Nonequilibrium ensemble average

In 1931, Onsager showed that an intimate relation exists between the time correlation functions and the dynamics of the nonequilibrium ensemble averages. Assume that some dynamical variable A has a nonzero average at time zero, at which time constraints are removed and the system begins to return to equilibrium then... 

Various phenomenological, linear laws for the time dependence of the nonequilibrium ensemble averages existed before any systematic theory of such laws was developed, Fourier s heat flow law. Pick s law, and the hydrodynamic equations are examples. By linear laws we mean laws of the form... 

For any problem, once we have isolated the correct set of variables, we now have a rule for writing the equation of motion for the nonequilibrium ensemble average of the linear variables. Of course, the same equation holds for the time correlation functions of the linear variables, and with the addition of the random force for the fluctuating linear variables [Eqs. (14) and (10)]. 

Due to the noncrystalline, nonequilibrium nature of polymers, a statistical mechanical description is rigorously most correct. Thus, simply hnding a minimum-energy conformation and computing properties is not generally suf-hcient. It is usually necessary to compute ensemble averages, even of molecular properties. The additional work needed on the part of both the researcher to set up the simulation and the computer to run the simulation must be considered. When possible, it is advisable to use group additivity or analytic estimation methods. 

The SES, ESP, and NES methods are particularly well suited for use with continuum solvation models, but NES is not the only way to include nonequilibrium solvation. A method that has been very useful for enzyme kinetics with explicit solvent representations is ensemble-averaged variational transition state theory [26,27,87] (EA-VTST). In this method one divides the system into a primary subsystem and a secondary one. For an ensemble of configurations of the secondary subsystem, one calculates the MEP of the primary subsystem. Thus the reaction coordinate determined by the MEP depends on the coordinates of the secondary subsystem, and in this way the secondary subsystem participates in the reaction coordinate. 

A more general case is that of a non-equilibrium distribution that is close to an equilibrium distribution. The parameters p, 77, E, I, v and A are all allowed to change, but (A) = N° is maintained. It is then found that d— (cr) increases as E — E increases, where the overbars indicate averages over the nonequilibrium ensemble. An interesting restriction is that /lA > i.e., the... 

There are two interesting regimes of time evolution in the probing/detection of dynamical nonequilibrium structures. In the regime of dynamics, the time evolution of atomic positions is detected on its intrinsic timescale, i.e., femtoseconds. Short X-ray pulses - on the timescale of atomic motion - are required in order to follow the dynamics of the chemical bond. In the regime of kinetics, which has to do with the time evolution of populations - and in the context of time-resolved X-ray diffraction -the time evolution in an ensemble average of different interatomic distances or the structural determination of short-lived chemical species is considered. 

Here S is the grand partition function and the equilibrium distribution is characterized by the probabilities PN.i. Let us consider the probabilities Pn, i defining the nonequilibrium distributions of the same system, at temperature T, generated through small perturbation of the bath parameters, viz. p and v(f). The average over any such nonequilibrium ensemble may be written as... 

Evidently the effect of the label change will be to increase the number density of labeled particles in the primary cell near the x=L boundary relative to that near the x = 0 boundary. In the long time limit, it is expected that the system will approach a one-dimensional steady state, in which a self-diffusion current ji of labeled particles will flow in the —e direction independent of r and t. The calculation depends on the establishment of this steady state and is to be contrasted with the use of an initial nonequilibrium ensemble in which one might study the number density and current as transients. Here the number density and current are to be evaluated as time averages, beginning at such a time that initial transients have vanished. 

In Eq. (90), the bilinear term is truly a product of linear terms and may be manipulated by standard techniques. The resulting expression for Ak(0 may then be averaged over a nonequilibrium ensemble to obtain >4k(t), or the expression may be multiplied on the left by >4-k(0) and averaged over an equilibrium ensemble to obtain (y4k(t)A-k) In the following, we shall concentrate on the calculation of the time correlation function, which is, of course, identical to the calculation of A it) in the limit of small deviations from equilibrium. 

There are numerous more advanced theories of transport coefficients in liquids, mostly based on nonequilibrium classical statistical mechanics. Some are based on approximate representations of the time-dependent reduced distribution function and others are based on the analysis of time correlation functions, which are ensemble averages of the product of a quantity evaluated at time 0 and the same quantity or a different quantity evaluated at time t For example, the self-diffusion coefficient of a monatomic liquid is given by " ... 

This identity [3, 15] between a weighted average of nonequilibrium trajectories (r.h.s.) and the equilibrium Boltzmann distribution (l.h.s.) is implicit in the work of Jarzynski [2], and is given explicitly by Crooks [16]. The average (... is over an ensemble of trajectories starting from the equilibrium distribution at / 0 and... 

The angular brackets ( ) in (7.33) denote an average over an ensemble of nonequilibrium transformation processes initiated from states z distributed according to a canonical distribution. The Jarzynski identity (7.33) is valid for nonequilibrium transformations carried out at arbitrary speed. 

Equation IV. 18 splits the heat of transport into two terms, the first of which is quasi-thermodynamic in that it involves only averages over equilibrium ensembles, and the second of which arises from the deviation of the distribution function in pair space from the equilibrium distribution function. Qn corresponds to the expressions for the heat of transport found by previous authors who have neglected the nonequilibrium perturbation to the distribution function. 

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. 

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). 

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