1. Revisiting Entropy Production Principles
Karel Netočný
Institute of Physics AS CR Seminar at ITF, K.U.Leuven, 14 November 2007

2. To be discussed
Min- and Max-entropy production principles: various examples
From variational principles to fluctuation laws: equilibrium case
Static versus dynamical fluctuations
Onsager-Machlup equilibrium dynamical fluctuation theory
Stochastic models of nonequilibrium
Conclusions, open problems, outlook,...
In collaboration with C. Maes,
B. Wynants, and S. Bruers,
K. U. Leuven, Belgium

3. Motivation: Modeling Earth climate [Ozawa et al, Rev. Geoph

4. Linear electrical networks explaining MinEP/MaxEP principles
Kirchhoff’s loop law:
Entropy production rate:
MinEP principle:
Stationary values of voltages
minimize the entropy
production rate
Not valid under inhomogeneous temperature! X k U j = E ( U ) = Q X j ; k 2 R U 2

5. Linear electrical networks explaining MinEP/MaxEP principles
Kirchhoff’s current law:
Entropy production rate:
Work done by sources:
(Constrained) MaxEP principle:
Stationary values of currents
maximize the entropy
production under constraint X j J k = ( J ) = Q X j ; k R 2 U 2 W ( J ) = X j k E Q ( J ) = W

6. Linear electrical networks summary of MinEP/MaxEP principles+
Current law
Current law +
Loop law U
Loop law +
MinEP principle
U, I
Generalized
variational principle

7. From principles to fluctuation laws Questions and ideas
How to go beyond approximate and ad hoc thermodynamical principles?
Inspiration from thermostatics:
Is there a nonequilibrium analogy of thermodynamical fluctuation theory?
Equilibrium variational principles are intimately related
to the structure of equilibrium fluctuations

8. From principles to fluctuation laws Equilibrium fluctuationsH ( x ) = N e H ( x ) = N e
add field P ( M x ) = N m e [ s ; q ] M ( x ) = N m e q
Probability of fluctuation
Typical value H h ( x ) = M N [ e m ]
The fluctuation made typical! s ( e ; m ) = h q

9. From principles to fluctuation laws Equilibrium fluctuations
functional
Variational
Thermodynamic
potential
Entropy
(Generalized)
free energy

10. From principles to fluctuation laws Static versus dynamical fluctuations
Empirical ergodic average:
Ergodic theorem:
Dynamical fluctuations:
Interpolating between static and dynamical fluctuations: m T = 1 R ( x t ) d m T ! e q ( ) ; 1 H ( x ) = N e P ( m T = ) e I S t a i c : ! 1 I ( 1 ) m = s e ; P 1 N k = m ( x ) e I D y n a m i c : !

11. Effective model of macrofluctuations Onsager-Machlup theory
Dynamics:
Equilibrium:
Path distribution: R d m t = s + q 2 N w P ( m 1 = ) / e 2 N s S ( m ) P ( ! ) = e x p N 4 R T 2 d m t + s

12. Effective model of macrofluctuations Onsager-Machlup theory
Dynamics:
Path distribution:
Dynamical fluctuations:
(Typical immediate) entropy production rate: R d m t = s + q 2 N P ( ! ) = e x p N 4 R T 2 d m t + s P ( m T = ) t ; e x p N s 2 8 R ( m ) = d S t N s 2 R

13. Effective model of macrofluctuations Onsager-Machlup theory
Dynamics:
Path distribution:
Dynamical fluctuations:
(Typical immediate) entropy production rate: R d m t = s + q 2 N P ( ! ) = e x p N 4 R T 2 d m t + s P ( m T = ) t ; e x p N s 2 8 R I ( m ) = 1 4 ( m ) = d S t N s 2 R

14. Towards general theory
Equilibrium
Nonequilibrium
Closed
Hamiltonian dynamics
Open
Stochastic dynamics
Mesoscopic
Macroscopic

15. Linear electrical networks revisited Dynamical fluctuations
Fluctuating dynamics:
Johnson-Nyquist noise:
Empirical ergodic average:
Dynamical fluctuation law: E = U + R 2 J f C _ 1 E U E f 1 E f 2 R 1 R 2 E f t = q 2 R C
white noise U T = 1 R t d 1 T l o g P ( U = ) 4 2 R + h E i

16. Linear electrical networks revisited Dynamical fluctuations
Fluctuating dynamics:
Johnson-Nyquist noise:
Empirical ergodic average:
Dynamical fluctuation law: E = U + R 2 J f C _ 1 E U E f 1 E f 2 R 1 R 2 E f t = q 2 R C
white noise
total dissipated
heat U T = 1 R t d 1 T l o g P ( U = ) 4 2 R + h E i

17. Stochastic models of nonequilibrium breaking detailed balance
Local detailed balance:
Global detailed balance generally broken:
Markov dynamics: x y k ( ; ) l o g k ( x ; y ) = s s ( x ; y ) = + d t ( x ) = X y k ;

18. Stochastic models of nonequilibrium breaking detailed balance
Local detailed balance:
Global detailed balance generally broken:
Markov dynamics: x y k ( ; ) l o g k ( x ; y ) = s
entropy change
in the environment s ( x ; y ) = + d t ( x ) = X y k ;

19. Stochastic models of nonequilibrium breaking detailed balance
Local detailed balance:
Global detailed balance generally broken:
Markov dynamics: x y k ( ; ) l o g k ( x ; y ) = s
entropy change
in the environment s ( x ; y ) = +
breaking term d t ( x ) = X y k ;

20. Stochastic models of nonequilibrium entropy production
Entropy of the system:
Entropy fluxes:
Mean entropy production rate: x y k ( ; ) S ( ) = X x l o g J ( x ; y ) = k | { z } e r o a t d i l b n c ( ) = d S t + 1 2 X x ; y J s k

21. Stochastic models of nonequilibrium entropy production
Entropy of the system:
Entropy fluxes:
Mean entropy production rate: x y k ( ; ) S ( ) = P x l o g J ( x ; y ) = k | { z } e r o a t d i l b n c ( ) = d S t + 1 2 X x ; y J s k
Warning:
Only for time-reversal
symmetric observables!

22. Stochastic models of nonequilibrium MinEP principle
(“Microscopic”) MinEP principle:
Can we again recognize entropy production as a fluctuation functional? x y k ( ; )
In the first order approximation
around detailed balance ( ) = m i n , s

23. Stochastic models of nonequilibrium dynamical fluctuations
Empirical occupation times:
Ergodic theorem:
Fluctuation law for occupation times?
Note: x y k ( ; ) p T ( x ) = 1 R Â ! t d p T ( x ) ! s ; 1 P ( p T = ) e I
natural variational functional I ( s ) =

24. Stochastic models of nonequilibrium dynamical fluctuations
Idea: Make the empirical distribution typical by modifying dynamics:
The “field” v is such that distribution p is stationary distribution for the modified dynamics:
Comparing both processes yields the fluctuation law: k ( x ; y ) ! v = e [ ] 2 X y p ( ) k v ; x = I ( p ) = X x ; y k v

25. Recall Equilibrium fluctuationsH ( x ) = N e H ( x ) = N e
add field P ( M x ) = N m e [ s ; q ] M ( x ) = N m e q
Probability of fluctuation
Typical value H h ( x ) = M N [ e m ]
The fluctuation made typical! s ( e ; m ) = h q

26. Stochastic models of nonequilibrium dynamical fluctuations
Idea: Make the empirical distribution typical by modifying dynamics:
The “field” v is such that distribution p is stationary distribution for the modified dynamics:
Comparing both processes yields the fluctuation law: k ( x ; y ) ! v = e [ ] 2 X y p ( ) k v ; x = I ( p ) = X x ; y k v

27. Stochastic models of nonequilibrium dynamical fluctuations
Traffic = mean dynamical activity:
Idea: Make the empirical distribution typical by modifying dynamics:
The “field” v is such that distribution p is stationary distribution for the modified dynamics:
Comparing both processes yields the fluctuation law: T = 1 2 X x ; y p ( ) k + k ( x ; y ) ! v = e [ ] 2 I ( p ) = e x c s i n t r a X y p ( ) k v ; x = I ( p ) = X x ; y k v

28. Stochastic models of nonequilibrium dynamical fluctuations close to equilibrium
General observation:
The variational functional is recognized as an approximate fluctuation functional
A consequence: A natural way how to go beyond MinEP principle is to study various fluctuation laws
In the first order approximation around
detailed balance I ( p ) = 1 4 s + o 2

29. Stochastic models of nonequilibrium dynamical fluctuations close to equilibrium
General observation:
The variational functional is recognized as an approximate fluctuation functional
A consequence: A natural way how to go beyond MinEP principle is to study various fluctuation laws
In the first order approximation around
detailed balance
Empirical currents: I ( ) = 1 4 s + o 2 J T ( x ; y ) = 1 # f j u m p s ! i n [ ] g y - + + x

30. Stochastic models of nonequilibrium dynamical fluctuations close to equilibrium
Typically, J T ( x ; y ) ! s k
Fluctuation law: J s ( x ; y )
General observation:
The variational functional is recognized as an approximate fluctuation functional
A consequence: A natural way how to go beyond MinEP principle is to study various fluctuation laws P ( J T = ) e G
In the first order approximation around
detailed balance
with the fluctuation functional G ( J ) = 1 4 _ S s + o 2
Empirical currents: I ( ) = 1 4 s + o 2
on stationary currents satisfying J T ( x ; y ) = 1 # f j u m p s ! i n [ ] g _ S ( J ) = D y - + + x

31. Stochastic models of nonequilibrium dynamical fluctuations close to equilibrium
Typically, J T ( x ; y ) ! s k
Fluctuation law:
General observation:
The variational functional is recognized as an approximate fluctuation functional
A consequence: A natural way how to go beyond MinEP principle is to study various fluctuation laws P ( J T = ) e G
Entropy flux 1 2 P x ; y J ( ) s
In the first order approximation around
detailed balance
with the fluctuation functional G ( J ) = 1 4 _ S s + o 2
Empirical currents: I ( ) = 1 4 s + o 2
on stationary currents satisfying J T ( x ; y ) = 1 # f j u m p s ! i n [ ] g _ S ( J ) = D
Onsager dissipation
function y - + + x

32. Stochastic models of nonequilibrium towards general fluctuation theory
It is useful to study the occupation time statistics and current statistics jointly
Joint occupation-current statistics has a canonical structure
Driving-parameterized dynamics
Current potential function k F ( x ; y ) = e 2 H ( p ; F ) = 2 [ T ]
anti-
symmetric
Reference equilibrium
Traffic

33. Joint occupation-current statistics has a canonical structure
Canonical equations H F ( x ; y ) p = J L e g n d r à ! G
It is useful to study the occupation time statistics and current statistics jointly
Joint occupation-current statistics has a canonical structure
Joint occupation-current fluctuation functional I F ( p ; J ) = 1 2 G + H _ S
Driving-parameterized dynamics
Current potential function k F ( x ; y ) = e 2 H ( p ; F ) = 2 [ T ]
anti-
symmetric
Reference equilibrium
Traffic

34. Stochastic models of nonequilibrium consequences of canonical formalism
Functional G describes (reference) equilibrium dynamical fluctuations
Fluctuation symmetry immediately follows:
Symmetric (p) and antisymmetric (J) fluctuations are coupled away from equilibrium, but: I F ( p ; J ) = _ S
for small fluctuations
close to equilibrium
Decoupling between p and J

35. General conclusions what we know
Both MinEP and MaxEP principles naturally follow from the fluctuation laws for empirical occupation times respectively for empirical currents
The validity of both principles is only restricted to the close-to-equilibrium and it is a consequence of
decoupling between time-symmetric and time-antisymmetric fluctuations
relation between traffic and entropy production for Markovian dynamics
Time-symmetric fluctuations are in general governed by traffic (nonperturbative result)
Joint occupation-current fluctuation has a general canonical structure

36. General conclusions what we would like to know
What is the operational meaning of new quantities (traffic,…) emerging in the dynamical fluctuation theory?
Are there useful computational schemes for the fluctuation functionals and can one systematically improve on the EP principles beyond equilibrium?
Is there a more general theory possible, also including velocity-like degrees of freedom and non-Markov dynamics?
What is the relation between static and dynamical fluctuations?
Could the dynamical fluctuation theory be a useful approach towards building nonequilibrium thermodynamics beyond close-to-equilibrium?
…and still many other things would be nice to know…

37. References C. Maes and K. Netočný, J. Math. Phys. 48:053306 (2007).
C. Maes and K. Netočný, Comptes Rendus – Physique 8: (2007).
S. Bruers, C. Maes, and K. Netočný, J. Stat. Phys. 129: (2007).
C. Maes and K. Netočný, cond-mat/
C. Maes, K. Netočný, and B. Wynants, cond-mat/
C. Maes, K. Netočný, and B. Wynants, condmat/

38. Thank You
for Your Attention!