The Microcanonical Ensemble

The microcanonical ensemble, which we have already gently introduced in a simplified setting in the previous chapter, is defined by constant number of particles N, volume V and total energy E. We assume that all systems of the ensemble evolve independently and are isolated from each other. If a Hamiltonian description is used, the first and third invariances are automatically maintained in a molecular 

The partition function Z, which normalizes the density, is effectively a function of N, V and E it represents the number of microstates available under given conditions. As this ensemble is associated to constant particle number N, volume V and energy E, it is often referred to as the NVE-ensemble, and when we speak of NVE simulation, we mean simulation that is meant to preserve the microcanonical distribution this, most often, would be based on approximating Hamiltonian dynamics, e.g. using the Verlet method or another of the methods introduced in Chaps. 2 and 3, and assuming the ergodic property. For a discussion of alternative stochastic microcanonical methods see [126]. 

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The microcanonical ensemble is a set of systems each having the same number of molecules N, the same volume V and the same energy U. In such an ensemble of isolated systems, any allowed quantum state is equally probable. In classical thennodynamics at equilibrium at constant n (or equivalently, N), V, and U, it is the entropy S that is a maximum. For the microcanonical ensemble, the entropy is directly related to the number of allowed quantum states C1(N,V,U) ... 

An explicit example of an equilibrium ensemble is the microcanonical ensemble, which describes closed systems with adiabatic walls. Such systems have constraints of fixed N, V and E < W< E + E. E is very small compared to E, and corresponds to the assumed very weak interaction of the isolated system with the surroundings. E has to be chosen such that it is larger than (Si )... 

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

In the last subsection, the microcanonical ensemble was fomuilated as an ensemble from which the equilibrium, properties of a dynamical system can be detennined by its energy alone. We used the postulate of... 

The definition of entropy and the identification of temperature made in the last subsection provides us with a coimection between the microcanonical ensemble and themiodynamics. 

For practical calculations, the microcanonical ensemble is not as useful as other ensembles corresponding to more connnonly occurring experimental situations. Such equilibrium ensembles are considered next. 

Figure A3.13.9. Probability density of a microcanonical distribution of the CH cliromophore in CHF within the multiplet with cliromophore quantum nmnber V= 6 (A. g = V+ 1 = 7). Representations in configuration space of stretching and bending (Q coordinates (see text following (equation (A3.13.62)1 and figure A3.13.10). Left-hand side typical member of the microcanonical ensemble of the multiplet with V= 6...

In the microcanonical ensemble, the signature of a first-order phase transition is the appearance of a van der Waals loop m the equation of state, now written as T(E) or P( ). The P( ) curve switches over from one... 

This section deals with the question of how to approximate the essential features of the flow for given energy E. Recall that the flow conserves energy, i.e., it maps the energy surface Pq E) = x e P H x) = E onto itself. In the language of statistical physics, we want to approximate the microcanonical ensemble. However, even for a symplectic discretization, the discrete flow / = (i/i ) does not conserve energy exactly, but only on... 

When g = 1 the extensivity of the entropy can be used to derive the Boltzmann entropy equation 5 = fc In W in the microcanonical ensemble. When g 1, it is the odd property that the generalization of the entropy Sq is not extensive that leads to the peculiar form of the probability distribution. The non-extensivity of Sq has led to speculation that Tsallis statistics may be applicable to gravitational systems where interaction length scales comparable to the system size violate the assumptions underlying Gibbs-Boltzmann statistics. [4]... 

In a canorrical ensemble the total temperature is constant. In the microcanonical ensemble, however, the temperature will fluctuate. The temperature is directly related to the kinetic energy of the system as follows ... 

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]. 

For an isolated system based on the microcanonical ensemble with phase points equally distributed in a narrow energy range, all systems exhibit the same long-term behaviour and spend the same fraction wt of time in different elementary regions. For a total number N of systems at time t,... 

For this value of the energy the exponential factor becomes a constant and the distribution a function of H only, like the microcanonical ensemble. As a matter of fact, as the number of systems in the ensemble approaches infinity, the canonical distribution becomes increasingly sharp, thus approaching a microcanonical surface ensemble. 

The microcanonical ensemble in quantum statistics describes a macroscopi-cally closed system in a state of thermodynamic equilibrium. It is assumed that the energy, number of particles and the extensive parameters are known. The Hamiltonian may be defined as... 

As indicated before (8.3.3.2) this formulation demonstrates that the canonical ensemble is made up of a large number of microcanonical ensembles. When calculating properties like the internal energy from the canonical ensemble, the values so obtained may reasonably be expected to be the same as those found from the microcanonical ensemble. Indeed, the two ensembles can be shown to be equivalent for large systems, in which case the sum... 

Differentiation of (41), using the formalism defined for the microcanonical ensemble, in terms of a generalized force operator, yields... 

Temperature effects are included explicitly in molecular dynamics simulations by including kinetic energy terms - the balls representing the atoms are now on the move The principles are simple. In the microcanonical ensemble (NVE) ... 

Figure 2). The calculations were done in the microcanonical ensemble at a temperature of 300K 5K. Energy was well conserved throughout the trajectories, and no overall drifts in molecular temperature were observed. Small ensembles of trajectories (12 for SI and 6 each for the other minima) were calculated for the averaging of system properties. Each trajectory was equilibrated by velocity reassignments during an initial period of 20ps, followed by another 20ps of dynamics used for data collection.

The lattice cluster theory (LCT) for glass formation in polymers focuses on the evaluation of the system s configurational entropy Sc T). Following Gibbs-DiMarzio theory [47, 60], Sc is defined in terms of the logarithm of the microcanonical ensemble (fixed N, V, and U) density of states 0( 7),... 

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