Functions in the Canonical Ensemble

Introduction 88. 2. Distribution Functions in the Canonical Ensemble 88. 3, Distribution Functions in the Grand Canonical Ensemble 90. 4. Idulticomponent 

In this chapter we shall briefly summarize a method due chiefly to McMiixan and Mayer [1945] and to Kirkwood and Buff [1951] which expresses the thermodjmamic properties of a multicomponent system in terms of integrals of the radial distribution function of the different types of molecular pairs present in the mixture. ) 

From the definition of the canonical ensemble (cf. Ch. II 2) it follows immediately that the probability of finding the N particles of the systemin the configuration element dii dr. .. drir about ri... rj is 

Ttie probability density (ri... r ) refers to a configuration in which molecule 1 is at ri, 2 at 12. .. A at r. Let us now consider the probability density corresponding to confignrations in whidi any set of h molecules arbitrarily chosen among the N molecules of the system are at the points ri.. r. Let us call (ri — r ) the probability density so defined. Since h objects may be chosen from N objects in N I N — h) ways we have simply 

The P are often called specific distribution functions and the generic distribution functions. From the definitions (5.2.2) and (5.2.3) results the recurrence relation 

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Hiroike K 1972 Long-range correlations of the distribution functions in the canonical ensemble J. Phys. Soc. Japan 32 904... 

The JE (Eq. (40)) indicates a way to recover free energy differences by measuring the work along all possible paths that start from an equihbrium state. Its mathematical form reminds one of the partition function in the canonical ensemble used to compute free energies in statistical mechanics. The formulas for the two cases are... 

Abstract The theoretical basis for the quantum time evolution of path integral centroid variables is described, as weU as the motivation for using these variables to study condensed phase quantum dynamics. The equihbrium centroid distribution is shown to be a well-defined distribution function in the canonical ensemble. A quantum mechanical quasi-density operator (QDO) can then be associated with each value of the distribution so that, upon the application of rigorous quantum mechanics, it can be used to provide an exact definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables can thus be defined and explored. Importantly, this perspective shows that the centroid constraint on the imaginary time paths introduces a non-stationarity in the equihbrium ensemble. This, in turn, can be proven to yield information on the correlations of spontaneous dynamical fluctuations. This exact formalism also leads to a derivation of Centroid Molecular Dynamics, as well as the basis for systematic improvements of that theory. 

Because the free energy is a thermodynamic state function, an alternative solution to the problem of rotational isomeric states of similar potential energy can be achieved. - For a system a with Hamiltonian 3 ", the fundamental thermodynamic state function in the canonical ensemble is the Helmholtz free energy. 

As a first step in the analysis of the thermodynamic Ijehavior of this system, we calculate the partition function in the canonical ensemble and in the position representation. A matrix element of the canonical density in the position representation is given by... 

While the partition function in the canonical ensemble is given by eq. (7.1), in the grand-canonical ensemble it is... 

The starting point for a theoretical description is the partition function in the canonical ensemble. Specifically, we consider wa polymers of species A and macromolecules of species B in a volume, V, at temperature, T. Both species of the symmetric blend are comprised of the same number, N = Na = Nb, of effective segments. 

The definition of the distribution function given above is valid in the canonical ensemble. This means that N is finite. Of course, N will, in general, be very large. Hence, g(ri,..., r/,) approaches 1 when aU the molecules are far apart but there is a term of order X/N that sometimes must be considered. This problem can be avoided by using the grand canonical ensemble. We will not pursue this point here but do wish to point it out. 

In the canonical ensemble (P2) = 3kBTM and p M. In the microcanonical ensemble (P2) = 3kgT i = 3kBTMNm/(M + Nm) [49]. If the limit M —> oo is first taken in the calculation of the force autocorrelation function, then p = Nm and the projected and unprojected force correlations are the same in the thermodynamic limit. Since MD simulations are carried out at finite N, the study of the N (and M) dependence of (u(t) and the estimate of the friction coefficient from either the decay of the momentum or force correlation functions is of interest. Molecular dynamics simulations of the momentum and force autocorrelation functions as a function of N have been carried out [49, 50]. 

This equation forms the fundamental connection between thermodynamics and statistical mechanics in the canonical ensemble, from which it follows that calculating A is equivalent to estimating the value of Q. In general, evaluating Q is a very difficult undertaking. In both experiments and calculations, however, we are interested in free energy differences, AA, between two systems or states of a system, say 0 and 1, described by the partition functions Qo and (), respectively - the arguments N, V., T have been dropped to simplify the notation ... 

As mentioned above, there are multiple ways to derive the PDT for the chemical potential. Here we utilize the older method in the canonical ensemble which says that 3/j,0 is just minus the logarithm of the ratio of two partition functions, one for the system with the distinguished atom or molecule present, and the other for the system with no solute. Using (11.7) we obtain [9, 48,49]... 

Just as there is a fundamental function that characterizes the microscopic system in quantum mechanics, i.e., the wave function, so too in statistical mechanics there is a fundamental function having equivalent status, and this is called the partition function. For the canonical ensemble, it is written as... 

Monte Carlo simulations are performed within a statistical ensemble. In the canonical ensemble (with the number of molecules, volume, and temperature fixed), the average value of a thermodynamic quantity, (T(x)), as a function of the states of system, x, is given by... 

Statistical thermodynamics has defined, in addition to the particle partition function z, the canonical ensemble partition function Zas follows ... 

An alternate procedure to calculate the distribution of an elastic membrane in a harmonic potential has as starting point the direct integration of the partition function of the canonical ensemble in Fourier space21... 

In the past few years, development of new theories have led to completely new ways of determining free energy changes. Traditionally, the difference in the free energy of two equilibrium state is (AFi 2) and the free energy change of a process can be obtained directly from the statistical mechanical definition of the free energy, F, in terms of the partition function. For the canonical ensemble F = —k T In J = —ksTln Z, where ka is Boltzmann s constant, //(F) is the phase... 

One also finds that fixing the director generates a new equilibrium ensemble where the Green-Kubo relations for the viscosities are considerably simpler compared to the conventional canonical ensemble. They become linear functions of time correlation function integrals instead of rational functions. The reason for this is that all the thermodynamic forces are constants of motion and all the thermodynamic fluxes are zero mean fluctuating phase functions in the constrained ensemble. 

The thermodynamic potential of the canonical ensemble, the Helmholtz free energy, is the first thermodynamic potential g=F, which is a function of the variables of state u 1 = T, x2=V, x3=N, and x4=z. It is obtained from the fundamental thermodynamic potential / =E (the energy) by the Legendre transform (Eq. (7)), exchanging the variable of state x1 =S of the fundamental thermodynamic potential with its conjugate variable u 1 = / . In the canonical ensemble, the first partial derivatives (Eq. (1)) of the fundamental thermodynamic potential are defined asu2=-p, u3=p, and u 4 = - S. The entropy (Eq. (46)) for the Tsallis and Boltzmann-Gibbs statistics in the canonical ensemble can be rewritten as... 

Rates in the canonical ensemble, that is, as a function of temperature rather than energy, can be obtained from Eq. (1.42) by Boltzmann weighting ... 

Barker [92-94] has presented a general formulation of the cell theory and we give a brief review of his approach here. We will restrict our discussion to single-component atomic solids and discuss the application to mixtures and nonspherical molecules later. Suppose we have a system of N molecules in the canonical ensemble. The configurational partition function, Eq. (2.205), may be rewritten by breaking the volume into N identical subvolumes or cells so that... 

The first of these difficulties can be avoided for symmetrical polymer mixtures (Na = Nb = N) by working in the semigrandcanonical ensemble of the polymer mixture [107] rather than keeping the volume fractions < )A, B and hence the numbers of chains nA, nB individually fixed, as one would do in experiment and in the canonical ensemble of statistical thermodynamics, we keep the chemical potential difference Ap = pA — pB between the two types of monomers fixed as the given independent variable. While the total volume fraction 1 — < )v taken by monomers is held constant, the volume fractions < )A, B of each species fluctuate and are not known beforehand, but rather are an output of the simulation. Thus in addition to the moves necessary to equilibrate the coil configuration (Fig. 16, upper part), one allows for moves where an A-chain is taken out of the system and replaced by a B-chain or vice versa. Note that for the symmetrical polymer mixture the term representing the contributions of the chemical potentials pA, pB to the grand-canonical partition function Z... 

In Section 4 we focused on the implementation of the MD method for the microcanonical (N, V, E) ensemble. Other ensembles are distinguished by the use of different independent variables which function as the control parameters during the simulation. In the canonical ensemble the independent variables are (N, V, T). The calculated value of the energy E is the same for both ensembles in the thermodynamic limit N - oo at constant Nj V. However, different formulae may apply for thermodynamic properties that are derivatives of a thermodynamic potential (Allen and Tildesley, 1987). 

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