Fluctuation theorem Totally Explained

Everything about The Fluctuation Theorem totally explained

The fluctuation theorem (FT) is a theorem from statistical mechanics dealing with the relative probability that the entropy of a system which is currently away from thermodynamic equilibrium (maximum entropy) will increase or decrease over a given amount of time. The second law of thermodynamics predicts that the entropy of an isolated system should tend to increase until it reaches equilibrium, but after the discovery of statistical mechanics physicists realized that the second law is only a statistical one, so that there should always be some nonzero probability that the entropy of an isolated system will spontaneously decrease; the fluctuation theorem precisely quantifies this probability.

Statement of the fluctuation theorem (roughly)

The fluctuation theorem is a statement concerning the probability distribution of the time-averaged irreversible entropy production [1], denoted

overline

the total electric current density J multiplied by the voltage drop across the circuit,

F_e

, and the system volume V, divided by the absolute temperature T, of the heat reservoir times Boltzmann's constant. Thus the dissipation function is easily recognised as the Ohmic work done on the system divided by the temperature of the reservoir. Close to equilibrium the long time average of this quantity is (to leading order in the voltage drop), equal to the average spontaneous entropy production per unit time - see de Groot and Mazur "Nonequilibrium Thermodynamics" (Dover), equation (61), page 348. However, the Fluctuation Theorem applies to systems arbitrariliy far from equilibrium where the definition of the spontaneous entropy production is problematic.

The fluctuation theorem and Loschmidt's paradox

The second law of thermodynamics, which predicts that the entropy of an isolated system out of equilibrium should tend to increase rather than decrease or stay constant, stands in apparent contradiction with the time-reversible equations of motion for classical and quantum systems. The time reversal symmetry of the equations of motion show that if one films a given time dependent physical process, then playing the movie of that process backwards doesn't violate the laws of mechanics. It is often argued that for every forward trajectory in which entropy increases, there exists a time reversed anti trajectory where entropy decreases, thus if one picks an initial state randomly from the system's phase space and evolves it forward according to the laws governing the system, decreasing entropy should be just as likely as increasing entropy. It might seem that this is incompatible with the second law of thermodynamics which predicts that entropy tends to increase. The problem of deriving irreversible thermodynamics from time-symmetric fundamental laws is referred to as Loschmidt's paradox.
The mathematical proof of the Fluctuation Theorem and in particular the Second Law Inequality shows that, given a non-equilibrium starting state, the probability of seeing its entropy increase is greater than the probability of seeing its entropy decrease - see The Fluctuation Theorem

from Advances in Physics 51: 1529. However, as noted in section 6 of that paper, one could also use the same laws of mechanics to extrapolate backwards from a later state to an earlier state, and in this case the same reasoning used in the proof of the FT would lead us to predict the entropy was likely to have been greater at earlier times than at later times. This second prediction would be frequently violated in the real world, since it's often true that a given nonequilibrium system was at an even lower entropy in the past (although the prediction would be correct if the nonequilibrium state were the result of a random fluctuation in entropy in an isolated system that had previously been at equilibrium - in this case, if you happen to observe the system in a lower-entropy state, it's most likely that you're seeing the minimum of the random dip in entropy, in which case entropy would be higher on either side of this minimum).
So, it seems that the problem of deriving time-asymmetric thermodynamic laws from time-symmetric laws can't be solved by appealing to statistical derivations which show entropy is likely to increase when you start from a nonequilibrium state and project it forwards. Many modern physicists believe the resolution to this puzzle lies in the low-entropy state of the universe shortly after the big bang, although the explanation for this initial low entropy is still debated.

Summary

The fluctuation theorem is of fundamental importance to nonequilibrium statistical mechanics. The FT (together with the Axiom of Causality) gives a generalisation of the second law of thermodynamics which includes as a special case, the conventional second law. It is then easy to prove the Second Law Inequality and the NonEquilibrium Partition Identity. When combined with the central limit theorem, the FT also implies the famous Green-Kubo relations for linear transport coefficients, close to equilibrium. The FT is however, more general than the Green-Kubo Relations because unlike them, the FT applies to fluctuations far from equilibrium. In spite of this fact, scientists have not yet been able to derive the equations for nonlinear response theory from the FT.
   The FT does not imply or require that the distribution of time averaged dissipation be Gaussian. There are many examples known where the distribution of time averaged dissipation is non-Gaussian and yet the FT (of course) still correctly describes the probability ratios.
   Lastly the theoretical constructs used to prove the FT can been applied to nonequilibrium transitions between two different equilibrium states. When this is done the so-called Jarzynski equality or nonequilibrium work relation, can be derived. This equality shows how equilibrium free energy differences can be computed or measured (in the laboratory), from nonequilibrium path integrals. Previously quasi-static (equilibrium) paths were required.
   The reason why the fluctuation theorem is so fundamental is that its proof requires so little. It requires:

knowledge of the mathematical form of the initial distribution of molecular states,
that all time evolved final states at time t, must be present with nonzero probability in the distribution of initial states (t = 0) - the so-called condition of ergodic consistency and,
an assumption of time reversal symmetry.

In regard to the latter "assumption", all the equations of motion for either classical or quantum dynamics are in fact time reversible.
For an alternative view on the same subject see http://www.scholarpedia.org/article/Fluctuation_theorem

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