1. Beyond Onsager-Machlup Theory of Fluctuations
Karel Netočný
Institute of Physics AS CR
Seminar, 13 May 2008

2. Why to worry about fluctuations?
Structure of equilibrium fluctuations is well understood and useful
Second law S = m a x
Thermodynamics
Fluctuations P = e x p S k S = k l n W
Entropy
thermodynamic potential
variational functional
fluctuation functional

3. Example: fluctuations of energy
System spontaneously exchanges energy with reservoir
Free energy and its variational extension determine fluctuations
Mathematical structure of large deviations
Reservoir
System E = 1 k T P ( E ) = e 1 k T f S F g l o g P F = m i n E f T S ( ) g 1 E E e q

4. Search for nonequilibrium ensembles
Reservoir
System E = 1 k T
Engine W Q
McLennan’s extension of Gibbs ensembles to weak nonequilibrium: P ( x ) ' 1 Z e H + S
Gibbs corrected by
“excess entropy”

5. Systems close to equilibrium
Linear response theory and fluctuation-dissipation relations (Kubo, Mori,…)
MaxEnt formalism (Jaynes) and Zubarev ensembles
Linear irreversible thermodynamics (Prigogine, Nicolis,…)
Local equilibrium methods (Lebowitz and Bergmann,…)
Dynamical fluctuation theory (Onsager and Machlup,…)

6. Landauer: essential role of dynamics
Blowtorch theorem says that noise plays a fundamental role in determining steady state
No “magic” universal principles are to be expected to describe open systems
One cannot avoid studying the full kinetics
Precist si poradne Landauera!

7. Onsager-Machlup fluctuation theory

8. Onsager-Machlup fluctuation theory
Modelling small macroscopic fluctuations around
exponential relaxation to equilibrium
Action has a generic structure: _ m i = L j + \ n o s e R k P ( ! ) = e A
path A ( ! ) = 1 2 R d t D _ S + E
Time locality D = R j k _ m S = s j k m E = L j k m

9. Onsager-Machlup fluctuation theory
Describes normal fluctuations away from criticality
It is consistent with the fluctuation-dissipation relation
It provides a “microscopic” basis for the least dissipation principle:
It provides a dynamical extension of Einstein’s theory of macroscopic fluctuations D 2 _ S = m i n

10. Nonlinear systems with weak noise
A method to study dynamical systems is to add a noise and to
analyze fluctuations
Friedlin-Wentzel theory:
Method of effective potential: _ x = Á ( ) + p
" : w h i t e n o s P ( ! ) = e x p 1
" A + c o r t i n s A = 1 4 R d t [ _ x Á ( ) ] 2
Field-theoretical methods to compute the effective potential ( x ) = m a 1 T R d t ! l o g P

11. Brillouin paradox _ x = Á ( ) + p " : w h i t e n o s
Nonlinear stochastic model
often leads to inconsistencies (e.g. to rectification of
fluctuations violating second law) _ x = Á ( ) + p
" : w h i t e n o s
N. G. van Kampen: “Statistical mechanics leads, on the macroscopic level to a stochastic description in terms of a master equation. Subsequently deterministic equations plus fluctuations can be extracted from it by suitable limiting processes. The misconception that one should start from the known macroscopic equations and then somehow add the fluctuations on to them is responsible for much confusion in the literature.”

12. Nonequilibrium macrostatistics
Initiated by Derrida, Jona-Lasinio, Bodineau,…
Many-body stochastic system analyzed in hydrodynamic limit
The scaling yields a diffusion equation + noise
Here it is possible to add white noise here because of
the diffusive regime @ t = r [ D ( ) + n o i s e ]

13. Some questions
Can one extend the Onsager-Machlup approach to mesoscopic systems possibly far from equilibrium? (strong noise!)
What would be the natural observables there?
Can one understand the mesoscopic stationary variational principles this way?
How to extend to quantum open systems?

14. Stochastic nonequilibrium network1 Q 2 S S
Dissipation modeled as the rate asymmetry
Local detailed balance condition z y y k ( x ! y ) = e S x

15. Some trivias Description using statistical distributions
Single-time distribution for Markov system evolves as
Local energy currents are
Stationary distribution and stationary currents observed as typical long-time averages d t ( x ) = X y k ; - x J ( x ; y ) = k | { z } e r o a t d i l b n c
Ergodic average of
empirical residence times
Ergodic average of
empirical directed number
of jumps

16. Result I. Natural fluctuation observables
It is useful to consider jointly all empirical residence times and all empirical currents as a collection of “canonical” observables
A natural expansion parameter of the fluctuation theory is the inverse observation time (remember that noise is no longer weak) P ( p T = ; j ) e I + c o r t i n s I
Mesoscopic
Onsager-Machlup
functional
stationary

17. Result II. Mesoscopic Onsager-Machlup functional
It takes the explicit form I ( p ; j ) = 1 2 G + H F _ S
Entropy current _ S = 1 2 X x ; y F ( ) j
Conjugated
dissipation functional @ H F = j G
Onsager-like
dissipation functional H = 2 ( T F )
Driving from reference
equilibrium k F ( x ; y ) = e 2
Traffic T = X x ; y p ( ) k +

18. Result III. Close-to-equilibrium decoupling
In the linear response regime, the time-symmetric and time-antisymmetric fluctuations mutually decouple
Moreover, the traffic excess equals to the total entropy production excess
Then the total entropy production fully
determines the structure of fluctuations I ( p ; j ) = 1 + 2 T F = E E = _ S + d s y t

19. Various remarks and outlook
Implies (mesoscopic) minimum and maximum entropy production principles close to equilibrium
The relation between the traffic and the entropy production excesses remains true in the diffusion regime far from equilibrium
Apart from the two cases, the traffic emerges as a novel relevant functional determining dynamical fluctuations
More coarse-grained fluctuation laws follow by solving corresponding variational problems
Extension of the Onsager-Machlup formalism to open systems with quantum coherence remains to be understood…

20. FCS approach to mesoscopic systems
More standard FCS methods yield a direct access to the (partial) current cummulants [See Flindt, Novotný, Braggio, Sassetti, and Jauho, PRL (2008).]
The FCS methods can also be adapted to study the time-symmetric fluctuations (e.g. the residence-time distribution)
Extensions to time-dependent (AC) driving

21. References
[1] C. Maes and K. Netočný, Europhys. Lett. 82 (2008)
[2] C. Maes, K. Netočný, and B. Wynants, Physica A 387 (2008) 2675.
[3] C. Maes, K. Netočný, and B. Wynants, arXiv:cond-mat/