Beyond Onsager-Machlup Theory of Fluctuations

1
Beyond Onsager-Machlup Theory of Fluctuations

• Karel Netocný
• 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

• 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

E
4
Search for nonequilibrium ensembles

• McLennans extension of Gibbs ensembles to weak
  nonequilibrium

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

7
Onsager-Machlup fluctuation theory
8
Onsager-Machlup fluctuation theory

• Modelling small macroscopic fluctuations around
• exponential relaxation to equilibrium
• Action has a generic structure

path
Time locality
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 Einsteins
  theory of macroscopic fluctuations

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

• Field-theoretical methods to compute the
  effective potential

11
Brillouin paradox

• Nonlinear stochastic model
• often leads to inconsistencies (e.g. to
  rectification of
• fluctuations violating second law)

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
13
Some questions

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

14
Stochastic nonequilibrium network
S
S

• Dissipation modeled as the rate asymmetry
• Local detailed balance condition

z
y
y
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

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)

Mesoscopic Onsager-Machlup functional
I
stationary
17
Result II. Mesoscopic Onsager-Machlup functional

• It takes the explicit form

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

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

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

21
References

• 1 C. Maes and K. Netocný, Europhys. Lett. 82
  (2008) 30003.
• 2 C. Maes, K. Netocný, and B. Wynants,
  Physica A 387 (2008) 2675.
• 3 C. Maes, K. Netocný, and B. Wynants,
  arXivcond-mat/0709.4327.