The Minimum Entropy Production Equilibrium Fluctuations Principle - - PowerPoint PPT Presentation

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

The Minimum Entropy Production Principle 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

MEP in Physics and Biology, Split 6-7th July 2006

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Thesis and Contents

The analysis of stationary fluctuations provides a natural framework for various variational principles

• f statistical mechanics.

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Thesis and Contents

The analysis of stationary fluctuations provides a natural framework for various variational principles

• f statistical mechanics.

◮ Introduction: Equilibrium fluctuations

◮ Closed versus open systems ◮ Macro- versus micro-description

◮ Dynamical fluctuations

◮ Law of empirical occupation times ◮ The entropy production as the fluctuation rate ◮ Equilibrium versus non-equilibrium Markov dynamics ◮ Onsager-Machlup process

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Thermodynamic Equilibrium

what did we learn in basic courses?

After long enough time any closed system is found in one

• f typical microstates.

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Thermodynamic Equilibrium

what did we learn in basic courses?

After long enough time any closed system is found in one

• f typical microstates.

In particular:

◮ Macroscopic observables take the values

corresponding to the highest number of microstates

= ⇒ Second law of thermodynamics, with entropy S(m) = kB log #{x : m(x) = m}

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Thermodynamic Equilibrium

what did we learn in basic courses?

After long enough time any closed system is found in one

• f typical microstates.

In particular:

◮ Macroscopic observables take the values

corresponding to the highest number of microstates

= ⇒ Second law of thermodynamics, with entropy S(m) = kB log #{x : m(x) = m}

◮ Macroscopic fluctuations occur with probability

Prob(m) ∝ eS(m) (kB = 1) and for large systems, S(m) ≃ Ns(m)

= ⇒ Einstein’s theory of equilibrium fluctuations

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Equilibrium of Open Systems

what did we learn in basic courses?

For a system in equilibrium with heat bath at temperature β−1, to any macrostate m is assigned the nonequilibrium free energy F(m) = E(m)

Energy

−β−1 S(m)

Entropy ◮ Second law: F(m) = min

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Equilibrium of Open Systems

what did we learn in basic courses?

For a system in equilibrium with heat bath at temperature β−1, to any macrostate m is assigned the nonequilibrium free energy F(m) = E(m)

Energy

−β−1 S(m)

Entropy ◮ Second law: F(m) = min ◮ Since ∆Stotal = ∆Ssystem + ∆Sbath = −β∆F,

(large) equilibrium fluctuations have the law Prob(m) ∝ e−βF(m) True for large N if F(m) ≃ Nf(m)

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Equilibrium of Open Systems

what did we learn in basic courses?

For a system in equilibrium with heat bath at temperature β−1, to any macrostate m is assigned the nonequilibrium free energy F(m) = E(m)

Energy

−β−1 S(m)

Entropy ◮ Second law: F(m) = min ◮ Since ∆Stotal = ∆Ssystem + ∆Sbath = −β∆F,

(large) equilibrium fluctuations have the law Prob(m) ∝ e−βF(m) True for large N if F(m) ≃ Nf(m)

◮ In specific models, F(m) is related to an equilibrium

thermodynamic potential

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Equilibrium of Open Systems

a microscopic version

◮ For a system with Hamiltonian H(x), the equilibrium

distribution dµeq(x) = ρeq(x)dx = 1 Z e−βH(x)dx

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Equilibrium of Open Systems

a microscopic version

◮ For a system with Hamiltonian H(x), the equilibrium

distribution dµeq(x) = ρeq(x)dx = 1 Z e−βH(x)dx is the minimizer of the functional F(µ) =

• H(x)ρ(x)dx − β−1S(µ)

◮ Gibbs variational principle ◮ S(µ) = −

R ρ(x) log ρ(x)dx is the Shannon entropy

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Equilibrium of Open Systems

a microscopic version

◮ For a system with Hamiltonian H(x), the equilibrium

distribution dµeq(x) = ρeq(x)dx = 1 Z e−βH(x)dx is the minimizer of the functional F(µ) =

• H(x)ρ(x)dx − β−1S(µ)

◮ Gibbs variational principle ◮ S(µ) = −

R ρ(x) log ρ(x)dx is the Shannon entropy

◮ F(ρ) is related to the fluctuations of empirical

distributions!

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Empirical Distributions

Construction of the empirical distribution νx

V in a cube V: ◮ For every configuration x consider all translates τix for

i ∈ V

◮ Let Nx(y) be the number of all the translates that

coincide with y

◮ Define the probability distribution

πx

V(·) = Nx(·)

|V|

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Empirical Distributions

Construction of the empirical distribution νx

V in a cube V: ◮ For every configuration x consider all translates τix for

i ∈ V

◮ Let Nx(y) be the number of all the translates that

coincide with y

◮ Define the probability distribution

πx

V(·) = Nx(·)

|V| Idea: Spatical ergodicity = ⇒ πx → µeq, for any typical x

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Empirical Distributions

Construction of the empirical distribution νx

V in a cube V: ◮ For every configuration x consider all translates τix for

i ∈ V

◮ Let Nx(y) be the number of all the translates that

coincide with y

◮ Define the probability distribution

πx

V(·) = Nx(·)

|V| Idea: Spatical ergodicity = ⇒ πx → µeq, for any typical x

• Theorem. (Olla, 1989) Asymptotically for large V,

Prob(x : πx

V = µV) ∝ exp[−β F(µV) ≃Vf(µV )

] for periodic distributions µ

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Equilibrium: Conclusion

The variational characterization of equilibrium states is a consequence of the structure

• f equilibrium fluctuations

As we have seen, this is true in a very general sense

◮ On both the macroscopic and microscopic levels ◮ For both closed and open equilibrium systems

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Equilibrium: Conclusion

The variational characterization of equilibrium states is a consequence of the structure

• f equilibrium fluctuations

As we have seen, this is true in a very general sense

◮ On both the macroscopic and microscopic levels ◮ For both closed and open equilibrium systems

Main Question. Can this strategy be put forward to study nonequilibrium steady states?

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Markovian Models

Consider a finite-state Markov process Xt:

◮ Transition rates k(x, y) ◮ Stationary distribution ρ: x ρ(x)k(x, y) = ρ(y) ◮ Ergodicity assumption: ρ(x) > 0 for all x ◮ Describes a general open system dynamics

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Markovian Models

Consider a finite-state Markov process Xt:

◮ Transition rates k(x, y) ◮ Stationary distribution ρ: x ρ(x)k(x, y) = ρ(y) ◮ Ergodicity assumption: ρ(x) > 0 for all x ◮ Describes a general open system dynamics

For a trajectory on [0, T], the empirical occupation times are LT(x) = 1 T T χ[Xs = x] ds

◮ The relative time spent in state x ◮ For an ergodic process, LT(x) → ρ(x), typically for

T → ∞

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Donsker-Varadhan Theory

The empirical occupation times have the asymptotic distribution Prob(LT = µ) ≃ e−TJ(µ) with the 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)

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Donsker-Varadhan Theory

The empirical occupation times have the asymptotic distribution Prob(LT = µ) ≃ e−TJ(µ) with the 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)

◮ Easy to check that J ≥ 0 and J(ρ) = 0

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Donsker-Varadhan Theory

The empirical occupation times have the asymptotic distribution Prob(LT = µ) ≃ e−TJ(µ) with the 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)

◮ Easy to check that J ≥ 0 and J(ρ) = 0 ◮ A special case of a more general result about large

deviations for Markov processes

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Donsker-Varadhan Theory

The empirical occupation times have the asymptotic distribution Prob(LT = µ) ≃ e−TJ(µ) with the 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)

◮ Easy to check that J ≥ 0 and J(ρ) = 0 ◮ A special case of a more general result about large

deviations for Markov processes

◮ The rate function J is a variational functional for the

stationary distribution ρ, but is it a useful one?

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Donsker-Varadhan Theory

The empirical occupation times have the asymptotic distribution Prob(LT = µ) ≃ e−TJ(µ) with the 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)

◮ Easy to check that J ≥ 0 and J(ρ) = 0 ◮ A special case of a more general result about large

deviations for Markov processes

◮ The rate function J is a variational functional for the

stationary distribution ρ, but is it a useful one?

◮ In general difficult to compute explicitly!

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Equilibrium Dynamics

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

◮ Describes an equilibrium dynamics (closed or open)

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Equilibrium Dynamics

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

◮ Describes an equilibrium dynamics (closed or open)

The rate function explicitly: J(µ) =

• x,y

µ(x)k(x, y)

• 1 −

µ(y)k(y, x) µ(x)k(x, y) 1

2

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Equilibrium Dynamics

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

◮ Describes an equilibrium dynamics (closed or open)

The rate function explicitly: J(µ) =

• x,y

µ(x)k(x, y)

• 1 −

µ(y)k(y, x) µ(x)k(x, y) 1

2

Theorem. J(µ) = 1

4σ(µ) + o(µ − ρ2)

where σ(µ) is the entropy production rate σ(µ) = dS(µt | ρ) dt

• t=0=
• x,y=x

µ(x)k(x, y) log µ(x)ρ(y) µ(y)ρ(x) Here, S(µ | ρ) = −

• x

µ(x) log µ(x) ρ(x) is the relative entropy.

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Remark: Why the Relative Entropy?

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Remark: Why the Relative Entropy?

◮ Closed: ρ(x) = 1 |Ω| – microcanonical distribution

S(µ | ρ) = S(µ)

• Shannon

− log |Ω|

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Remark: Why the Relative Entropy?

◮ Closed: ρ(x) = 1 |Ω| – microcanonical distribution

S(µ | ρ) = S(µ)

• Shannon

− log |Ω|

◮ Open: ρ(x) = 1 Z e−βH(x) – canonical distribution

S(µ | ρ) = S(µ) − β

• x

µ(x)H(x)

• µ−mean energy

− log Z

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Remark: Why the Relative Entropy?

◮ Closed: ρ(x) = 1 |Ω| – microcanonical distribution

S(µ | ρ) = S(µ)

• Shannon

− log |Ω|

◮ Open: ρ(x) = 1 Z e−βH(x) – canonical distribution

S(µ | ρ) = S(µ) − β

• x

µ(x)H(x)

• µ−mean energy

− log Z Its changes: ∆S(µ | ρ) = ∆S(µ) −β∆Hµ

• entropy flux

= ∆Stotal(µ)

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Remark: Why the Relative Entropy?

◮ Closed: ρ(x) = 1 |Ω| – microcanonical distribution

S(µ | ρ) = S(µ)

• Shannon

− log |Ω|

◮ Open: ρ(x) = 1 Z e−βH(x) – canonical distribution

S(µ | ρ) = S(µ) − β

• x

µ(x)H(x)

• µ−mean energy

− log Z Its changes: ∆S(µ | ρ) = ∆S(µ) −β∆Hµ

• entropy flux

= ∆Stotal(µ) Remember: Equilibrium dynamics so far!

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

The entropy production governs the small fluctuations of occupational times

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

The entropy production governs the small fluctuations of occupational times

⇓?

Can the known nonequilibrium variational principles get a more fundamental justification in a similar way?

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Case 2: Close-to-Equilibrium Dynamics

Given:

◮ Reference detailed-balanced dynamics with rates

k0(x, y) and invariant distribution ρ0(x)

◮ A perturbed Markov process with rates

kǫ(x, y) = k0(x, y)eǫψ(x,y)/2

◮ Truly nonequilibrium dynamics in general!

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Case 2: Close-to-Equilibrium Dynamics

Given:

◮ Reference detailed-balanced dynamics with rates

k0(x, y) and invariant distribution ρ0(x)

◮ A perturbed Markov process with rates

kǫ(x, y) = k0(x, y)eǫψ(x,y)/2

◮ Truly nonequilibrium dynamics in general!

Total entropy production rate: σǫ(µ) = dS(µt) dt

• t=0
• system

+

• x,y

µ(x)k(x, y) log kǫ(x, y) kǫ(y, x)

• entropy flux into reservoirs

=

• x,y

µ(x)kǫ(x, y) log µ(x)kǫ(x, y) µ(y)kǫ(y, x)

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Case 2: Close-to-Equilibrium Dynamics

Given:

◮ Reference detailed-balanced dynamics with rates

k0(x, y) and invariant distribution ρ0(x)

◮ A perturbed Markov process with rates

kǫ(x, y) = k0(x, y)eǫψ(x,y)/2

◮ Truly nonequilibrium dynamics in general!

Total entropy production rate: σǫ(µ) = dS(µt) dt

• t=0
• system

+

• x,y

µ(x)k(x, y) log kǫ(x, y) kǫ(y, x)

• entropy flux into reservoirs

=

• x,y

µ(x)kǫ(x, y) log µ(x)kǫ(x, y) µ(y)kǫ(y, x)

◮ the most general formula for entropy production

in Markov systems!

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Minimum entropy production principle

a microscopic version

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

ρǫ be the minimizer of the entropy production rate σǫ

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Minimum entropy production principle

a microscopic version

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

ρǫ be the minimizer of the entropy production rate σǫ

• 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

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Minimum entropy production principle

a microscopic version

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

ρǫ be the minimizer of the entropy production rate σǫ

• 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

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Minimum entropy production principle

a microscopic version

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

ρǫ be the minimizer of the entropy production rate σǫ

• 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 MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Fluctuations of empirical distribution

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

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

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

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

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Fluctuations of empirical distribution

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

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

4[σǫ(µǫ) − infν σǫ(ν)] + 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!

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Fluctuations of empirical distribution

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

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

4[σǫ(µǫ) − infν σǫ(ν)] + 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 MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

A Toy: Onsager-Machlup Process

from micro to macro

How to describe small fluctuations around a macroscopic law of motion d¯

Xt dt = φ(¯

Xt)?

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

A Toy: Onsager-Machlup Process

from micro to macro

How to describe small fluctuations around a macroscopic law of motion d¯

Xt dt = φ(¯

Xt)?

◮ Linearize and add the normalized white noise

→ deviation mt = Xt − ¯ Xt satisfies R dmt dt = −smt +

• 2R

N wt

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

A Toy: Onsager-Machlup Process

from micro to macro

How to describe small fluctuations around a macroscopic law of motion d¯

Xt dt = φ(¯

Xt)?

◮ Linearize and add the normalized white noise

→ deviation mt = Xt − ¯ Xt satisfies R dmt dt = −smt +

• 2R

N wt

◮ Equilibrium: Prob(m∞ = m) ∝ exp[−N

2 sm2

• S(m)

]

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

A Toy: Onsager-Machlup Process

from micro to macro

How to describe small fluctuations around a macroscopic law of motion d¯

Xt dt = φ(¯

Xt)?

◮ Linearize and add the normalized white noise

→ deviation mt = Xt − ¯ Xt satisfies R dmt dt = −smt +

• 2R

N wt

◮ Equilibrium: Prob(m∞ = m) ∝ exp[−N

2 sm2

• S(m)

]

◮ Linearized law of motion: dmt dt = − s R mt

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

A Toy: Onsager-Machlup process

from micro to macro

◮ Distribution on trajectories:

Prob((mt)0≤t≤T

• ω

) ∝ exp

• −N

4 T dt R 2 dmt dt + s R mt 2

• A(ω)

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

A Toy: Onsager-Machlup process

from micro to macro

◮ Distribution on trajectories:

Prob((mt)0≤t≤T

• ω

) ∝ exp

• −N

4 T dt R 2 dmt dt + s R mt 2

• A(ω)
• ◮ Empirical average distribution:

Prob 1 T T mt dt = m

• ≈ Prob(∀t : mt = m)

◮ the probability dominated by the constant profile

∝ exp

• −NT

4 s2 2R m2 = exp

• −T

4 σ(m)

• where

σ(m) = dS(mt) dt

• t=0 = Ns2

2R m2 is the entropy production rate, Q.E.D.

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Conclusions

◮ Similar arguments can be given for diffusions and for

fluctuating hydrodynamics models

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Conclusions

◮ Similar arguments can be given for diffusions and for

fluctuating hydrodynamics models

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

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

• − T

4 (σ(µ) + o(ǫ2))

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Conclusions

◮ Similar arguments can be given for diffusions and for

fluctuating hydrodynamics models

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

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

• − T

4 (σ(µ) + o(ǫ2))

• ◮ This gives an independent and deeper explanation for

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

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Open Problems

◮ 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 MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

Open Problems

◮ 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?

◮ Can also the maximum entropy production principle

be explained from the point of view of fluctuations?

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations

Closed System Open System Microscopic Level Conclusion

Dynamical Fluctuations

Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process

Conclusions

• 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

. H. E. Meijer. Principle of minimum entropy production. Phys. Rev., 96:250–255 (1954).

• C. Maes and K. Netoˇ

cný. Minimum entropy production principle from a dynamical fluctuation law. In preparation.

• C. Maes and K. Netoˇ

cný. Time-reversal and entropy,

• J. Stat. Phys., 110: 269–310 (2003).
• L. Onsager and S. Machlup. Fluctuations and

irreversible processes, Phys. Rev., 91:1505 ˝ U-1512 (1953).

• I. Prigogine. Introduction to Non-Equilibrium
• Thermodynamics. Wiley-Interscience, New York

(1962).