Nonequilibrium ensemble

Sharp and Lohr proposed recently a somewhat different point of view on the relation between the electron spin relaxation and the PRE (126). They pointed out that the electron spin relaxation phenomena taking a nonequilibrium ensemble of electron spins (or a perturbed electron spin density operator) back to equilibrium, described in Eqs. (53) and (59) in terms of relaxation superoperators of the Redfield theory, are not really relevant for the PRE. In an NMR experiment, the electron spin density operator remains at, or very close to, thermal equilibrium. The pertinent electron spin relaxation involves instead the thermal decay of time correlation functions such as those given in Eq. (56). The authors show that the decay of the Gr(T) (r denotes the electron spin vector components) is composed of a sum of contributions... 

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... 

In other words, if we look at any phase-space volume element, the rate of incoming state points should equal the rate of outflow. This requires that Cbe a function of the constants of the motion, and especially C=C(W). Equilibrium also implies d(x)/dt = 0 for any %. The extension of the above equations to nonequilibrium ensembles requires a consideration of entropy production, the method of controlling energy dissipation (thermostatting) and the consequent non-Liouville nature of the time evolution [35]. 

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... 

Bredenbeck J, et al. 2003. Transient 2D IR spectroscopy snapshots of the nonequilibrium ensemble during the picosecond conformational transition of a small peptide. J Phys Chem B 107(33) 8654 8660. 

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 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)]. 

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 is also another important problem concerning the nature of ki-netically nonequilibrium states of chemical systems. This problem can be formulated in the following way. What are we dealing with, a nonequilibrium mixture of equilibrium molecules or nonequilibrium molecules In any chemical process there appear molecules in nonequilibrium states. For low-molecular compounds, the electronic and vibrational relaxation after the elementary chemical act takes little time (as a rule, less than 10" -10" s). Therefore, there are relaxed atoms, ions, free radicals, and molecules, i.e., the particles in their equilibrium states, that take part in the subsequent chemical acts of chemical transformations. If a system consisting of low-molecular compounds is removed from the state of chemical equilibrium, then, as a rule, we can speak of a nonequilibrium ensemble of equilibrium molecules. 

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