Steady-state fluctuation theorems

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). 

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. 

As was shown in Ref [42], the single component FPM system yields the Gibbs distribution as the steady-state solution to the Fokker-Planck equation under the condition of detailed balance. Consequently, it obeys the fluctuation-dissipation theorem, which defines the relationship between the normalized weight functions, which are chosen such that... 

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 ... 

Various transient or steady-state forms of Eqn (15.50) are known as integral fluctuation theorems. Fluctuation theorems express universal properties of the probability distribution p Qi) for functionals Q[x(t)], like work, heat or entropy change, evaluated along the fluctuating trajectories taken from ensembles with well-specified... 

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... 

Due to the Lagrangian of the functional (99) is the sum of the dissipation potentials, which is equal to the entropy production in case of every real steady-state physical processes, this extremum theorem involves the minimum principle of global entropy production (MPGEP). The physical meaning of MPGEP needs a clarification. Consider the variations of the fluxes and of the intensive parameters as fluctuations of the system around their stationer state values. When these fluctuations are small, the fluctuation of the global entropy production of the system is equal to its first approximations and it has a form... 

The minimum entropy production theorem dictates that, for a system near equilibrium to achieve a steady state, the entropy production must attain the least possible value compatible with the boundary conditions. Near equilibrium, if the steady state is perturbed by a small fluctuation (8), the stability of the steady state is assured if the time derivative of entropy production (P) is less than or equal to zero. This may be expressed mathematically as dPIdt 0. When this condition pertains, the system will develop a mechanism to damp the fluctuation and return to the initial state. The minimum entropy production theorem, however, may be viewed as providing an evolution criterion since it implies that a physical system open to fluxes will evolve until it reaches a steady state which is characterized by a minimal rate of dissipation of energy. Because a system on the thermodynamic branch is governed by the Onsager reciprocity relations and the theorem of minimum entropy production, it cannot evolve. Yet as a system is driven further away from equilibrium, an instability of the thermodynamic branch can occur and new structures can arise through the formation of dissipative structures which requires the constant dissipation of energy. 

Another form of the fluctuation-dissipation theorem involves time correlation functions. If the external perturbation is applied at t = —oo and turned off at t = 0, the system will move from a steady state back to equilibrium. The dynamic relaxation of property A from its original steady state, (A(0)), to equilibrium, A)o, can be shown to be congruent to the normalized time autocorrelation function of equilibrium fluctuations. 

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