Stochastic Driving and Algebraic Topology · Topology John R. Klein Markov Processes Stat-Mech...
Transcript of Stochastic Driving and Algebraic Topology · Topology John R. Klein Markov Processes Stat-Mech...
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
.
......Stochastic Driving and Algebraic Topology
John R. Klein
Stanford Symposium
July 24, 2012
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
Embodiment of Mathematical Taste
Figure: Sideburns and hair illustrate gradient dynamics
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
Collaborators
Vladimir Chernyak (Wayne State)
Nikolai Sinitsyn (Los Alamos)
Mike Catanzaro (Student, Wayne State)
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
Collaborators
Vladimir Chernyak (Wayne State)
Nikolai Sinitsyn (Los Alamos)
Mike Catanzaro (Student, Wayne State)
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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
John R. Klein Stochastic Driving and Algebraic Topology
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MarkovProcesses
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Main Results
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TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
Key Feature
Nano-mechanical motion is known to be noisy (stochastic)
So we should use the language of probabilities.
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
Key Feature
Nano-mechanical motion is known to be noisy (stochastic)
So we should use the language of probabilities.
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
Example: Kinesin, a Molecular Motor
Figure: Molecular motors operate in the thermal bath, so fluctuationsdue to thermal noise are significant.
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
3-Catenane
Figure: A 3-catenane molecule
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
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MarkovProcesses
Stat-MechTools
Main Results
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TheQuantizationTheorem
The RealizationTheorem
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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.
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
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TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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).
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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).
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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).
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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).
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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).
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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).
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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.
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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.
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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.
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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.
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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.
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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.
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
The Master Equation
We will be working with a version of the Fokker-Planckequation on graphs.
It is called the master equation.
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
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TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
The Master Equation
We will be working with a version of the Fokker-Planckequation on graphs.
It is called the master equation.
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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, . . . .
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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, . . . .
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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, . . . .
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
Populations and Currents
The population space is
C0(Γ) ≡ C0(Γ;R)
and the current space is
C1(Γ) ≡ C1(Γ;R)
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
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TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
Populations and Currents
The population space is
C0(Γ) ≡ C0(Γ;R)
and the current space is
C1(Γ) ≡ C1(Γ;R)
John R. Klein Stochastic Driving and Algebraic Topology
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TheQuantizationTheorem
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References
. . . . . .
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) .
John R. Klein Stochastic Driving and Algebraic Topology
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TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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) .
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
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MarkovProcesses
Stat-MechTools
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TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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) .
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
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TheQuantizationTheorem
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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) .
John R. Klein Stochastic Driving and Algebraic Topology
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TheQuantizationTheorem
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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) .
John R. Klein Stochastic Driving and Algebraic Topology
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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) .
John R. Klein Stochastic Driving and Algebraic Topology
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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 .)
John R. Klein Stochastic Driving and Algebraic Topology
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TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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 .)
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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 .)
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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.
John R. Klein Stochastic Driving and Algebraic Topology
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TheQuantizationTheorem
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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.
John R. Klein Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
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TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
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.
John R. Klein Stochastic Driving and Algebraic Topology
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TheQuantizationTheorem
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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.
John R. Klein Stochastic Driving and Algebraic Topology
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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.
John R. Klein Stochastic Driving and Algebraic Topology
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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.
John R. Klein Stochastic Driving and Algebraic Topology
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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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The RealizationTheorem
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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..
......
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..
......
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)..
......
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
.Theorem (Adiabatic Theorem)..
......
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
.Theorem (Adiabatic Theorem)..
......
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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. . . . . .
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
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
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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. . . . . .
Intrinsically Robust Loops
.Definition (Intrinsically Robust Loops)..
......
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)..
......
The image of the map
Q : LMΓ → H1(Γ;R)
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
Q : LMΓ → H1(Γ;R)
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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. . . . . .
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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. . . . . .
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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. . . . . .
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..
......
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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Weak Quantization
.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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Weak Quantization
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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Weak Realization
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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Weak Realization
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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Weak Realization
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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Weak Realization
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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Weak Realization
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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Weak Realization
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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Weak Realization
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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Weak Realization
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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Weak Realization
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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Weak Realization
.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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Weak Realization
Figure: A graph with a total ordering of its edges and its associatedσ-spanning tree.
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Weak Realization
.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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Weak Realization
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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Weak Realization
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 Stochastic Driving and Algebraic Topology
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. . . . . .
Weak Realization
.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 Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
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TheQuantizationTheorem
The RealizationTheorem
References
. . . . . .
Weak Realization
.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 Stochastic Driving and Algebraic Topology
StochasticDriving andAlgebraicTopology
John R. Klein
MarkovProcesses
Stat-MechTools
Main Results
Proofs
TheQuantizationTheorem
The RealizationTheorem
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
John R. Klein Stochastic Driving and Algebraic Topology