From elementary classical statistical mechanics for the canonical ensemble (constant NVT), we can relate the free energy G of any system to an integral of the Boltzmann factor over coordinate q) and momentum (p) phase spaces [Pg.1037]

Under very general conditions, it follows from classical statistical mechanics that the equilibrium behavior of our fluid system is adequately described % the behavior of a Gibbskn ensemble of systems characterized by a canonical distribution (in energy) in phase space. This has two immediate consequences. First it specifies the spatial distribution of our N molecule system. The simultaneous probability that some first molecule center hes in the volume element dr whose center is at and etc., and the Nih molecule center lies in the volume element dr f whose center is at is [Pg.232]

One must distinguish between this situation and the statistical theories described under the general heading of statistical mechanics in classical statistical mechanics, the laws which govern the evolution of the system are extremely well known, but the path is so complex that it cannot be followed in detail. Here, the number of states is large, but finite, and the randomness arises from an ensemble of possible Hamiltonians. [Pg.373]

There are numerous more advanced theories of transport coefficients in liquids, mostly based on nonequilibrium classical statistical mechanics. Some are based on approximate representations of the time-dependent reduced distribution function and others are based on the analysis of time correlation functions, which are ensemble averages of the product of a quantity evaluated at time 0 and the same quantity or a different quantity evaluated at time t For example, the self-diffusion coefficient of a monatomic liquid is given by " [Pg.1193]

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density [Pg.435]

Let us consider an ensemble of N molecules in a fixed volume V with a fixed total energy E. This is a microcanonical ensemble of classical statistical mechanics. Typical values for N used in these simulations of chemical interest is of the order of hundreds to a few thousands. In order to simulate an infinite system, periodic boundary conditions are invariably imposed. Thus a typical MD system would consist of N molecules enclosed in a cubic box with each side equal to length L. MD solves the equations of motion for a molecule i [Pg.96]

We consider only the equilibrium case so that the distribution of these points phase space is time-independent. In quantum statistical mechanics, we had a discrete list of possible states. In classical statistical mechanics, we have coordinates and momentum components that can range continuously. We denote the probability disttibution (probability density) for the ensemble by / and define the probability that the phase point of a randomly selected system of the ensemble will lie in the 6A -dimensional volume element d tNci pi to be [Pg.1134]

Is the temperature 1/0 related to the variance of the momentum distribution as in the classical equipartition theorem It happens that there is no simple generalization of the equipartition theorem of classical statistical mechanics. For the 2N dimensional phase space F = (xi. .. XN,pi,.. -Pn) the ensemble average for a harmonic system is [Pg.199]

The series of studies of molecular liquids presented herein collect results on a diverse set of chemically relevant systems from a uniform theoretical point of view ab initio classical statistical mechanics on the (T,V,N) ensemble with potential functions representative of ab initio quantum mechanical calculations of pairwise interactions and structural analysis carried out in terms of quasicomponent distribution functions. The level of agreement between calculated and observed quantities is quoted to indicate the capabilities and limitations to be expected of these calculations and in that perspective we find a number of structural features of the systems previously discussed on [Pg.214]

Other ensembles, notably NVT (canonical), NPE, and NPT, are also used routinely. Since energy is constantly flowing back and forth from potential to kinetic terms in any MD simulation, it is necessary to regularly readjust the kinetic energy in the ensembles that hold the temperature constant. This is required because the temperature is deflned by the classical statistical mechanical equation [Pg.4802]

No system is exactly unifomi even a crystal lattice will have fluctuations in density, and even the Ising model must pemiit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothemial compressibility K j,and the number of molecules Ain volume V [Pg.647]

Variational Monte Carlo (or VMC, as it is now commonly called) is a method that allows one to calculate quantum expectation values given a trial wavefunction [1,2]. The actual Monte Carlo methodology used for this is almost identical to the usual classical Monte Carlo methods, particularly those of statistical mechanics. Nevertheless, quantum behavior can be studied with this technique. The key idea, as in classical statistical mechanics, is the ability to write the desired property <0> of a system as an average over an ensemble [Pg.38]

A t) ical Anneal-Flex run on a molecule such as the vitamin D3 ketone 1 consists of 20 runs of 1000 steps per temperature at 30 temperatures. Since the acceptance rate is usually around 30%, there are about 180,000 accepted steps or 9,000 lines of data for each 20-run file. In classical statistical mechanics, one Anneal-Flex run can be considered as one member of an ensemble [30]. The collection of twenty runs is the ensemble. In this type of formulation, the numerical value of the quantity of interest is obtained by calculating averages over this ensemble. While the quantities that we are interested in are too complicated to be represented by a single number, the same statistical mechanical principles can be used to create the distribution functions which accurately represent dihedral space. [Pg.360]

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p. s). This corresponds to a set of systems separated by diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function [Pg.375]

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