Ensembles classical statistical mechanics

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... 

In addition to the study of atomic motion during chemical reactions, the molecular dynamics technique has been widely used to study the classical statistical mechanics of well-defined systems. Within this application considerable progress has been made in introducing constraints into the equations of motion so that a variety of ensembles may be studied. For example, classical equations of motion generate constant energy trajectories. By adding additional terms to the forces which arise from properties of the system such as the pressure and temperature, other constants of motion have been introduced. 

In the general approach to classical statistical mechanics, each particle is considered to occupy a point in phase space, i.e., to have a definite position and momentum, at a given instant. The probability that the point corresponding to a particle will fall in any small volume of the phase space is taken proportional to die volume. The probability of a specific arrangement of points is proportional to the number of ways that the total ensemble of molecules could be permuted to achieve the arrangement. When this is done, and it is further required that the number of molecules and their total energy remain constant, one can obtain a description of the most probable distribution of the molecules in phase space. Tlie Maxwell-Boltzmann distribution law results. 

Classical statistical mechanics is concerned with the probability distribution of phase points. In a classical microcanonical ensemble the phase space density is constant. Loosely speaking, aU phase points with the same energy are equally likely. In consequence the number of states of the classical system in a given energy range E to E + dE is proportional to the volume of the phase space shell defined by this energy range. 

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. 

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. 

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... 

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... 

Monte Carlo techniques are methods of estimating the values of manydimensional integrals by sampling with the help of random numbers/ It is obvious that this makes them methods appropriate to equilibrium statistical mechanics. Among the integrals of interest in classical statistical mechanics are ensemble averages of any mechanical quantity M( ),... 

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 ... 

In normal classical statistical mechanics, it is assumed that all states which are fixed by the same external constraints, e.g., total volume V, average energy < ), average particle number N), are equally probable. All possible states of the system are generated and are assigned weight unity if they are consistent with these constraints and, zero otherwise. Thus in the case of an iV-particle system with classical Hamiltonian //j, the microcanonical ensemble entropy S E) is obtained from the total number of states ( ) via the definition... 

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... 

These MC methods then are based on the Metropolis algorithm [251], by which one constructs a stochastic trajectory through the crmfiguration space (X) of the system, performing transitions W(X X ). The transition probability must be chosen such that it satisfies the detailed balance principle with the probability distribution that one wishes to study. For example, for classical statistical mechanics, the canonical ensemble distribution is given in terms of the total potential energy t/(X), where X = ( i, 2,..., iV) stands symbolically for a point in configuration space [17] ... 

Linear response theory [152] is perfectly suited to the study of fluid structures when weak fields are involved, which turns out to be the case of the elastic scattering experiments alluded to earlier. A mechanism for the relaxation of the field effect on the fluid is just the spontaneous fluctuations in the fluid, which are characterized by the equilibrium (zero field) correlation functions. Apart from the standard technique used to derive the instantaneous response, based on Fermi s golden rule (or on the first Bom approximation) [148], the functional differentiation of the partition function [153, 154] with respect to a continuous (or thermalized) external field is also utilized within this quantum context. In this regard, note that a proper ensemble to carry out functional derivatives is the grand ensemble. All of this allows one to gain deep insight into the equilibrium structures of quantum fluids, as shown in the works by Chandler and Wolynes [25], by Ceperley [28], and by the present author [35, 36]. In doing so, one can bypass the dynamics of the quantum fluid to obtain the static responses in k-space and also make unexpected and powerful connections with classical statistical mechanics [36]. 

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... 

Statistical mechanics can also be based on classical mechanics, and a brief introduction to this subject was included in the chapter, based on the canonical ensemble. Since classical states are specified by values of coordinates and momentum components, the probability distribution for classical statistical mechanics is a probability density... 

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 " ... 

While free energies can be theoretically determined in a number of ways, free energy calculations is generally understood to refer to a class of simulations that relate, through equations of classical statistical mechanics, the free energy difference between two different molecular states or conformations to a thermodynamic ensemble average that depends on potential energy properties of those states or conformations. That is,... 

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 ... 

Let be a function of coordinates only, so a point in is specified by [3(A(ab - 1) - 1] coordinates and 3(A(ab - ) momenta. Identify the missing coordinate as the reaction coordinate s (so s becomes a coordinate normal to the hypersurface), and identify the momentum conjugate to s as p. Let C denote the [6(A(ab — 1) - 2]-dimensional hyperface in in which ps = 0. Assume that the % region of phase space is populated according to a Boltzmann equilibrium distribution then Liouville s theorem of classical statistical mechanics shows it will evolve into a Boltzmann equilibrium distribution at and hence also at C. Consider the one-way flux of this equilibrium ensemble of phase points through in the 5 —> P direction. This flux may be calculated quite generally, and using this calculation plus equation (2) yields... 

In view of the formal identity of the expressions for the Gibbs entropy of quantum and classical microcanonical ensembles, the statistical mechanical expressions given by Eqs. (44)-(46) also apply to quantum systems. One need only reinterpret the quantity 2( , F, A AE) appearing in these expressions as the number of eigenstates of the system Hamiltonian H lying in the energy shell [E,E + AE]. 

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... 

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... 

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... 

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 ... 

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