Stochastic Driving and Algebraic Topology · Topology John R. Klein Markov Processes Stat-Mech Tools Main Results Proofs The Quantization Theorem The Realization Theorem References

StochasticDriving andAlgebraicTopology

John R. Klein

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The RealizationTheorem

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......Stochastic Driving and Algebraic Topology

John R. Klein

Stanford Symposium

July 24, 2012

John R. Klein Stochastic Driving and Algebraic Topology


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Embodiment of Mathematical Taste

Figure: Sideburns and hair illustrate gradient dynamics



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Collaborators

Vladimir Chernyak (Wayne State)

Nikolai Sinitsyn (Los Alamos)

Mike Catanzaro (Student, Wayne State)



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Collaborators

Vladimir Chernyak (Wayne State)

Nikolai Sinitsyn (Los Alamos)

Mike Catanzaro (Student, Wayne State)



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Goal

Study periodically driven stochastic systems using themachinery of algebraic topology.

Possible applications to

chemical kinetics (molecular motors)

electrical grids

cell locomotion

Brownian ratchets and turnstiles



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Key Feature

Nano-mechanical motion is known to be noisy (stochastic)

So we should use the language of probabilities.



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Key Feature

Nano-mechanical motion is known to be noisy (stochastic)

So we should use the language of probabilities.



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Example: Kinesin, a Molecular Motor

Figure: Molecular motors operate in the thermal bath, so fluctuationsdue to thermal noise are significant.



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3-Catenane

Figure: A 3-catenane molecule



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Example: A Brownian Ratchet

Figure: The gadget is immersed in a heat bath, and moleculesundergoing Brownian motion hit the paddle wheel at T1, causing theratchet at T2 to turn.



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States and Transitions

The states of the system are the totality of possibleconfigurations.

The transitions between states are represented by edges; thisgives a graph (in the above examples, cyclic).

So we have a Markov process on a finite graph Γ.

We will be interested in periodic one-parameter families ofthese (= loops of Markov processes).



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The transitions between states are represented by edges; thisgives a graph

(in the above examples, cyclic).





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We will be interested in periodic one-parameter families ofthese

(= loops of Markov processes).



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The Average Current Map

We shall describe an invariant of such a family. The invariant isdefined using ideas from statistical mechanics.

The invariant is a smooth map

Q : LMΓ → H1(Γ;R)

called the average current map.

Here LMΓ is the free loop space of the “space of parameters”MΓ, which is a vector space whose vectors define the data fora Markov process on Γ.



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Quantum vs. Statistical Mechanics

Equations describing the evolution of stochastic and quantummechanical systems are mathematically similar.

While quantum dynamics is modeled by the Schrodingerequation, stochastic (Langevin) dynamics is governed by theFokker-Planck equation.

It’s not uncommon to treat statistical mechanics as quantummechanics in imaginary time.



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The Master Equation

We will be working with a version of the Fokker-Planckequation on graphs.

It is called the master equation.



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The Master Equation

We will be working with a version of the Fokker-Planckequation on graphs.

It is called the master equation.



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Graphs

Fix a finite connected graph

Γ = (Γ0, Γ1)

where Γ0 is the set of vertices and Γ1 is the set of edges.

Choose a linear ordering of the set of vertices. This enables usto describe the attaching data as a map

d = (d0, d1) : Γ1 → Γ0 × Γ0 .

(The graph is allowed to have multiple edges and loop edges.)

Vertices are denoted by lower case roman letters i , j , . . . andedges by lower case greek letters α, . . . .



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Graphs


Γ = (Γ0, Γ1)



d = (d0, d1) : Γ1 → Γ0 × Γ0 .





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Graphs


Γ = (Γ0, Γ1)



d = (d0, d1) : Γ1 → Γ0 × Γ0 .





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Populations and Currents

The population space is

C0(Γ) ≡ C0(Γ;R)

and the current space is

C1(Γ) ≡ C1(Γ;R)



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Populations and Currents

The population space is

C0(Γ) ≡ C0(Γ;R)

and the current space is

C1(Γ) ≡ C1(Γ;R)



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Reduced Populations

A population vector p is reduced if∑

i pi = 0.

I.e., it lies in the space of 0-boundaries.

The linear space of reduced population vectors is

C0(Γ) .



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Reduced Populations


i pi = 0.



C0(Γ) .



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Reduced Populations


i pi = 0.



C0(Γ) .



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Normalized Populations

A population vector p is normalized if∑

i pi = 1.

I.e., p is a probability distribution on Γ0.

The (open simplex) of normalized population vectors is

C0(Γ) .



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Normalized Populations

A population vector p is normalized if∑

i pi = 1.

I.e., p is a probability distribution on Γ0.

The (open simplex) of normalized population vectors is

C0(Γ) .



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Conserved Currents

A current vector J is conserved if ∂J = 0, where

∂ :C1(Γ) → C0(Γ)

is the boundary operator.

In this instanceJ ∈ H1(Γ;R) .



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Conserved Currents

A current vector J is conserved if ∂J = 0, where

∂ :C1(Γ) → C0(Γ)

is the boundary operator.

In this instanceJ ∈ H1(Γ;R) .



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The Space Of Parameters

The space of parameters

MΓ

is the vector space of pairs (E ,W ), where

E : Γ0 → R (well energies) and

W : Γ1 → R (barrier energies)

are functions.



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Example: A Graph With Parameters



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The Boltzmann Distribution

Given

a finite set T , and

a positive real number β (inverse temperature),

the Boltzmann distribution is the smooth map

RT → ∆[T ]

given by

E 7→ Z−1∑j∈T

e−βEj j Z ≡∑j∈T

e−βEj

(It takes functions on T to probability distributions on T .)



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Given




RT → ∆[T ]

given by

E 7→ Z−1∑j∈T


e−βEj




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Given




RT → ∆[T ]

given by

E 7→ Z−1∑j∈T


e−βEj




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Example

Set T = Γ0 and fix β. The Boltzmann distribution in this caseis

C 0(Γ) → C0(Γ) .

Composing with the projection (E ,W ) 7→ E , we obtain

ρB :MΓ → C0(Γ) .

This only depends on the well energies.



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Example

Set T = Γ0 and fix β. The Boltzmann distribution in this caseis

C 0(Γ) → C0(Γ) .

Composing with the projection (E ,W ) 7→ E , we obtain

ρB :MΓ → C0(Γ) .

This only depends on the well energies.



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Driving Protocols

A driving protocol is pair (τ, γ) in which τ > 0 and

γ : [0, τ ] → MΓ

is a smooth map. Here, τ denotes driving time.

It is periodic if γ(0) = γ(τ) and the induced loop is smooth.



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Driving Protocols

A driving protocol is pair (τ, γ) in which τ > 0 and

γ : [0, τ ] → MΓ

is a smooth map. Here, τ denotes driving time.

It is periodic if γ(0) = γ(τ) and the induced loop is smooth.



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Driving Protocols

So a periodic driving protocol just a smooth, unbased Mooreloop of MΓ.



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The Master Operator

For fixed (E ,W ) ∈ MΓ and β > 0, the master operator

H :C0(Γ) → C0(Γ)

is−∂g−1∂∗κ ,

where

∂∗ is the formal adjoint to ∂,

κ is the diagonal matrix with κii = eβEi , and

g is the diagonal matrix with gαα = eβWα .



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The Master Operator

The master operator is the discrete analogue of theFokker-Planck operator in Langevin dynamics, which governsdiffusion and advection processes.

One can think of H as a specific perturbation of the graphLaplacian.



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The Master Operator

The master operator is the discrete analogue of theFokker-Planck operator in Langevin dynamics, which governsdiffusion and advection processes.

One can think of H as a specific perturbation of the graphLaplacian.



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The Master Equation

Let (τ, γ) be a periodic driving protocol and β > 0. Themaster equation is the differential equation on given by

p(t) = τH(γ(t))p(t)

with p(t) ∈ C0(Γ).

The master equation models probability flux of distributions.



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Remark

If γ is constant with value (E ,W ), then the Boltzmanndistribution is an equilibrium solution to the master equation.

So it describes the ground state.



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Remark

If γ is constant with value (E ,W ), then the Boltzmanndistribution is an equilibrium solution to the master equation.

So it describes the ground state.



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Formal Solution

The master equation has a (periodic) formal solution

ρ(t) := T exp(τ

∫ t

0dt ′H(γ(t)))ρ(0) ,

where T is the time ordering operator.

Here ρ(0) ∈ C0(Γ) is a choice of initial distribution.



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Formal Solution

The master equation has a (periodic) formal solution

ρ(t) := T exp(τ

∫ t

0dt ′H(γ(t)))ρ(0) ,

where T is the time ordering operator.

Here ρ(0) ∈ C0(Γ) is a choice of initial distribution.



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Instantaneous Current

For fixed β, the instantaneous current of (τ, γ) at t ∈ [0, 1] is

J(t) := τ g−1∂∗κρ(t) ∈ C1(Γ)

where ρ(t) is a formal solution to the master equation.

In other words, J(t) is the unique current satisfying thecontinuity equation

∂J(t) = −ρ(t) .

So J(t) is just probability flux.



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Instantaneous Current

For fixed β, the instantaneous current of (τ, γ) at t ∈ [0, 1] is

J(t) := τ g−1∂∗κρ(t) ∈ C1(Γ)

where ρ(t) is a formal solution to the master equation.

In other words, J(t) is the unique current satisfying thecontinuity equation

∂J(t) = −ρ(t) .

So J(t) is just probability flux.



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Average Current

The average current of (τ, γ) is

Q ≡∫ 1

0J(t) dt .

It is an element of H1(Γ;R).

SoQ : LMΓ → H1(Γ;R)

(it’s a smooth map).



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Average Current

The average current of (τ, γ) is

Q ≡∫ 1

0J(t) dt .

It is an element of H1(Γ;R).

SoQ : LMΓ → H1(Γ;R)

(it’s a smooth map).



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The Adiabatic Limit

The instantaneous current J(t) as well as the average currentQ, depend on the driving time τ as well as the inversetemperature β.

Taking the limit τ → ∞, we obtain what is referred to as theadiabatic limit.



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The Adiabatic Limit

The instantaneous current J(t) as well as the average currentQ, depend on the driving time τ as well as the inversetemperature β.

Taking the limit τ → ∞, we obtain what is referred to as theadiabatic limit.



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The Operator A

For given (E ,W , β), the operator

−∂ : im(g−1∂∗κ) → C0(Γ)

is an isomorphism.

LetA : C0(Γ) → C1(Γ)

be its left inverse.



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The Operator A

For given (E ,W , β), the operator

−∂ : im(g−1∂∗κ) → C0(Γ)

is an isomorphism.Let

A : C0(Γ) → C1(Γ)

be its left inverse.



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The Operator A

.Remark..

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A is the solution to Kirchhoff’s network problem for Γ withrespect to branch resistances eβW .

In particular, A can be expressed as a linear combination ofspanning trees.



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The Operator A

.Remark..

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A is the solution to Kirchhoff’s network problem for Γ withrespect to branch resistances eβW .

In particular, A can be expressed as a linear combination ofspanning trees.



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Adiabatic Theorem

.Theorem (Adiabatic Theorem)..

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For fixed β, we have

limτ→∞

Q =

∫ 1

0JB dt

where JB := A(γ(t), ρB(γ(t)).

In particular, the adiabatic limit of Q is a linear combination ofspanning trees.



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Adiabatic Theorem


......


limτ→∞

Q =

∫ 1

0JB dt





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Adiabatic Theorem


......


limτ→∞

Q =

∫ 1

0JB dt





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The Payoff

In the adiabatic limit, we do not need to refer to solutions ofthe master equation.

Instead, we can work with the operator A and the Boltzmanndistribution.



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The Payoff

In the adiabatic limit, we do not need to refer to solutions ofthe master equation.

Instead, we can work with the operator A and the Boltzmanndistribution.



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Recapitulation

Two numbers were involved in defining Q:

A positive real number τ which is (periodic) drivingtime, and

A positive real number β which is inverse temperature.

SoQ = Q(τ, β) .



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Recapitulation

The limit τ → ∞ is called the adiabatic limit (slow driving).It is always well-defined.

Henceforth, we pass to the adiabatic limit.

The limit β → ∞ is called the low temperature limit. It isnot always defined.



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Recapitulation






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Recapitulation






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Intrinsically Robust Loops

.Definition (Intrinsically Robust Loops)..

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A loop γ ∈ LMΓ is said to be intrinsically robust if there is anopen neighborhood U of γ such that the low temperature limit

Q := limβ→∞

Q(β,−)

is well-defined and constant on U. The subspace of LMΓ

consisting of the intrinsically robust loops is denoted by LMΓ.



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Quantization

.Theorem (Pumping Quantization Theorem)..

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The image of the map


is contained in the integer lattice H1(Γ;Z) ⊂ H1(Γ;R).

It is non-trivial whenever H1(Γ) = 0.



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Quantization

.Theorem (Pumping Quantization Theorem)..

......

The image of the map


is contained in the integer lattice H1(Γ;Z) ⊂ H1(Γ;R).

It is non-trivial whenever H1(Γ) = 0.



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Representability

.Theorem (Representability Theorem)..

......

There is a topological subspace D ⊂ MΓ such that

LMΓ = L(MΓ \ D) .

Consequently, the space of intrinsically robust loops is a loopspace.



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The Discriminant

The subspace D is called the discriminant, and its complementMΓ := MΓ \ D is called the space of robust parameters.



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The Discriminant

.Theorem (Discriminant Theorem)..

......

The one point compactification of the discriminant, i.e., D+,has the structure of a finite regular CW complex of dimensiond −2. In particular, the inclusion MΓ ⊂ MΓ is open and dense.



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Realization

The complement to the discriminant, MΓ \ D, is called thespace of robust parameters. Denote it by MΓ..Theorem (Realization)..

......

There is a weak map

q :MΓ → |Γ|

which induces Q by sending a loop γ to the homology class ofq(γ).



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Example: 3-Catenane

Figure: A 3D cross section of the space of parameters. Thediscriminant appears as a one dimensional subspace. Integer currentsare generated by linking, yielding quantization.



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Proofs

It’s not so easy to outline the proofs in the general case.

To give the flavor of the ideas, we’ll sketch proofs of theQuantization and the Realization Theorems in a weak case.

We first define a space MΓ which turns out to be a subset ofthe space of the robust parameters MΓ.

We then prove versions of the main results on this subspace.



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Good Parameters

.Definition..

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The space of good parameters

MΓ

is the set of pairs (E ,W ) ∈ MΓ such that

E : Γ0 → R has a unique minimum, or

W : Γ1 → R is one-to-one.



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Weak Quantization

We will establish quantization for loops inside MΓ.



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Weak Quantization

The main idea of the proof is to use a van Kampen typeargument. We have a decomposition

MΓ = U ∪ V ,

where

U is the open set of (E ,W ) such that E has a uniqueminimum,

V is the open set of (E ,W ) such that W is one-to-one.



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Weak Quantization

Idea: Any loop γ : S1 → MΓ can be decomposed into closedarcs of type U and closed arcs of type V and these alternate:



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Weak Quantization

.Lemma..

......

The contribution of an arc I of type U to the average current iszero in the low temperature limit.



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Weak Quantization

Proof: The relevant contribution is given by∫IJBds ,

and on I the function E : Γ0 → R has a unique minimum.ρB along I tends to the E -minimal vertex, say ℓ. That’sbecause the j-component of ρB is given by

e−βEj∑i e

−βEj

and this tends to zero unless j = ℓ. The rest of the argumentessentially follows from the fact that JB is defined using ρB,and ρB is trivial in the low temperature limit.



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.Lemma..

......

The contribution of an arc I of type V to the average currenttends to an integral current in the low temperature limit.



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Weak Quantization

Proof: Fix a basepoint i ∈ Γ0. By Kirchoff’s networktheorem and integration by parts, we get

JB =∑T ,j

QTij ϱ

BT ρ

Bj ,

where

T ranges over all spanning trees of Γ, j ranges over allvertices;

QTij is the integral current defined by the unique path in

T from i to j (with suitable signs);

ϱBT is the T -component of the Boltzmann distribution forthe set of spanning trees, where the energy is given by∑

α∈T Wα.

(Note: QTij doesn’t depend on γ(t).)



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So it’s enough to prove that the real number∫IϱBT ρ

Bj

tends to an integer in the low temperature limit.Integration by parts identifies this with

ϱBTρ

Bj ]∂I −

∫IϱBTρ

Bj .

Rough Idea: The first term tends to an integer by the previousLemma and the fact that ∂I is of type U.

The second term tends to zero because the condition that W isone-to-one implies that ϱB

T tends to zero in the lowtemperature limit.



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Weak Realization

RecallMΓ = U ∪ V .

Idea: define a subspace N ⊂ MΓ × |Γ| such that

MΓ Np1∼

oop2 // |Γ|

defines the desired weak map.



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N is defined asNU ∪ NV ,

where NU → U, NU ∩ NV → U ∩ V and NV → V arehomotopy equivalences.

The conclusion will then follow from the gluing lemma.



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John R. Klein

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Recall that for (E ,W ) ∈ U, the function E : Γ0 → R has aunique minimum vE .

Let BE be the open neighborhood vE in |Γ| consisting of pointshaving distance < 1/3 to vE .

Then NU is the space of ((E ,W ), x) consisting of (E ,W ) ∈ Uand x ∈ BE .

It is easy to see that NU → U is a weak equivalence.



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John R. Klein

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John R. Klein

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John R. Klein

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The construction of NV requires an auxiliary definition.

For each total ordering σ of the set of edges Γ1, we may definea spanning tree Tσ for Γ by sequentially removing the edgeswith the highest possible value in the ordering such that theremaining graph remains connected.



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The construction of NV requires an auxiliary definition.

For each total ordering σ of the set of edges Γ1, we may definea spanning tree Tσ for Γ by sequentially removing the edgeswith the highest possible value in the ordering such that theremaining graph remains connected.



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.Definition..

......

The tree Tσ given by the above procedure is called thespanning tree associated with σ, or simply the σ-spanningtree.



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Figure: A graph with a total ordering of its edges and its associatedσ-spanning tree.



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.Remark..

......

If (E ,W ) ∈ V , then W : Γ1 → R is one-to-one, so it determinesa preferred partial ordering σ and therefore a σ-spanning tree.



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Given W nondegenerate, and any spanning tree T for Γ, definelet

w := w(T ,W ) =∑

α∈Γ1\T1

Wα

.Lemma (Characterization of Tσ)..

......

With σ defined by W : Γ1 → R non-degenerate, the σ-spanningtree Tσ is the unique absolute maximum for the functionw(−,W ).



John R. Klein

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. . . . . .

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Given W nondegenerate, and any spanning tree T for Γ, definelet

w := w(T ,W ) =∑

α∈Γ1\T1

Wα

.Lemma (Characterization of Tσ)..

......

With σ defined by W : Γ1 → R non-degenerate, the σ-spanningtree Tσ is the unique absolute maximum for the functionw(−,W ).



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.Definition..

......

NV ⊂ MΓ × |Γ|

is the subspace consisting of

((E ,W ), x)

in which

(E ,W ) ∈ V ;

x ∈ BW , where BW is an open metric neighborhood ofradius 1/3 containing |Tσ|.

Then the projection NV → V is clearly a weak equivalence.



John R. Klein

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.Definition..

......

NV ⊂ MΓ × |Γ|

is the subspace consisting of

((E ,W ), x)

in which

(E ,W ) ∈ V ;

x ∈ BW , where BW is an open metric neighborhood ofradius 1/3 containing |Tσ|.

Then the projection NV → V is clearly a weak equivalence.



John R. Klein

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References

. . . . . .

References

Chernyak, V.Y., Klein, J.R., Sinitsyn, N.A.:

1. Algebraic topology and the quantization of fluctuating currents, submitted. arXiv:1204.2011

2. Quantization and Fractional Quantization of Currents in Periodically Driven Stochastic Systems I:Average Currents. Jour. Chem. Physics 136, 154107 (2012)

3. Quantization and Fractional Quantization of Currents in Periodically Driven Stochastic Systems II: FullCounting Statistics, Jour. Chem. Physics 136, 154108 (2012)

4. Bollobas, Bela: Modern graph theory. Graduate Texts in Mathematics, 184. Springer-Verlag, New York,1998

5. van Kampen, N.G.: Stochastic processes in physics and chemistry, 3rd ed. North-Holland Personal Library,Elsevier, Amsterdam, 2007


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Stochastic Driving and Algebraic Topology · Topology John R. Klein Markov Processes Stat-Mech Tools Main Results Proofs The Quantization Theorem The Realization Theorem References