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** Classical statistical mechanics **

** Distribution functions statistics **

The starting point of classical statistical mechanics is the exact equation of evolution of the distribution function p in phase space the Liouville equation, which Prigogine always wrote in the form [Pg.28]

In statistical mechanics we do not measure observables directly. Instead we observe an average over all possible values. The averaging is done by means of a probability distribution function, which in classical mechanics is averaged over all of phase space. Let us compare an equilibrium ensemble with grand potential Q, and an arbitrary nearby ensemble prepared by a small perturbation, AQ. Let the equilibrium probability distribution function be f and that for the nearby [Pg.106]

The rules of classical statistical mechanics give the orientational distribution function in terms of the potential function V as [Pg.35]

Percus, J. K., The pair distribution function in classical statistical mechanics. In The Equilibrium Theory of Classical Fluids, pp. II33-II170, New York Benjamin [Pg.223]

Equation (1.24) is very similar to that of the single-particle distribution function of classical statistical mechanics. In the limit h—>0 we get the first equation of the BBGKY hierarchy. [Pg.184]

In the subsequent chapters in which we will be investigating the thermal, electrical, optical, and magnetic properties of materials, it will be necessary to be able to determine the energy distribution of electrons, holes, photons, and phonons. To do this, we need to introduce some quantum statistical mechanical concepts in order to develop the distribution fimc-tions needed for this purpose. We will develop the Bose-Einstein (B-E) distribution function that applies to all particles except electrons and holes (and other fermions) that obey the Pauli exclusion principle and show how this function becomes the Maxwell-Boltzmann (M-B) distribution in the classical limit. Also, we will show how the Planck distribution results by relaxing the requirement that particles be conserved. Next we develop the Fermi-Dirac (F-D) distribution that applies to electrons and holes and becomes the basis for imderstanding semiconductors and photonic systems. [Pg.297]

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]

Derivation of the Boltzmann distribution function is based on statistical mechanical considerations and requires use of Stirling s approximation and Lagrange s method of undetermined multipliers to arrive at the basic equation, (N,/No) = (g/go)exp[-A Ae/]. The exponential term /3 defines the temperature scale of the Boltzmann function and can be shown to equal t/ksT. In classical mechanics, this distribution is defined by giving values for the coordinates and momenta for each particle in three-coordinate space and the lin- [Pg.95]

Molecular dynamics consists of examining the time-dependent characteristics of a molecule, such as vibrational motion or Brownian motion within a classical mechanical description [13]. Molecular dynamics when applied to solvent/solute systems allow the computation of properties such as diftiision coefficients or radial distribution functions for use in statistical mechanical treatments. In this calculation a number of molecules are given some initial position and velocity. New positions are calculated a short time later based on this movement, and the process is iterated for thousands of steps in order to bring the system to an equilibrium. Next the data are Fourier transformed into the frequency domain. A given peak can be chosen and transformed back to the time domain, to see the motion at that frequency. [Pg.321]

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]

The approximate solution to the Schrodinger equation, defined by the effective Hamiltonian in Eq. (9-1), with either method described in the previous section, associates to every vector of molecule coordinates, R, together with the solvent-solvent interaction potential, an energy (R). From basic classical statistical mechanics an N-particle distribution function (PDF) n(R) is thus obtained [Pg.231]

Many of the equilibrium properties of such systems can be obtained through the two-body reduced coordinate distribution function and the radial distribution function, defined in Eqs. (27.6-5) and (27.6-7). There are a number of theories that are used to calculate approximate radial distribution functions for liquids, using classical statistical mechanics. Some of the theories involve approximate integral equations. Others are perturbation theories similar to quantum mechanical perturbation theory (see Section 19.3). These theories take a hard-sphere fluid or other fluid with purely repulsive forces as a zero-order system and consider the attractive part of the forces to be a perturbation. [Pg.1184]

For gas-phase molecules the assumption of electronic adiabaticity leads to the usual Bom-Oppenheimer approximation, in which the electronic wave function is optimized for fixed nuclei. For solutes, the situation is more complicated because there are two types of heavy-body motion, the solute nuclear coordinates, which are treated mechanically, and the solvent, which is treated statistically. The SCRF procedures correspond to optimizing the electronic wave function in the presence of fixed solute nuclei and for a statistical distribution of solvent coordinates, which in turn are in equilibrium with the average electronic structure. The treatment of the solvent as a dielectric material by the laws of classical electrostatics and the treatment of the electronic charge distribution of the solute by the square of its wave function correctly embodies the result of [Pg.64]

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]

** Classical statistical mechanics **

** Distribution functions statistics **

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