The Canonical Distribution

While the equilibrium kinetic theory permits us to develop in fairly simple manner the properties of a dilute hard sphere gas, it becomes progressively more complicated and difficult to apply to both dense systems and systems in which there are forces acting between particles. To deal with such systems, we shall here outline briefly the very powerful statistical methods of Gibbs.  

From the classical mechanical point of view discussed in Sec. VI. 1, any system of N particles is uniquely defined by a knowledge of 6N independent pieces of information together with the description of the components of the system (masses, force fields, etc.). These QN quantities may be looked upon as the QN constants of integration implicit in Newton s differential equations of motion. 

We have seen that a statistical description of such a system is possible if we can express its average behavior in terms of a distribution function. Such a distribution function will in general be a function of all the individual variables needed to define the molecular system and also of the time. When the system is in a state of equilibrium, the distribution function which describes it is no longer a function of time but only of the molecular variables. 

In that case the distribution function cannot be any arbitrary function of the variables but only a function of the combinations of variables that allow it to be independent of time. Such combinations are called invariants of the system, and for any complex system only seven are known the three components of total linear momentum, the three components of total angular momentum, and the total energy H. If we select a system at equilibrium and not in motion with respect to some set of fixed axes, only the total energy H remains as an invariant of interest. 

If we select the position coordinates (total 3AT) and the components of momentum (total N) of each particle in the system as our variables, we can write the total energy H as  

As a first example, we consider the problem in which the subsystem and reservoir are free to exchange only energy, as is indicated schematically in fig. 3.19. In this case, the number of microstates available to the subsystem is Q,s(Es), where the subscript x refers to the subsystem. Similarly, the number of microstates available to the reservoir is given by Qr(E — Es). From the perspective of the composite universe made up of the system and its reservoir, the total energy is 

Within the microcanonical formalism presented in the previous section we asserted that all microstates are equally likely. In the present situation, we center our interest just on the subsystem. The price we pay for restricting our interest to the nonisolated subsystem is that we no longer find all states are of equal probability. Though the states of the composite universe are all equally likely, we will find fhat there is a calculable energy dependence to the probability for a given energy state for the subsystem. 

This equation notes that the ratio of the probabilities of finding the subsystem with two particular energies is equal to the ratios of the corresponding number of states available to the reservoir at the two energies of interest. The remainder of our analysis can be built around the presumed smallness of Ea with respect to E itself, which affords an opportunity for Taylor expansion of the slowly varying logarithms in the form 

Coupled with the thermodynamic identity that relates the temperature to the energy dependence of the entropy, namely, 

This expression states that the partition function itself can be determined from 

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The above derivation leads to the identification of the canonical ensemble density distribution. More generally, consider a system with volume V andA particles of type A, particles of type B, etc., such that N = Nj + Ag +. . ., and let the system be in themial equilibrium with a much larger heat reservoir at temperature T. Then if fis tlie system Hamiltonian, the canonical distribution is (quantum mechanically)... 

The canonical distribution corresponds to the probability density for the system to be in a specific microstate with energy E- H, from it one can also obtain the probability P( ) that the system has an energy between E and E + AE i the density of states D E) is known. This is because, classically. 

As an alternative to sampling the canonical distribution, it is possible to devise equations of motion for which the iiiechanicaT temperature is constrained to a constant value [84, 85, 86]. The equations of motion are... 

Given a size N lattice (thought of now as a heat-bath), consider some subsystem of size n. An interesting question is whether the energy distribution of the subsystem, Pn E), is equal to the canonical distribution of a thermodynamic system in equilibrium. That is, we are interested in comparing the actual energy distribution... 

This section is used to introduce the momentum-enhanced hybrid Monte Carlo (MEHMC) method that in principle converges to the canonical distribution. This ad hoc method uses averaged momenta to bias the initial choice of momenta at each step in a hybrid Monte Carlo (HMC) procedure. Because these average momenta are associated with essential degrees of freedom, conformation space is sampled effectively. The relationship of the method to other enhanced sampling algorithms is discussed. 

For a system with many degrees of freedom the canonical distribution of energy closely approximates the microcanonical distribution about the most... 

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. 

Another way of looking at the Boltzmann distribution assumes that the energy spectrum consists of closely spaced, but fixed energy levels, e . The probability that level u is populated is specified in terms of the canonical distribution... 

In view of the ergodic hypothesis the average value of an observable property may be regarded as the quantity measured under specified conditions. In this way the internal energy of a system corresponds to the average energy of the canonical distribution ... 

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... 

This dynamical formulation of the equilibrium correlations in an interacting system will be the starting point of our analysis of equilibrium electrolytes. Of course, this method gives results analogous to the more usual methods based on the canonical distribution 40... 

The corresponding formulation was made by von Neumann2 for quantum mechanics. This entropy-maximizing (or information-minimizing) principle is the most direct path to the canonical distribution and thus to the whole equilibrium theory. It is understood that the extremalizing is conditional, i.e., certain expected values, such as that of the Hamiltonian, are fixed. 

If these ideas are combined with the concept of a stable distribution, i.e., one which is but minimally altered by any slight changes at the boundary, one is led to seek the S for which these effects have operated to their fullest extent, i.e., to the one which minimizes /[J ] among all those that have the same total expected mass, momentum, and energy. This of course is the method of obtaining the canonical distribution noted in Section I. 

As the next stage beyond the zero fluxes and stability underlying the canonical distribution, we consider more general initial (t = 0) basic macroscopic data (p, uE) and their basic macroscopic fluxes (t 0) on the boundary of Y, but under restrictions which may be described intuitively as follows Y is of reasonable shape and proportions—not much more irregular than a potato. The initial data (p, ux, E) vary slowly (percentagewise) with (x). On the boundary of Y the flux of p is zero those of ua and E are piecewise slowly varying (percentagewise) with position and time. 

This defines the macrocanonical ensemble the canonical distribution can be derived from it although it is often postulated as an a priori distribution in its own right. 

The extraction of a homogeneous process from a stationary Markov process is a familiar procedure in the theory of linear response. As an example take a sample of a paramagnetic material placed in a constant external magnetic field B. The magnetization Y in the direction of the field is a stationary stochastic process with a macroscopic average value and small fluctuations around it. For the moment we assume that it is a Markov process. The function Px (y) is given by the canonical distribution... 

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. 

A and B (and thus of the controlling Hamiltonian ) at a point in w -space, averaged over that space. In the numerator this switch is from S h to A and the average is with respect to the canonical distribution in ensemble B in the denominator the roles of A and B are reversed. This is Bennett s acceptance ratio formula [61]. 

Thus, from the point of view of Boltzmann s presentation, the introduction of the canonical distribution seems to be an analytical trick reminiscent of Dirichlet s discontinuous factor. 183 In the calculation of the average value f(q, p) the integrations (see Eq. 56) always remain extended over the infinite T-space. However, if the modulus and the parameters r , , rm are chosen in the proper way, the decisive majority of all gas models will lie in those parts of T-space given by that M.-B. 

A. Wassmuth (1908) shows that among all distributions of the form p=F(E) only the canonical distribution satisfies the following requirement Let us consider only those (7-points of the ensemble which give a certain definite configuration (gi, , gf) to the molecules of the gas model for arbitrary values of the velocities. Now let us form for these particles the average of the square of a momentoid (see note 179). We require that this average... 

For obtaining the canonical distributions, the multiple-histogram reweighting techniques [9,10] are particularly suitable. Suppose we have made a single... 

One way to arrive at the canonical distribution is via maximizing the number W under the constraints imposed by Equations (12.28) and (12.29). At the maximum value,... 

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