Equivalently, it is just the formal adjoint to ∂ in the modifedinner product on C∗(Γ;R) determined by g and κ.

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Transition Rates

If(α, j)

is a pair consisting of an edge α and a vertex j contained in theboundary of α, then the (α, j) matrix element of g−1∂∗κ is, upto sign,

hα,j := eβ(Ej−Wα) .

This number is called the transition rate of the pair (α, j).

(Note: it depends on the time parameter t ∈ [0, τ ].)

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Escape Rates

If (ψ, t) is a trajectory containing a vertex ij with jump time tj ,then its escape rate at ij is is

uij (tj , tj−1) := e−

∑α

∫ tjtj−1

hα,ij dt ,

where the summation is over all edges α that contain the givenvertex ij .

Observe: the escape rate at a vertex is determined by thetransitions rates of the edges at that vertex.

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Probability of a Trajectory

Fix an initial probability distribution p ∈ C0(Γ;R), and graphenergies γ = (E•,W•).

Then the probability of a trajectory .. (ψ, t) is:

PΓ(ψ, t) := pi1ui1(t1, 0)hα1,i1(t1)ui2(t2, t1) · · ·· · · hαk ,ik (tk)uik+1

(tk+1, tk)

:= pi1 ·k+1∏j=1

uij (tj , tj−1) ·k∏

j=1

hαj ,ij (tj) .

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Schematic

Figure: The probability of a trajectory is obtained by multiplying itsrates.

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Proposition

PΓ is a probability measure on the space of trajectories ΩΓ.

(The proof uses perturbation theory.)

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Base Change

Let ΠΓ denote the set of paths. Forgetting jump times definesa function

h : ΩΓ → ΠΓ .

Integration of h along the fibers (= jump time coordinates)induces a measure ω on ΠΓ.

So if f : ΠΓ → R is an observable, we have a base change rule∫ΠΓ

f dω =

∫ΩΓ

f h dP .

Note that ΠΓ is discrete, so the path integral on the left is aninfinite series.

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Trajectory Probability When d > 1

In the d > 1 case, we start with (a one parameter family of)

γ = (E•,W•)

where E• consists of real numbers labelling the (d − 1)-cellsand W• consists of real numbers labelling the d-cells.

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Trajectory Probability When d > 1

For an edge of ΓX labelled by (e, f ), where e is a d-cell and fis a (d − 1)-cell, the transition rate is

he,f := eβ(Ef −We) .

Up to sign, this is the (e, f ) matrix element of the biaseddivergence operator

g−1∂∗κ :Cd−1(X ;R) → Cd(X ;R) .

We use these transition rates to determine the escape rate ateach vertex in a trajectory, just as in the d = 1 case.

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..5 The Current

..6 Quantization

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We will now define a linear transformation

J :Zd(X ;R) → F (ΩΓX ;R)

called the current density.

Again, by availing ourselves of the cycle incidence graph ΓX ,we can reduce to the d = 1 case.

By linearity, we only need to define J on the edges of Γ.

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Letδα ∈ Z 1(Γ;R)

be the 1-cocycle determined by an edge α.

Then, given a trajectory (ψ, t) ∈ ΩΓ, its current densityacross α is

Jα(ψ, t) :=n+ − n−

τ,

where n± is the number of times α appears withpositive/negative orientation in ψ.

The integern+ − n− = δα(ψ)

is just the net flow across α.

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Q

QB

B

Figure: Depiction of the net flow n+ − n− across an oriented edge.

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So we get a combinatorial observable

J :Z 1(Γ;R) → F (ΩΓ,R) .

whose expected value ⟨J⟩ is the flux or average current.

Remark: The path integral converges because δα(ψ) is anapproximately linear function of path length, whereas theprobability of ψ decays exponentially in path length.

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Does J give a homological theory?

No!

But there’s an easy modification that works in the d = 1 case:we can iterate γ and take a limit.

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Iterated Driving

For a positive integer N, let γN be the periodic driving protocolgiven by

I.e.,t 7→ γ(t − jτ), jτ ≤ t < (j + 1)τ, j ∈ N

for t ∈ [0,Nτ ].

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Let’s clarify notation: recall that

ΩτΓ

is the set of trajectories of time duration τ .

Then γN gives rise to a measure on ΩNτΓ , and we have a

combinatorial observable

JN :Z 1(Γ;R) → F (ΩNτΓ ,R) .

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Proposition

Assume d = 1. Then the sequence

limN→∞

⟨JN⟩

pointwise converges and its limit is homological.

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Proof

I will omit the proof of convergence, which is technical.

Step 1: Think of an edge α as a 1-cochain. Let ψ be a path,considered as a 1-chain. Then

NτJN(δα)(ψ) = n+ − n− = δα(ψ) .

So NτJN is an observable on paths and

⟨JN(u)⟩ =1

Nτ

∫ψ∈ΠΓ

dωN(ψ)u(ψ) .

for any 1-cochain u. Here ωN is the measure on ΠΓ that arisesfrom the forgetful map ΩNτ

Γ → ΠΓ.

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Proof

Step 2: For a pair of vertices x , y , let Πx ,y ⊂ ΠΓ be the set ofpaths starting at x and ending at y . Then

ΠΓ =⨿x ,y

Πx ,y .

By linearity, we may assume u = ∂∗v is the coboundary of asingle vertex v of Γ. Fix a simple path ψ0 from x to y . Ifψ ∈ Πx ,y , then

|u(ψ)| = |v(∂ψ)| = |v(∂ψ0)| ≤ 1 .

Hence,

|⟨JN(u)⟩| ≤C

Nτ

where C is the number of pairs of distinct vertices (x , y).Taking N → ∞ finishes the argument.

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Problems when d > 1.

The argument we gave falls apart when d > 1. First of all, wedon’t know if limN→∞⟨JN⟩ exists.

Secondly, the argument we gave for coboundaries relied on thefact that the graph has finitely many vertices. This is notgenerally true for ΓX .

Nevertheless, we are optimistic that the proposition is still validfor d > 1.

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..5 The Current

..6 Quantization

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The following results were motivated by observations made instatistical mechanics.

The results say that average current quantizes in the limit ofslow driving and low temperature.

In the literature, such results are called pumping quantizationtheorems.

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Parameter Dependence

The definition of the probability distribution on depended onthe parameters

(τ, γ(t), β)

in which

• [0, τ ] is the domain of the periodic driving protocol γ(t),and

• β > 0 is inverse temperature.

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Slow Driving

Let τ ′ > τ . Then time rescaling is the function

r : [0, τ ′] → [0, τ ]

given by r(t) = (τ/τ ′)t.

Replace (τ, γ(t), β) by

(τ ′, γ(r(t)), β)

to slow down the system.

The adiabatic limit takes τ ′ → ∞.

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Low Temperature

Taking β → ∞ is called the low temperature limit.

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Integer Quantization for Graphs

Theorem (Chernyak-K.-Sinitsyn (2013))

Take N → ∞, then the adiabatic limit, and thereafter the lowtemperature limit of the average current q. Then for genericdriving the limit of the average current exists.

Furthermore, it lies in the integer lattice

H1(Γ;Z) ⊂ H1(Γ;R) .

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Generic Driving

A set of energy parameters (E•,W•) is nondegenerate ifeither

• The function E• : Γ0 → R is one-to-one, or

• The function W• : Γ1 → R is one-to-one.

A driving protocol γ(t) = (E•(t),W•(t)) is generic if γ(t) isnondegenerate for all t.

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The d > 1 Case

As mentioned above, we don’t know if ⟨JN⟩ converges fordimX > 1.

However, by a very different method, we are still able to definea real homology class

qH ∈ Hd(X ;R) ,

which seems to be a good substitute.

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The d > 1 Case

The class qH is given by the biased divergence of the“Boltzmann cycle” ρB ∈ Zd−1(X ;R):

qH :=

∫ τ

0g−1∂∗κ︸︷︷︸biased

divergence

ρB︸︷︷︸Boltzmanncycle

dt ∈ Zd(X ;R) = Hd(X ;R) .

The Boltzmann cycle ρB represents x ∈ Hd−1(X ;Z) and is thesolution to the combinatorial Hodge theory problem associatedwith the biased Laplacian

∂g−1∂∗κ .

When d = 1, it turns out that qH coincides with the currentq, as described using the path integral formulation.

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Fractional Quantization for d > 1

Theorem (Catanzaro-Chernyak-K. (2014))

Let X be a finite connected CW complex of dimension d > 1.Take N → ∞, then the adiabatic limit, and thereafter the lowtemperature limit of qH .

Then for generic driving the limit exists and it lies in theimage of the homomorphism

Hd(X ;Z[ 1R ]) → Hd(X ;R) ,

where R which is a non-zero multiple of the order of thetorsion subgroup of Hd−1(X ;Z).

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