Phase Space Time Correlation Functions

We consider a simple classical fluid system containing N particles confined in a volume n at an equilibrium temperature T = It is assumed that the 

The dynamical variable fundamental in our study is the classical phase space density 

The equilibrium averages of products of/ s at equal times give a complete description of the static correlations in the system. If we use one of the canonical ensembles we can show that 

We see, for example, that by combining these results one has, for the total energy, 

One can go on to define higher-order static correlation functions beyond but we will introduce them only when they are needed. It is important to note that in dense fluids static correlation functions play an important role in determining the dynamical properties. Consequently, detailed knowledge of these functions is essential in developing any dynamical theory. In our discussions we will regard static correlation functions as known quantities and rely on other theories for their calculations. This means that g(r) is understood to be 

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In Sections 2 and 3, we set up a formalism for dealing with the dynamics of dense fluids at the molecular level. We begin in Section 2 by focusing attention on the phase space density correlation function from which the space-time correlation functions of interest in scattering experiments and computer simulations can be obtained. The phase space correlation function obeys a kinetic equation that is characterized by a memory function, or generalized collision kernel, that describes all the effects of particle interactions. The memory function plays the role of an effective one-body potential and one can regard its presence as a renormalization of the motions of the particles. 

The basic quantity that we wish to calculate is the time-dependent phase space density correlation function defined by... 

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. 

The details of the pair potential used in the simulations are given in Table I. This consists of an -trans model of the sec-butyl chloride molecule with six moieties. The intermolecular pair potential is then built up with 36 site-site terms per molecular pair. Each site-site term is compost of two parts Lennard-Jones and charge-charge. In this way, chiral discrimination is built in to the potential in a natural way. The phase-space average R-R (or S-S) potential is different from the equivalent in R-S interactions. The algorithm transforms this into dynamical time-correlation functions. 

This time correlation function is the conditional probability that a phase space point in A at time 0 will be found in B at time t. This correlation function for the overall states has the advantage that for regions A and B that are sufficiently far apart recrossings of the phase space hypersurface separating the overall states are essentially eliminated and the reaction rate constant kAB can be identified with the slope of C t) at time 0 ... 

The phase space trajectory r (Z), p (Z) is uniquely determined by the initial conditions r (Z = 0) = r p (Z = 0) = p. There are therefore no probabilistic issues in the time evolution from Z = 0 to Z. The only uncertainty stems from the fact that our knowledge of the initial condition is probabilistic in nature. The phase space definition of the equilibrium time correlation function is therefore. 

An important property of time correlation functions is derived from the time reversal symmetry of the equations of motion. The time reversal operation, that is, inverting simultaneously the sign of time and of all momenta, reverses the direction of the system s trajectory in phase space. At the moment the transformation is made each dynamical variable is therefore transformed according to... 

These normal modes evolve independently of each other. Their classical equations of motion are Uk = —co Uk, whose general solution is given by Eqs (6.81). This bath is assumed to remain in thermal equilibrimn at all times, implying the phase space probability distribution (6.77), the thermal averages (6.78), and equilibrium time correlation functions such as (6.82). The quantum analogs of these relationships were discussed in Section 6.5.3. 

Compared to crystalline materials, the production and handling of amorphous substances are subject to serious complexities. Whereas the formation of crystalline materials can be described in terms of the phase rule, and solid-solid transformations (polymorphism) are well characterised in terms of pressure and temperature, this is not the case for glassy preparations that, in terms of phase behaviour, are classified as unstable . Their apparent stability derives from their very slow relaxations towards equilibrium states. Furthermore, where crystal structures are described by atomic or ionic coordinates in space, that which is not possible for amorphous materials, by definition, lack long-range order. Structurally, therefore, positions and orientations of molecules in a glass can only be described in terms of atomic or molecular distribution functions, which change over time the rates of such changes are defined by time correlation functions (relaxation times). 

In Paper I, general imaginary-time correlation functions were expressed in terms of an averaging over the coordinate-space centroid density p (qj and the centroid-constrained imaginary-time-position correlation function Q(t, qj. This formalism was extended in Paper III to the phase-space centroid picture so that the momentum could be treated as an independent variable. The final result for a general imaginary-time correlation function is found to be given approximately by [5,59]... 

In the early papers [4,8], the development of the CMD method was guided in part by the effective harmonic analysis and, in part, by physical reasoning. In Paper III, however, a mathematical justification of CMD was provided. In the latter analysis, it was shown that (1) CMD always yields a mathematically well-defined approximation to the quantum Kubo-transformed position or velocity correlation function, and (2) the equilibrium path centroid variable occupies an important role in the time correlation function because of the nature of the preaveraging procedure in CMD. Critical to the analysis of CMD and its justification was the phase-space centroid density formulation of Paper III, so that the momentum could be treated as an independent dynamical variable. The relationship between the centroid correlation function and the Kubo-transformed position correlation function was found to be unique if the centroid is taken as a dynamical variable. The analysis of Paper III will now be reviewed. For notational simplicity, the equations are restricted to a two-dimensional phase space, but they can readily be generalized. 

Here the symbol )c denotes the centroid-constrained average with the phase-space imaginary time path integral and /= -dV/dq. With the centroid trajectories in hand from the equations above, the CMD time correlation function is given by [4-6, 8]... 

The cumulant expansion with CMD approach for general correlation functions can be extended to calculate correlation functions having momentum-dependent operators via the phase-space perspective of Paper III. Only the final expressions will be given here, so the reader is referred to the original paper for the details [5]. The approximate result for the real-time correlation function C g(t) in this approach is given by [5]... 

G. J. Martyna, J. Chem. Phys. (in press, 1996). In this paper, an effective set of molecular dynamics equations are specified that provide an alternative path-integral approach to the calculation of position and velocity time correlation functions. This approach is essentially based on the Wigner phase-space function. For general nonlinear systems, the appropriate MD mass in this approach is not the physical mass, but it must instead be a position-dependent effective mass. 

The most prominent feature of this auto-correlation function is that it does not show the exponential decay as one expects from single-phase experiments (3). The curve is divided into two parts, reflecting two mechanisms influencing the signal. The decrease at higher time displacements reflects the correlation within the velocity data of the liquid bulk, and from the steep decrease at small time displacements we believe that it is due to the bubbles influence on the velocity signal. One reason for this assumption is that the time at which the turn of the curve occurs corresponds well with the time needed by bubbles of mean diameter to pass the probe with mean velocity. Other indications result from measurements of space-time correlations. 

The description of chemical reactions as trajectories in phase space requires that the concentrations of all chemical species be measured as a function of time, something that is rarely done in reaction kinetics studies. In addition, the underlying set of reaction intennediates is often unknown and the number of these may be very large. Usually, experimental data on the time variation of the concentration of a single chemical species or a small number of species are collected. (Some experiments focus on the simultaneous measurement of the concentrations of many chemical species and correlations in such data can be used to deduce the chemical mechanism [7].)... 

There is one paradoxical subtlety here, insofar as the dimensionality one computes will usually be based on some finite-time sampling. Consequently if a system is truly ergodic but requires a very long time interval to exhibit its ergodicity, the dimensionality one obtains can be no better than a lower bound. We shall return to this topic later, when we examine local properties of several-body systems. Meanwhile, we just quote the results of computations of the dimensionality of the phase space for Ar3, as a function of energy [4]. The calculations were carried out by the method introduced by Grassberger and Procaccia [6,7]. Values of the dimension, specifically the correlation... 

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