Macrostates

The basic chemical description of rare events can be written in terms of a set of phenomenological equations of motion for the time dependence of the populations of the reactant and product species [6-9]. Suppose that we are interested in the dynamics of a conformational rearrangement in a small peptide. The concentration of reactant states at time t is N-n(t), and the concentration of product states is N-pU). We assume that we can define the reactants and products as distinct macrostates that are separated by a transition state dividing surface. The transition state surface is typically the location of a significant energy barrier (see Fig. 1). 

When it is possible to recognize distinct macrostates, we can write the phenomenological rate equations... 

If a confined fluid is thermodynamically open to a bulk reservoir, its exposure to a shear strain generally gives rise to an apparent multiplicity of microstates all compatible with a unique macrostate of the fluid. To illustrate the associated problem, consider the normal stress which can be computed for various substrate separations in grand canonical ensemble Monte Carlo simulations. A typical curve, plotted in Fig. 16, shows the oscillatory decay discussed in Sec. IV A 2. Suppose that instead... 

In equation (1.17), S is entropy, k is a constant known as the Boltzmann constant, and W is the thermodynamic probability. In Chapter 10 we will see how to calculate W. For now, it is sufficient to know that it is equal to the number of arrangements or microstates that a molecule can be in for a particular macrostate. Macrostates with many microstates are those of high probability. Hence, the name thermodynamic probability for W. But macrostates with many microstates are states of high disorder. Thus, on a molecular basis, W, and hence 5, is a measure of the disorder in the system. We will wait for the second law of thermodynamics to make quantitative calculations of AS, the change in S, at which time we will verify the relationship between entropy and disorder. For example, we will show that... 

This is a law about the equilibrium state, when macroscopic change has ceased it is the state, according to the law, of maximum entropy. It is not really a law about nonequilibrium per se, not in any quantitative sense, although the law does introduce the notion of a nonequilibrium state constrained with respect to structure. By implication, entropy is perfectly well defined in such a nonequilibrium macrostate (otherwise, how could it increase ), and this constrained entropy is less than the equilibrium entropy. Entropy itself is left undefined by the Second Law, and it was only later that Boltzmann provided the physical interpretation of entropy as the number of molecular configurations in a macrostate. This gave birth to his probability distribution and hence to equilibrium statistical mechanics. 

The present theory can be placed in some sort of perspective by dividing the nonequilibrium field into thermodynamics and statistical mechanics. As will become clearer later, the division between the two is fuzzy, but for the present purposes nonequilibrium thermodynamics will be considered that phenomenological theory that takes the existence of the transport coefficients and laws as axiomatic. Nonequilibrium statistical mechanics will be taken to be that field that deals with molecular-level (i.e., phase space) quantities such as probabilities and time correlation functions. The probability, fluctuations, and evolution of macrostates belong to the overlap of the two fields. 

Moving downward to the molecular level, a number of lines of research flowed from Onsager s seminal work on the reciprocal relations. The symmetry rule was extended to cases of mixed parity by Casimir [24], and to nonlinear transport by Grabert et al. [25] Onsager, in his second paper [10], expressed the linear transport coefficient as an equilibrium average of the product of the present and future macrostates. Nowadays, this is called a time correlation function, and the expression is called Green-Kubo theory [26-30]. 

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... 

Macrostates are collections of microstates [9], which is to say that they are volumes of phase space on which certain phase functions have specified values. The current macrostate of the system gives its structure. Examples are the position or velocity of a Brownian particle, the moments of energy or density, their rates of change, the progress of a chemical reaction, a reaction rate, and so on. Let x label the macrostates of interest, and let x(r) be the associated phase function. The first entropy of the macrostate is... 

The macrostates can have either even or odd parity, which refers to their behavior under time reversal or conjugation. Let e, = 1 denote the parity of the th microstate, so that = e r). (It is assumed that each state is... 

The unconditional transition probability between macrostates in time r for the isolated system satisfies... 

This uses the fact that dr = dT. For macrostates all of even parity, this says that for an isolated system the forward transition x > x will be observed as frequently as the reverse x —> x. This is what Onsager meant by the principle of dynamical reversibility, which he stated as in the end every type of motion is just as likely to occur as its reverse [10, p. 412]. Note that for velocity-type variables, the sign is reversed for the reverse transition. 

Maximizing the second entropy with respect to x for fixed x yields the most likely terminal position x(x, x) = x, and hence the most likely coarse velocity x(x,x). Alternatively, differentiating the most likely terminal position with respect to x yields the most likely terminal velocity, x(x, x). So constraining the system to be in the macrostate x at a time x after it was in the state x is the same as constraining the coarse velocity. 

For simplicity, it is assumed that the equilibrium value of the macrostate is zero, x = 0. This means that henceforth x measures the departure of the macrostate from its equilibrium value. In the linear regime, (small fluctuations), the first entropy may be expanded about its equilibrium value, and to quadratic order it is... 

If one now adds a reservoir with thermodynamic force Xr, then the subsystem macrostate x can change by internal processes A°x, or by exchange with the reservoir, Arx = Axr. Imagining that the transitions occur sequentially,... 

The generic case is a subsystem with phase function x(T) that can be exchanged with a reservoir that imposes a thermodynamic force Xr. (The circumflex denoting a function of phase space will usually be dropped, since the argument T distinguishes the function from the macrostate label x.) This case includes the standard equilibrium systems as well as nonequilibrium systems in steady flux. The probability of a state T is the exponential of the associated entropy, which is the total entropy. However, as usual it is assumed (it can be shown) [9] that the... 

As mentioned, xA(r) is half of the total adiabatic change in the subsystem macrostate associated with the current phase space point T. The factor of is used to compensate for double counting of the past and future changes. In the steady state, the subsystem most likely does not change macrostate, and hence this change has to be compensated by the change in the reservoir, Axr = xA(r). 

The final approximation is valid if the adiabatic change in the macrostate is relatively negligible, x — x internal force from the reservoir force is relatively negligible, X( Xr -c Xr. ... 

The first term on the right-hand side is independent of the subsystem and may be neglected. With this, the entropy of the total system constrained to be in the macrostate Es is the sum of that of the isolated subsystem in that macrostate and that of the reservoirs,... 

Onsager and Machlup [32] gave an expression for the probability of a path of a macrostate, p[x]. The exponent may be maximized with respect to the path for fixed end points, and what remains is conceptually equivalent to the constrained second entropy used here, although it differs in mathematical detail. The Onsager-Machlup functional predicts a most likely terminal velocity that is exponentially decaying [6, 42] ... 

Of course, depending on the system, the optimum state identified by the second entropy may be the state with zero net transitions, which is just the equilibrium state. So in this sense the nonequilibrium Second Law encompasses Clausius Second Law. The real novelty of the nonequilibrium Second Law is not so much that it deals with the steady state but rather that it invokes the speed of time quantitatively. In this sense it is not restricted to steady-state problems, but can in principle be formulated to include transient and harmonic effects, where the thermodynamic or mechanical driving forces change with time. The concept of transitions in the present law is readily generalized to, for example, transitions between velocity macrostates, which would be called an acceleration, and spontaneous changes in such accelerations would be accompanied by an increase in the corresponding entropy. Even more generally it can be applied to a path of macrostates in time. 

Symmetric-matrix valued function, transition state trajectory, colored noise, 208-209 Symmetry rules, linear thermodynamics, macrostates, 11... 

I = / i T) —1. In this expression, the macrostate probabilities at a given temperature are easy to identify - the probability that each energy will be visited is proportional to the integrand. 

Note that the expression in (3.1) is a continuous probability distribution in that p(U T)dU gives the probability of macrostates with energy U dU/2. In an NVT simulation, we measure this distribution to a finite precision by employing a nonzero bin width All. Letting f(U) be the number of times an energy within the range [U,U I All] is visited in the simulation, the normalized observed energy distribution... 

If we wish to generate a uniform distribution in all of the macrostates that fluctuate during the simulation (in this case both N and U), the same arguments necessitate the following microstate sampling scheme ... 

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