Canonical distribution function

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

It is instructive to see this in temis of the canonical ensemble probability distribution function for the energy, NVT - Referring to equation B3.3.1 and equation (B3.3.2I. it is relatively easy to see that... 

To obtain thermodynamic averages over a canonical ensemble, which is characterized by the macroscopic variables (N, V, T), it is necessary to know the probability of finding the system at each and every point (= state) in phase space. This probability distribution, p(r, p), is given by the Boltzmann distribution function. 

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. 

All averages of the form (3-96) can be calculated in terms of a canonical set of averages called joint distribution functions by means of an extension of the theorem of averages proved in Section 3.3. To this end, we shall define the a order distribution function of X for time spacings rx < r2 < < by the equation,... 

This canonical ensemble is also a one-parameter distribution function for various values of 9, independent of q and p. 

Usually when closed, isothermal systems (N,V,T) are studied, the canonical distribution function is chosen ... 

Expression (21) tells us that if II is the canonical distribution function (7), given that... 

We first consider a Hamiltonian, thus deterministic, system. Denoting by oo the set of all phase space coordinates of a point in phase space (which determines the instantaneous state of the system), the motion of this point is determined by a canonical transformation evolving in time, 7), with Tq = I. The function of time TfCO thus represents the trajectory passing through co at time zero. The evolution of the distribution function is obtained by the action on p of a unitary transformation Ut, related to 7) as follows ... 

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. 

For an arbitrary canonical density operator, the phase space centroid distribution fimction is imiquely defined. However, this function does not directly contain any dynamical information from the quantum ensemble because such information has been lost in the course of the trace operation. The lost information may be recovered by associating to each value of the centroid distribution function the following normalized operator ... 

Abbreviations MD, molecular dynamics TST, transition state theory EM, energy minimization MSD, mean square displacement PFG-NMR, pulsed field gradient nuclear magnetic resonance VAF, velocity autocorrelation function RDF, radial distribution function MEP, minimum energy path MC, Monte Carlo GC-MC, grand canonical Monte Carlo CB-MC, configurational-bias Monte Carlo MM, molecular mechanics QM, quantum mechanics FLF, Hartree-Fock DFT, density functional theory BSSE, basis set superposition error DME, dimethyl ether MTG, methanol to gasoline. 

Where T is the initial phase point of the system, L is the Liouville operator, y(tf)(F) is the canonical distribution function, and Bk(T) and k(T) are the values of the classical properties Bk and iLk when the system is in the classical state T. Much work has been done to determine how the quantum-mechanical functions approach the corresponding classical functions. 

Briefly, we recall some basic definitions involving the short-order structural functions typical of the liquid state and their relationships with thermodynamic quantities. Considering a homogenous fluid of N particles, enclosed in a definite volume V at a given temperature T (canonical ensemble), the two-particles distribution function [7, 9, 17, 18] is defined as... 

This statistical mechanical expression for surface tension depends explicitly on the potentials of inteimolecular force and molecular distribution functions. Upon recognition that the two-phase system under consideration is thermodynamically open, it follows that the distribution functions must be represented in the grand canonical ensemble. Thus, the dependence of y on temperature, T, and chemical potentials, enters through the implicit dependence of the distribution func-... 

Such a distribution of systems is called a canonical ensemble, and the distribution function is If C is written in the form then it... 

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