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The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

The Min/Max-Entropy Production Principles from a Dynamical Fluctuation Law

• C. Maes1
• K. Netoˇ

cný2

1Instituut voor Theoretische Fysica

• K. U. Leuven

2Institute of Physics

Academy of Sciences of the Czech Republic

Mathematical Physics Days XIII, Leuven 28-29 September 2006

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Introduction

◮ Thermal equilibrium distribution is characterized by the

Gibbs variational principle

◮ proves useful in analyzing infinite-volume systems,

phase transitions,...

◮ No general variational characterization of

nonequilibrium stationary states is known!

◮ Yet, there are various approximative principles:

◮ Minimum & maximum entropy production principles ◮ Either restricted to close-to-equilibrium regime or

uncontrollable

◮ Even close to equilibrium counterexamples are known!

◮ What is the status of these principles? ◮ Can we understand more or even go beyond?

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Introduction

◮ Thermal equilibrium distribution is characterized by the

Gibbs variational principle

◮ proves useful in analyzing infinite-volume systems,

phase transitions,...

◮ No general variational characterization of

nonequilibrium stationary states is known!

◮ Yet, there are various approximative principles:

◮ Minimum & maximum entropy production principles ◮ Either restricted to close-to-equilibrium regime or

uncontrollable

◮ Even close to equilibrium counterexamples are known!

◮ What is the status of these principles? ◮ Can we understand more or even go beyond?

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Introduction

◮ Thermal equilibrium distribution is characterized by the

Gibbs variational principle

◮ proves useful in analyzing infinite-volume systems,

phase transitions,...

◮ No general variational characterization of

nonequilibrium stationary states is known!

◮ Yet, there are various approximative principles:

◮ Minimum & maximum entropy production principles ◮ Either restricted to close-to-equilibrium regime or

uncontrollable

◮ Even close to equilibrium counterexamples are known!

◮ What is the status of these principles? ◮ Can we understand more or even go beyond?

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Introduction

◮ Thermal equilibrium distribution is characterized by the

Gibbs variational principle

◮ proves useful in analyzing infinite-volume systems,

phase transitions,...

◮ No general variational characterization of

nonequilibrium stationary states is known!

◮ Yet, there are various approximative principles:

◮ Minimum & maximum entropy production principles ◮ Either restricted to close-to-equilibrium regime or

uncontrollable

◮ Even close to equilibrium counterexamples are known!

◮ What is the status of these principles? ◮ Can we understand more or even go beyond?

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Introduction

◮ Thermal equilibrium distribution is characterized by the

Gibbs variational principle

◮ proves useful in analyzing infinite-volume systems,

phase transitions,...

◮ No general variational characterization of

nonequilibrium stationary states is known!

◮ Yet, there are various approximative principles:

◮ Minimum & maximum entropy production principles ◮ Either restricted to close-to-equilibrium regime or

uncontrollable

◮ Even close to equilibrium counterexamples are known!

◮ What is the status of these principles? ◮ Can we understand more or even go beyond?

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Introduction

◮ Thermal equilibrium distribution is characterized by the

Gibbs variational principle

◮ proves useful in analyzing infinite-volume systems,

phase transitions,...

◮ No general variational characterization of

nonequilibrium stationary states is known!

◮ Yet, there are various approximative principles:

◮ Minimum & maximum entropy production principles ◮ Either restricted to close-to-equilibrium regime or

uncontrollable

◮ Even close to equilibrium counterexamples are known!

◮ What is the status of these principles? ◮ Can we understand more or even go beyond?

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov chains

Notation

Consider a finite-state continuous-time Markov process ηt:

◮ Rates λ(x, y) ◮ Ergodicity assumption =

⇒ unique stationary distribution ρ:

• x

[ρ(x)λ(x, y) − ρ(y)λ(y, x)] = 0

◮ Alternatively, the process given by the generator:

(Lf)(η) =

• y=x

λ(x, y)[f(ηxy) − f(η)] so that Eµ[f(ηt)] = µ(etLf) = µt(f)

◮ Describes a general nonequilibrium dynamics of a

thermodynamically open system

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov chains

Notation

Consider a finite-state continuous-time Markov process ηt:

◮ Rates λ(x, y) ◮ Ergodicity assumption =

⇒ unique stationary distribution ρ:

• x

[ρ(x)λ(x, y) − ρ(y)λ(y, x)] = 0

◮ Alternatively, the process given by the generator:

(Lf)(η) =

• y=x

λ(x, y)[f(ηxy) − f(η)] so that Eµ[f(ηt)] = µ(etLf) = µt(f)

◮ Describes a general nonequilibrium dynamics of a

thermodynamically open system

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov chains

Equilibrium dynamics

Detailed balance assumption: ρ(x)λ(x, y) = ρ(y)λ(y, x)

◮ Describes an equilibrium dynamics

◮ closed system for ρ the counting measure ◮ Relaxing to an equilibrium with a bath for ρ a Gibbs

measure

Defining the entropy S(µ) = −S(µ | ρ) = −

• x

µ(x) log µ(x) ρ(x) the entropy production rate is E(µ) = dS(µt) dt

• t=0=
• x,y=x

µ(x)λ(x, y) log µ(x)ρ(y) µ(y)ρ(x)

◮ By convexity, σ(µ) ≥ 0 and the equality is only for µ = ρ

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov chains

Equilibrium dynamics

Detailed balance assumption: ρ(x)λ(x, y) = ρ(y)λ(y, x)

◮ Describes an equilibrium dynamics

◮ closed system for ρ the counting measure ◮ Relaxing to an equilibrium with a bath for ρ a Gibbs

measure

Defining the entropy S(µ) = −S(µ | ρ) = −

• x

µ(x) log µ(x) ρ(x) the entropy production rate is E(µ) = dS(µt) dt

• t=0=
• x,y=x

µ(x)λ(x, y) log µ(x)ρ(y) µ(y)ρ(x)

◮ By convexity, σ(µ) ≥ 0 and the equality is only for µ = ρ

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov chains

Equilibrium dynamics

Detailed balance assumption: ρ(x)λ(x, y) = ρ(y)λ(y, x)

◮ Describes an equilibrium dynamics

◮ closed system for ρ the counting measure ◮ Relaxing to an equilibrium with a bath for ρ a Gibbs

measure

Defining the entropy S(µ) = −S(µ | ρ) = −

• x

µ(x) log µ(x) ρ(x) the entropy production rate is E(µ) = dS(µt) dt

• t=0=
• x,y=x

µ(x)λ(x, y) log µ(x)ρ(y) µ(y)ρ(x)

◮ By convexity, σ(µ) ≥ 0 and the equality is only for µ = ρ

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov Chains

equilibrium dynamics

= ⇒ Two variational principles:

◮ Gibbs principle: S(µ) = max

Think of ρ(x) ∝ e−βH(x), in which case S(µ) = −

• x

µ(x) log µ(x) − β

• x

µ(x)H(x) = −β × “Free energy(µ)”

◮ Minimum entropy production principle: σ(µ) = min

Not very exciting since the reversible distribution ρ is known!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov Chains

equilibrium dynamics

= ⇒ Two variational principles:

◮ Gibbs principle: S(µ) = max

Think of ρ(x) ∝ e−βH(x), in which case S(µ) = −

• x

µ(x) log µ(x) − β

• x

µ(x)H(x) = −β × “Free energy(µ)”

◮ Minimum entropy production principle: σ(µ) = min

Not very exciting since the reversible distribution ρ is known!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov Chains

equilibrium dynamics

= ⇒ Two variational principles:

◮ Gibbs principle: S(µ) = max

Think of ρ(x) ∝ e−βH(x), in which case S(µ) = −

• x

µ(x) log µ(x) − β

• x

µ(x)H(x) = −β × “Free energy(µ)”

◮ Minimum entropy production principle: σ(µ) = min

Not very exciting since the reversible distribution ρ is known!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov Chains

nonequilibrium dynamics

◮ Take an arbitrary (e.g. counting or Gibbs) reference

distribution ρ0 Then σ(x, y) = log ρ0(x)λ(x, y) ρ0(y)λ(y, x) = −σ(y, x) is the entropy flux to the environment for jump x → y

◮ Define entropy as S(µ) = −S(µ | ρ0) ◮ Total entropy production rate:

E(µ) = dS(µt) dt

• t=0
• system

+

• x,y

µ(x)λ(x, y)σ(x, y)

• entropy flux into environment

=

• x,y

µ(x)λ(x, y) log µ(x)λ(x, y) µ(y)λ(y, x)

◮ the most general formula for entropy production

in Markov systems!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov Chains

nonequilibrium dynamics

◮ Take an arbitrary (e.g. counting or Gibbs) reference

distribution ρ0 Then σ(x, y) = log ρ0(x)λ(x, y) ρ0(y)λ(y, x) = −σ(y, x) is the entropy flux to the environment for jump x → y

◮ Define entropy as S(µ) = −S(µ | ρ0) ◮ Total entropy production rate:

E(µ) = dS(µt) dt

• t=0
• system

+

• x,y

µ(x)λ(x, y)σ(x, y)

• entropy flux into environment

=

• x,y

µ(x)λ(x, y) log µ(x)λ(x, y) µ(y)λ(y, x)

◮ the most general formula for entropy production

in Markov systems!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov Chains

nonequilibrium dynamics

◮ Take an arbitrary (e.g. counting or Gibbs) reference

distribution ρ0 Then σ(x, y) = log ρ0(x)λ(x, y) ρ0(y)λ(y, x) = −σ(y, x) is the entropy flux to the environment for jump x → y

◮ Define entropy as S(µ) = −S(µ | ρ0) ◮ Total entropy production rate:

E(µ) = dS(µt) dt

• t=0
• system

+

• x,y

µ(x)λ(x, y)σ(x, y)

• entropy flux into environment

=

• x,y

µ(x)λ(x, y) log µ(x)λ(x, y) µ(y)λ(y, x)

◮ the most general formula for entropy production

in Markov systems!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov chains

Nonequilibrium dynamics

Is the entropy production E(µ) minimal at µ = ρ?

◮ Not in general but approximately true close to

equilibrium!

◮ Given a reference detailed balanced dynamics with

rates λ0(x, y) and reversible distribution ρ0(x)

◮ Consider a parametric Markov process with rates

λǫ(x, y) = λ0(x, y)eǫψ(x,y)/2

◮ ρǫ be the (ǫ-dependent) stationary distribution ◮ ¯

ρǫ be the minimizer of the entropy production rate σǫ, i.e. min

µ Eǫ(µ) = Eǫ(¯

ρǫ)

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov chains

Nonequilibrium dynamics

Is the entropy production E(µ) minimal at µ = ρ?

◮ Not in general but approximately true close to

equilibrium!

◮ Given a reference detailed balanced dynamics with

rates λ0(x, y) and reversible distribution ρ0(x)

◮ Consider a parametric Markov process with rates

λǫ(x, y) = λ0(x, y)eǫψ(x,y)/2

◮ ρǫ be the (ǫ-dependent) stationary distribution ◮ ¯

ρǫ be the minimizer of the entropy production rate σǫ, i.e. min

µ Eǫ(µ) = Eǫ(¯

ρǫ)

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov chains

Nonequilibrium dynamics: MinEP principle

• Theorem. (Klein and Meier, 1954, for a particular model)

ρǫ = ¯ ρǫ + O(ǫ2) = ρ0 + O(ǫ)

◮ Up to the leading order, the minimizer ¯

ρǫ and the stationary distribution ρǫ coincide

◮ A microscopic (= distribution level) version of

Prigogine’s MinEP principle

◮ To prove it, solve perturbatively both the Master

equation and the variational problem

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov chains

Nonequilibrium dynamics: MinEP principle

• Theorem. (Klein and Meier, 1954, for a particular model)

ρǫ = ¯ ρǫ + O(ǫ2) = ρ0 + O(ǫ)

◮ Up to the leading order, the minimizer ¯

ρǫ and the stationary distribution ρǫ coincide

◮ A microscopic (= distribution level) version of

Prigogine’s MinEP principle

◮ To prove it, solve perturbatively both the Master

equation and the variational problem

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 1: Markov chains

Nonequilibrium dynamics: MinEP principle

• Theorem. (Klein and Meier, 1954, for a particular model)

ρǫ = ¯ ρǫ + O(ǫ2) = ρ0 + O(ǫ)

◮ Up to the leading order, the minimizer ¯

ρǫ and the stationary distribution ρǫ coincide

◮ A microscopic (= distribution level) version of

Prigogine’s MinEP principle

◮ To prove it, solve perturbatively both the Master

equation and the variational problem

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 2: Electrical Circuits

◮ Consider two resistors R1 and R2 in series with a

voltage source E

◮ Let U be the variable voltage on resistor R1, hence

E − U is the voltage on R2

◮ The total dissipated (Joule) heat power is

Q(U) = U2 R1 + (E − U)2 R2

◮ Trivially, Q(U) = min for U = R1 R1+R2 E in accordance

with Kirchhoff laws

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 2: Electrical Circuits

◮ Consider two resistors R1 and R2 in series with a

voltage source E

◮ Let U be the variable voltage on resistor R1, hence

E − U is the voltage on R2

◮ The total dissipated (Joule) heat power is

Q(U) = U2 R1 + (E − U)2 R2

◮ Trivially, Q(U) = min for U = R1 R1+R2 E in accordance

with Kirchhoff laws

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 2: Electrical Circuits

◮ Consider two resistors R1 and R2 in series with a

voltage source E

◮ Let U be the variable voltage on resistor R1, hence

E − U is the voltage on R2

◮ Entropy production rate is

E(U) = β1 U2 R1 + β2 (E − U)2 R2

◮ In the linear irreversible regime, β1,2 = β + O(ǫ),

E = O(ǫ) and U = O(ǫ), then also min

U Eǫ(U) = Eǫ(¯

Uǫ) with ¯ Uǫ =

R1 R1+R2 E + O(ǫ2) ◮ It can be generalized to an arbitrary network with

resistors and capacitors

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 2: Electrical Circuits

◮ Consider two resistors R1 and R2 in series with a

voltage source E

◮ Let U be the variable voltage on resistor R1, hence

E − U is the voltage on R2

◮ Entropy production rate is

E(U) = β1 U2 R1 + β2 (E − U)2 R2

◮ In the linear irreversible regime, β1,2 = β + O(ǫ),

E = O(ǫ) and U = O(ǫ), then also min

U Eǫ(U) = Eǫ(¯

Uǫ) with ¯ Uǫ =

R1 R1+R2 E + O(ǫ2) ◮ It can be generalized to an arbitrary network with

resistors and capacitors

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Example 2: Electrical Circuits

◮ Consider two resistors R1 and R2 in series with a

voltage source E

◮ Let U be the variable voltage on resistor R1, hence

E − U is the voltage on R2

◮ Entropy production rate is

E(U) = β1 U2 R1 + β2 (E − U)2 R2

◮ In the linear irreversible regime, β1,2 = β + O(ǫ),

E = O(ǫ) and U = O(ǫ), then also min

U Eǫ(U) = Eǫ(¯

Uǫ) with ¯ Uǫ =

R1 R1+R2 E + O(ǫ2) ◮ It can be generalized to an arbitrary network with

resistors and capacitors

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

(Counter)example 3: Landauer’s

◮ Consider now resistor R and inductor L in series with

voltage source E − → dynamical equation for current I: RI + LdI dt = E

◮ The Joule heat in the resistor is now Q(I) = RI2 whose

minimum apparently differs from the stationary current I = E

R ! ◮ Compare with the Langevin dynamics for a driven

particle, dvt = (f − γvt)dt + 2γ β 1

2 dWt

for which the (mean) dissipated heat reads Q(v) = (fv − v ˙ v)| ˙

v=f−γv = γv2

− → the MinEP principle holds here neither!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

(Counter)example 3: Landauer’s

◮ Consider now resistor R and inductor L in series with

voltage source E − → dynamical equation for current I: RI + LdI dt = E

◮ The Joule heat in the resistor is now Q(I) = RI2 whose

minimum apparently differs from the stationary current I = E

R ! ◮ Compare with the Langevin dynamics for a driven

particle, dvt = (f − γvt)dt + 2γ β 1

2 dWt

for which the (mean) dissipated heat reads Q(v) = (fv − v ˙ v)| ˙

v=f−γv = γv2

− → the MinEP principle holds here neither!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

(Counter)example 3: Landauer’s

What is wrong?

◮ Note that both the current I and the velocity v are

time-reversal odd observables!

◮ Realize that for the fine-state Markov chains the

functional E(µ) =

• x,y

µ(x)λ(x, y) log µ(x)λ(x, y) µ(y)λ(y, x) is always an (approximative) variational functional, however, E(µ) is not the physical entropy production anymore!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

(Counter)example 3: Landauer’s

What is wrong?

◮ Note that both the current I and the velocity v are

time-reversal odd observables!

◮ Realize that for the fine-state Markov chains the

functional E(µ) =

• x,y

µ(x)λ(x, y) log µ(x)λ(x, y) µ(y)λ(y, x) is always an (approximative) variational functional, however, E(µ) is not the physical entropy production anymore!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

General Considerations

◮ Remember in equilibrium there is a deep connection

between the variational characterization of equilibrium states and the structure of equilibrium fluctuations

◮ On a heuristic level, the probability of macroscopic

fluctuations is given by the Einstein law P(m) ∝ eS(m) which has a precise formulation through the theory of large deviations

◮ This has formulations on various levels, from

macroscopic functions to empirical processes

◮ Generally, the equilibrium variational functionals

coincide with certain large deviation rate functions

Main Question. Can a similar explanation be given to to the MinEP (and possibly other) principles?

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

General Considerations

◮ Remember in equilibrium there is a deep connection

between the variational characterization of equilibrium states and the structure of equilibrium fluctuations

◮ On a heuristic level, the probability of macroscopic

fluctuations is given by the Einstein law P(m) ∝ eS(m) which has a precise formulation through the theory of large deviations

◮ This has formulations on various levels, from

macroscopic functions to empirical processes

◮ Generally, the equilibrium variational functionals

coincide with certain large deviation rate functions

Main Question. Can a similar explanation be given to to the MinEP (and possibly other) principles?

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Donsker-Varadhan Theory

Markov chains revisited

For any trajectory (ηt, 0 ≤ t ≤ T) define the empirical

• ccupation times as

LT(x) = 1 T T χ[ηs = x] ds

◮ The relative time spent at state x ◮ Law of large numbers: for an ergodic process,

limT LT(x) = ρ(x), Pρ-a.s.

◮ The empirical occupation times satisfy the large

deviations Prob(LT = µ) ≃ e−TJ(µ) with the rate function J(µ) = − inf

ν>0

• dµd(νL)

dν =

• x,y=x

µ(x)λ(x, y) − inf

ν

• x,y=x

ν(x)µ(y) ν(y) λ(x, y)

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Markov Chains Revisited

Donsker-Varadhan theory

◮ This is a special case of a more general result about

large deviations for Markov processes studied by Donsker and Varadhan

◮ The rate function J is manifestly a variational functional

for the stationary distribution ρ, but is it a useful one?

◮ In general difficult to compute explicitly because J(µ)

itself given by a variational principle!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Markov Chains Revisited

Donsker-Varadhan theory

◮ This is a special case of a more general result about

large deviations for Markov processes studied by Donsker and Varadhan

◮ The rate function J is manifestly a variational functional

for the stationary distribution ρ, but is it a useful one?

◮ In general difficult to compute explicitly because J(µ)

itself given by a variational principle!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Markov Chains Revisited

Donsker-Varadhan theory

◮ This is a special case of a more general result about

large deviations for Markov processes studied by Donsker and Varadhan

◮ The rate function J is manifestly a variational functional

for the stationary distribution ρ, but is it a useful one?

◮ In general difficult to compute explicitly because J(µ)

itself given by a variational principle!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Markov Chains Revisited

equilibrium dynamics

The rate function can be computed explicitly under detailed balance assumption: J(µ) =

• x,y

µ(x)λ(x, y)

• 1 −

µ(y)λ(y, x) µ(x)λ(x, y) 1

2

= 1 8

• x,y=x

ρ(x)λ(x, y) µ(x) ρ(x) − µ(y) ρ(y) 2 + o(µ − ρ2) When compared to the entropy production rate, we get Theorem. J(µ) = 1

4E(µ) + o(µ − ρ2)

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Markov Chains Revisited

equilibrium dynamics

The rate function can be computed explicitly under detailed balance assumption: J(µ) =

• x,y

µ(x)λ(x, y)

• 1 −

µ(y)λ(y, x) µ(x)λ(x, y) 1

2

= 1 8

• x,y=x

ρ(x)λ(x, y) µ(x) ρ(x) − µ(y) ρ(y) 2 + o(µ − ρ2) When compared to the entropy production rate, we get Theorem. J(µ) = 1

4E(µ) + o(µ − ρ2)

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

The entropy production governs the small fluctuations of occupational times

⇓?

Can the close-to-equilibrium MinEP principle be obtained in a similar way?

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

The entropy production governs the small fluctuations of occupational times

⇓?

Can the close-to-equilibrium MinEP principle be obtained in a similar way?

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Markov Chains Revisited

nonequilibrium dynamics

Recall:

◮ Given a reference detailed balanced dynamics with

rates λ0(x, y) and reversible distribution ρ0(x)

◮ Consider a parametric Markov process with rates

λǫ(x, y) = λ0(x, y)eǫψ(x,y)/2

◮ ρǫ be the (ǫ-dependent) stationary distribution

• Theorem. In the small fluctuations regime, µǫ = ρ0 + O(ǫ),

the rate function of fluctuations is Jǫ(µǫ) = 1

4[Eǫ(µǫ) − Eǫ(ρǫ)] + o(ǫ2) ◮ For a close-to-equilibrium dynamics and for small

fluctuations, the rate function is intimately related to the entropy production rate

◮ Hence, the entropy production plays a role similar to

the entropy in equilibrium, but only approximatively!

◮ Not valid far from equilibrium, yet the rate function J(µ)

provides a variational functional for the stationary distribution – not explicit however!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Markov Chains Revisited

nonequilibrium dynamics

Recall:

◮ Given a reference detailed balanced dynamics with

rates λ0(x, y) and reversible distribution ρ0(x)

◮ Consider a parametric Markov process with rates

λǫ(x, y) = λ0(x, y)eǫψ(x,y)/2

◮ ρǫ be the (ǫ-dependent) stationary distribution

• Theorem. In the small fluctuations regime, µǫ = ρ0 + O(ǫ),

the rate function of fluctuations is Jǫ(µǫ) = 1

4[Eǫ(µǫ) − Eǫ(ρǫ)] + o(ǫ2) ◮ For a close-to-equilibrium dynamics and for small

fluctuations, the rate function is intimately related to the entropy production rate

◮ Hence, the entropy production plays a role similar to

the entropy in equilibrium, but only approximatively!

◮ Not valid far from equilibrium, yet the rate function J(µ)

provides a variational functional for the stationary distribution – not explicit however!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Markov Chains Revisited

nonequilibrium dynamics

Recall:

◮ Given a reference detailed balanced dynamics with

rates λ0(x, y) and reversible distribution ρ0(x)

◮ Consider a parametric Markov process with rates

λǫ(x, y) = λ0(x, y)eǫψ(x,y)/2

◮ ρǫ be the (ǫ-dependent) stationary distribution

• Theorem. In the small fluctuations regime, µǫ = ρ0 + O(ǫ),

the rate function of fluctuations is Jǫ(µǫ) = 1

4[Eǫ(µǫ) − Eǫ(ρǫ)] + o(ǫ2) ◮ For a close-to-equilibrium dynamics and for small

fluctuations, the rate function is intimately related to the entropy production rate

◮ Hence, the entropy production plays a role similar to

the entropy in equilibrium, but only approximatively!

◮ Not valid far from equilibrium, yet the rate function J(µ)

provides a variational functional for the stationary distribution – not explicit however!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Markov Chains Revisited

nonequilibrium dynamics

Recall:

◮ Given a reference detailed balanced dynamics with

rates λ0(x, y) and reversible distribution ρ0(x)

◮ Consider a parametric Markov process with rates

λǫ(x, y) = λ0(x, y)eǫψ(x,y)/2

◮ ρǫ be the (ǫ-dependent) stationary distribution

• Theorem. In the small fluctuations regime, µǫ = ρ0 + O(ǫ),

the rate function of fluctuations is Jǫ(µǫ) = 1

4[Eǫ(µǫ) − Eǫ(ρǫ)] + o(ǫ2) ◮ For a close-to-equilibrium dynamics and for small

fluctuations, the rate function is intimately related to the entropy production rate

◮ Hence, the entropy production plays a role similar to

the entropy in equilibrium, but only approximatively!

◮ Not valid far from equilibrium, yet the rate function J(µ)

provides a variational functional for the stationary distribution – not explicit however!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Macroscopic Limit Approach

Introduction

◮ We want to understand the above relation in a more

elucidating way

◮ Towards a general scheme for macroscopic dynamical

systems

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

A Toy Model: Free Particles

◮ Let us study the fluctuating dynamics of N identical and

independent Markov chains η1, . . . , ηN through

◮ empirical densities

pN

t (x) = 1

N

N

❳

k=1

χ[X k

t = x]

x = 1, . . . , n

◮ empirical currents

1 N

N

❳

k=1

χ[X k

t = x] χ[Xt+δt = y] − χ[X k t = y] χ[Xt+δt = x]

= jN

t (x, y)δt + o(δt)

◮ Balance equation:

dpN

t (x)

dt = ❳

y=x

jN

t (y, x)

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Free Particles Model

◮ Large deviations for the probability of a macroscopic

trajectory ω = (pt, jt), 0 ≤ t ≤ T: Pµ{ωN = ω} . = e−N[S(p0 | µ)+

❘ T

0 dt L(pt,jt)]

◮ For simplicity, we restrict only to the quadratic

approximation around a detailed balance process:

◮ Let

p(x) = ρ0[1 + u(x)] ρ0(x)λ(x, y) = γ(x, y)e

1 2 σ(x,y)

with γ(x, y) = ρ0(x)λ0(x, y) = γ(y, x)

◮ Assume u, j, σ = O(ǫ) ◮ The balance equation:

ρ0(x) ˙ u(x) = ❳

y=x

j(y, x)

◮ Entropy:

S(u) = −S(p | ρ0) = −1 2 ❳

x

ρ0(x)u(x)2 + o(ǫ2)

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Free Particles Model

Onsager-Machlup formalism

Lagrangian, up to order O(ǫ2): L(u, j) =

• (xy)

1 4γ(x, y){j(x, y) − γ(x, y)[u(x) − u(y) + σ(x, y)]}2 Defining the dissipation functions D(j) =

• (xy)

j2(x, y) γ(x, y) E(u) =

• (xy)

γ(x, y)[u(x) − u(y) + σ(x, y)]2 and the total entropy production ˙ S(u, j) =

• (xy)

j(x, y)[u(x) − u(y)]

• −ρ0(x)ρ0u(x) ˙

u(x)

+

• (xy)

j(x, y)σ(x, y) the Lagrangian gets the Onsager-Machlup form L(u, j) = 1 2 1 2D(j) + 1 2E(u) − ˙ S(u, j)

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Free Particles Model

least dissipation principle

◮ The typical value ju of the currents given u solves the

least dissipation principle 1 2D(j) − ˙ S(u, j) = min

◮ Alternatively, under the extra constraint

D(j) = ˙ S(u, j) the variational principle gets the form ˙ S(u, j) = max

= ⇒ Transient MaxEP principle

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Free Particles Model

MinEP principle

◮ Contraction (u, j) → u leads to the Lagrangian

L(1)(u, ˙ u) = inf

j {L(u, j) | ρ0(x) ˙

u(x) +

• y=x

j(x, y) = 0 (∀x)}

◮ The minimizer j = j ˙ u satisfies the equations

j ˙

u(x, y) = γ(x, y)[u ˙ u(x) − u ˙ u(y) + σ(x, y)]

ρ(x) ˙ u(x) =

• y=x

j ˙

u(y, x) ◮ In particular, for ˙

u = 0 the typical configuration u ˙

u=0

and the typical current j ˙

u=0 are the stationary ones and

˙ S(u, j0) = ˙ S(u0, j0) = E(u0)

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Free Particles Model

MinEP principle

◮ This gives

L(1)(u, 0) = 1 4[E(u) − E(u0)]

◮ Can L(u, 0) be recognized as a large deviation rate? ◮ The empirical average

¯ uN

T = 1

T T uN

t dt

has the fluctuation law PN,T{¯ uN

T = u} .

= e−NAT (u) with the rate AT(u) = inf

(ut)0≤t≤T

T dt L(1)(ut, ˙ ut)

• ¯

uT = u

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Free Particles Model

MinEP principle

◮ This gives

L(1)(u, 0) = 1 4[E(u) − E(u0)]

◮ Can L(u, 0) be recognized as a large deviation rate? ◮ The empirical average

¯ uN

T = 1

T T uN

t dt

has the fluctuation law PN,T{¯ uN

T = u} .

= e−NAT (u) with the rate AT(u) = inf

(ut)0≤t≤T

T dt L(1)(ut, ˙ ut)

• ¯

uT = u

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Free Particles Model

MinEP principle

◮ For a large time-span T the minimizer is essentially

constant, ut ≡ u. In fact, lim

T→∞

AT(u) T = L(1)(u, 0)

◮ This yields the fluctuation law

PN,T{¯ uN

T = u} .

= e− 1

4 TN[E(u)−E(u0)] ◮ Recall E(u) is the (expected or typical) entropy

production given u

◮ Compare with the result for a single Markov chain

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Free Particles Model

MaxEP principle

◮ Analogously, consider contraction (u, j) → j. The

effective Lagrangian is now L(2)(j) = inf

u L(u, j) ◮ The minimizer u = uj solves

• y=x

Jj(x, y) =

• y=x

j(x, y) ≡ −ρ(x) ˙ u(x) (∀x) Jj(x, y) = γ(x, y)[uj(x) − uj(y) + σ(x, y)]

◮ Impose the next two restrictions:

• y=x

j(x, y) = 0 (∀x) D(j) = ˙ S(uj, j) ≡ ˙ S(u0, j) then E(uj) = E(u0)

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Free Particles Model

MaxEP principle

◮ Under the imposed conditions,

L(2)(j) = 1 4[D(j0) − D(j)]

◮ By a similar argument as for the MinEP principle, we

finally get the fluctuation law PN,T{¯ jN

T = j} .

= e− 1

4 TN[D(2)(j0)−D(2)(j)]

for the empirical time-averaged current ¯ jN

T = 1

T T jN

t dt ◮ The associated variational principle is called a

(stationary) maximum entropy production principle

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Electrical Circuits Revisited

RR + C

◮ Consider two resistors R1 and R2 in series with a

voltage source E

◮ Add a capacitor C in parallel with resistor R1 ◮ Let U be the variable voltage on resistor R1, hence

E − U is the voltage on R2

◮ Thermal fluctuations modeled by the Johnson-Nyquist

noise voltages: Uf1

t dt =

• 2R1

β1 dW 1

t

Uf2

t dt =

• 2R2

β2 dW 2

t ◮ From Kirchhoff laws

E = Ut + R2It + U2f

t

It = C dU dt + Ut − Uf1

t

R1

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Electrical Circuits Revisited

RR + C

◮ The process Ut satisfies the Langevin-type equation

dUt = 1 C E R2 − 1 R1 + 1 R2

• Ut
• dt +
• 2

C2

• 1

β1R1 + 1 β2R2

• dWt

◮ Dynamical large deviations:

Pµ{ω = (Ut, 0 ≤ t ≤ T)} . = eS(µ)+

❘ T

0 dt L(Ut, ˙

Ut)

with the Lagrangian L(U, ˙ U) = C2 4 β1β2R1R2 β1R1 + β2R2

• ˙

U − 1 C E R2 − 1 R1 + 1 R2

• U

2

◮ Cheating but it can be made precise within the

Friedlin-Wentzel theory!

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Electrical Circuits Revisited

RR + C

◮ The large deviations for the empirical voltage

¯ UT = 1 T T dt Ut given by P{¯ UT = U} . = e−AT (U) with the rate AT given by the contraction principle AT(U) = inf

(UT ,0≤t≤T)

T dt L(Ut, ˙ Ut)

• ¯

UT = U

• ◮ For large T, the minimizing trajectory is essentially

constant, i.e. Ut ≡ U. Hence, lim

T→∞

AT(U) T = L(U, 0) = 1 4 β1β2(R1 + R2) β1R1 + β2R2 U2 R1 + (E − U)2 R2

• Q(U)

− E2 R1 + R2

• Q(U0)

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Conclusions and Open Problems

◮ In a high generality, for small fluctuations and close to

equilibrium, the (expected) entropy production governs the law of certain empirical time-averages: Prob(LT = µ) . = exp

• − T

4 (E(µ) + o(ǫ2))

• ◮ This gives an independent and possibly deeper

explanation of the minimum entropy production principle, and also indicates the origin of its restriction to close to equilibrium dynamics

◮ The Donsker-Varadhan rate function

J(µ) =

• x,y=x

µ(x)k(x, y) − inf

ν

• x,y=x

ν(x)µ(y) ν(y) k(x, y) is itself given by a variational problem!

◮ Can it still get useful far from equilibrium?

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

Conclusions and Open Problems

◮ In a high generality, for small fluctuations and close to

equilibrium, the (expected) entropy production governs the law of certain empirical time-averages: Prob(LT = µ) . = exp

• − T

4 (E(µ) + o(ǫ2))

• ◮ This gives an independent and possibly deeper

explanation of the minimum entropy production principle, and also indicates the origin of its restriction to close to equilibrium dynamics

◮ The Donsker-Varadhan rate function

J(µ) =

• x,y=x

µ(x)k(x, y) − inf

ν

• x,y=x

ν(x)µ(y) ν(y) k(x, y) is itself given by a variational problem!

◮ Can it still get useful far from equilibrium?

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

• L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio,
• C. Landim. Large deviation approach to non equilibrium

processes in stochastic lattice gases. Preprint math.PR0602557 (2006).

• M. D. Donsker, S. R. Varadhan. Asymptotic evaluation
• f certain Markov process expectations for large time, I.
• Comm. Pure Appl. Math., 28:1–47 (1975).
• E. T. Jaynes. The minimum entropy production
• principle. Ann. Rev. Phys. Chem., 31:579–601 (1980).
• M. J. Klein and P

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The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains Electrical Networks Landauer’s Counterex. Dynamical Fluctuations General considerations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium Macroscopic Limit Introduction Free Particles Model

• El. Circuits Revisited

MinEP principle Conclusions

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