Nonequilibrium steady state fluctuations

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). 

A very remarkable property exhibited by non equilibrium systems is the appearance of coherence and long range order [1] Once subjected to a constraint which maintains a system in a nonequilibrium steady state, fluctuations which occur in its different parts, are not independent. The correlation extends over the size of the sample [2-5]. We shall here study the origin of this behaviour in a simple example and describe a simulation set up in order to demonstrate this property. 

It is most remarkable that the entropy production in a nonequilibrium steady state is directly related to the time asymmetry in the dynamical randomness of nonequilibrium fluctuations. The entropy production turns out to be the difference in the amounts of temporal disorder between the backward and forward paths or histories. In nonequilibrium steady states, the temporal disorder of the time reversals is larger than the temporal disorder h of the paths themselves. This is expressed by the principle of temporal ordering, according to which the typical paths are more ordered than their corresponding time reversals in nonequilibrium steady states. This principle is proved with nonequilibrium statistical mechanics and is a corollary of the second law of thermodynamics. Temporal ordering is possible out of equilibrium because of the increase of spatial disorder. There is thus no contradiction with Boltzmann s interpretation of the second law. Contrary to Boltzmann s interpretation, which deals with disorder in space at a fixed time, the principle of temporal ordering is concerned by order or disorder along the time axis, in the sequence of pictures of the nonequilibrium process filmed as a movie. The emphasis of the dynamical aspects is a recent trend that finds its roots in Shannon s information theory and modem dynamical systems theory. This can explain why we had to wait the last decade before these dynamical aspects of the second law were discovered. 

H. Qian. Open-system nonequilibrium steady state statistical thermodynamics, fluctuations and chemical oscillations. J. Phys. Chem. B., 110 15063-15074, 2006. 

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. 

However, despite numerical, and recent experimental evidence, that the steady state relation does hold under these assumptions, it must be said that there are some subtleties regarding convergence times and the decay of fluctuations in nonequilibrium steady states. Recently a much more rigorous derivation of the steady state ES FR has been given. This derivation demonstrated that the underlying assumption necessary to obtain (2.4) is that time-correlations decay sufficiently quickly. ... 

We intend the term steady state to be understood in the statistical thermodynamic sense of aUowing stationary concentration fluctuations around a steady mean concentration This mean concentration could be established either in equilibrium or in a nonequilibrium steady state. If several steady states are compatible with a given set of concentration constraints (as is discussed later), there will be concentration fluctuations centered around each of the steady states. 

The fluctuating variables aie thereby projected onto pair-density fluctuations, whose time-dependence follows from that of the transient density correlators q(,)(z), defined in (12). Tliese describe the relaxation (caused by shear, interactions and Brownian motion) of density fluctuations with equilibrium amplitudes. Higher order density averages are factorized into products of these correlators, and the reduced dynamics containing the projector Q is replaced by the full dynamics. The entire procedure is written in terms of equilibrium averages, which can then be used to compute nonequilibrium steady states via the ITT procedure. The normalization in (10a) is given by the equilibrium structure factors such that the pair density correlator with reduced dynamics, which does not couple linearly to density fluctuations, becomes approximated to ... 

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

Qian H. 2006. Open-system nonequilibrium steady state Statistical thermodynamics, fluctuations, and chemical oscillations. Journal of Physical Chemistry B 110 15063. 

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... 

Figure 17.5 The time variation of the entropy production P = diS/dt— Y i FkJk for equilibrium and near equilibrium states, (a) For a lluctpation from the equilibrium state, the initial nonzero value of P decreases to its equilibrium value of zero, (b) In the Unear regime, a fluctuation from a nonequilibrium steady state can only increase the value of P above the stationary value Pgu irreversible processes drive P back to its minimum value Pst...

Space and time-correlation functions of macrovariables around nonequilibrium steady states have recently been calculated from nonequilibrium statistical mechanics [2]. Despite this progress, fluctuation theory for nonequilibrium states remains based, in its essential aspects, on stochastic theory. Specifically, it is assumed that one can define an appropriate set of discrete variables... 

This alone is an advantage over the earlier G-K approach, which could calculate only the linear transport coefficients which describe transport processes arbitrarily close to equilibrium. Evans Morriss (1990), however, have shown that the G-K relations are simple examples of more general nonlinear, fluctuation dissipation relations which are valid for nonequilibrium steady states. 

As the reaction proceeds, the diffusion layer extends into the bulk of the solution outside the double layer. When the diffusion-layer thickness increases much more than the autocorrelation distance of the asymmetrical nonequilibrium fluctuation, a steady state emerges. In contrast to Eq. (103), in this case the following condition holds,... 

In Sections IVA, VA, and VI the nonequilibrium probability distribution is given in phase space for steady-state thermodynamic flows, mechanical work, and quantum systems, respectively. (The second entropy derived in Section II gives the probability of fluctuations in macrostates, and as such it represents the nonequilibrium analogue of thermodynamic fluctuation theory.) The present phase space distribution differs from the Yamada-Kawasaki distribution in that... 

Figure 8 shows the r-dependent thermal conductivity for a Lennard-Jones fluid (p = 0.8, 7o = 2) [6]. The nonequilibrium Monte Carlo algorithm was used with a sufficiently small imposed temperature gradient to ensure that the simulations were in the linear regime, so that the steady-state averages were equivalent to fluctuation averages of an isolated system. 

Nonequilibrium effects. In applying the various formalisms, a Boltzmann distribution over the vibrational energy levels of the initial state is assumed. The rate constant calculated on the basis of the equilibrium distribution, keq, is the maximum possible value of k. If the electron transfer is very rapid then the assumption of an equilibrium distribution over the energy levels is not valid, and it is more appropriate to treat the nuclear fluctuations in terms of a steady-state rather than an equilibrium formalism. Although a rigorous treatment of this problem has not yet appeared, intuitively it seems that since the slowest nuclear fluctuation will generally be a solvent orientational motion, ke will equal keq when vout keq and k will tend to vout when vout keq (a simple treatment gives l/kg - 1/ vout + 1/keq). These considerations are... 

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