Fluctuation theorems nonequilibrium states

Fluctuation-dissipation theorem, transition state trajectory, white noise, 203—207 Fluctuation theorem, nonequilibrium thermodynamics, 6—7... 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

Here, <... > denotes an average over the equilibrium ensemble of initial conditions. C t) is the conditional probability to find the system in state B at time t provided it was in state A at time 0. According to the fluctuation-dissipation theorem [63], dynamics of equilibrium fluctuations are equivalent to the relaxation from a nonequilibrium state in which only state A is populated. At long timescales, these nonequilibrium dynamics are described by the phenomenology of macroscopic kinetics. Thus, the asymptotic behavior of C(t) is determined by rate constants and kg. At long times, and provided that a single dynamical bottleneck separating A from B causes simple two-state kinetics. 

Assume that a finite system is in contact with a heat bath at constant temperature and driven away from equilibrium by some external time-dependent force. Many nonequilibrium statistical analyses are available for the systems in the vicinity of equilibrium. The only exception is the fluctuation theorems, which are related to the entropy production and valid for systems far away from global equilibrium. The systems that are far from global equilibrium are stochastic in nature with varying spatial and timescales. The fluctuation theorem relates to the probability distributions of the time-averaged irreversible entropy production a. The theorem states that, in systems away from equilibrium over a finite time t, the ratio between the probability that CT takes on a value A and the probability that it takes the opposite value, —A, will be exponential in At. For nonequilibrium system in a finite time, the fluctuation theorem formulates that entropy will flow in a direction opposite to that dictated hy the second law of thermodynamics. Mathematically, the fluctuation theorem is expressed as ... 

The fluctuation theorem deals with fluctuations. Since the statistics of fluctuations will be different in different statistical ensembles, the fluctuation theorem is a set of closely related theorems. Also some theorems consider nonequilibrium steady-state fluctuations, while others consider transient fluctuations. One of the fluctuation theorems state that in a time-reversible dynamic system in contact with constant temperature heat bath, the fluctuations in the time-averaged irreversible entropy productions in a nonequilibrium steady state satisfy Eqn (15.49) (Evans and Searles, 2002). 

In general, for a system driven by a time symmetric process, the resulting nonequilibrium steady-state ensemble would be invariant under time reversal. This symmetry ensures that the forward and backward processes become indistinguishable and the entropy production is odd under a time reversal, the fluctuation theorem will be valid for any integer number of cycles ... 

At nonequilibrium steady state, a net flux in the species occurs if it is possible to adjust the concentrations. Hence, the stationary state violates the detailed balance condition PnJnm = PmJmn where J is the rate of transformation and p is the probability. For such nonequilibrium steady states, a detailed fluctuation theorem is... 

By relating the fluctuations to the reaction coordinate v, de = SNh Eqs. (12.10) and (12.12) imply that if a system is stable to fluctuations in diffusion, it is also stable to fluctuations in chemical reactions, which is called the Duhem-Jougeut theorem (Kondepudi and Prigogine, 1999). However, a nonequilibrium steady state involving chemical reactions may be unstable even if the system is stable with respect to diffusion. 

Speck, T, Seifert, U. Restoring a fluctuation-dissipation theorem in a nonequilibrium steady state. Europhys. Lett. 2006, 74,391. 

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