Generalized canonical ensemble

To construct an ensemble conjugate to the MD ensemble, we have to introduce a second parameter which gives 3 a nonzero average value. This is the generalized canonical ensemble, whose statistical weight function is... 

The generalized canonical ensemble was previously studied by Grad, and by Lado. The discussion here is based on the paper of Wallace and Straub. 

Thus when b > 0, the generalized-canonical ensemble reduces to the ordinary-canonical ensemble. The generalized-canonical average of A Is , which we will abbreviate by A ... 

It is now possible to construct a generalized canonical ensemble from a collection of MD ensembles. The MD statistical... 

Since PV is of order N, the correction term in (68) is of relative order. In the generalized canonical ensemble, fluctuation averages are related to thermodynamic functions, as in (48) and (49). Such results, together with the fluctuation correction equation (64), can be used to evaluate fluctuation averages in the MD ensemble. Two results of interest are. 

The resulting expression for the distribution function of the generalized canonical ensemble reads... 

Mavrantzas, V.G. and Ottinger, H.C. (2002) Atomistic Monte Carlo simulations of polymer melt elasticity their nonequilibrium thermodynamics GENERIC formulation in a generalized canonical ensemble. Macromolecules, 35, 960-975. 

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)... 

Since the averaging operator is not normalized and in general (1), 1 for g 7 1, it is necessary to compute Zq to determine the average. To avoid this difficulty, we employ a different generalization of the canonical ensemble average... 

Consider the generalized distribution Pq(r ) to be generated in the Gibbs-Boltzmann canonical ensemble (9 = 1) by an effective potential W,(r /3) which is defined... 

If a confined fluid is thermodynamically open to a bulk reservoir, its exposure to a shear strain generally gives rise to an apparent multiplicity of microstates all compatible with a unique macrostate of the fluid. To illustrate the associated problem, consider the normal stress which can be computed for various substrate separations in grand canonical ensemble Monte Carlo simulations. A typical curve, plotted in Fig. 16, shows the oscillatory decay discussed in Sec. IV A 2. Suppose that instead... 

The definition of the distribution function given above is valid in the canonical ensemble. This means that N is finite. Of course, N will, in general, be very large. Hence, g(ri,..., r/,) approaches 1 when aU the molecules are far apart but there is a term of order X/N that sometimes must be considered. This problem can be avoided by using the grand canonical ensemble. We will not pursue this point here but do wish to point it out. 

This equation forms the fundamental connection between thermodynamics and statistical mechanics in the canonical ensemble, from which it follows that calculating A is equivalent to estimating the value of Q. In general, evaluating Q is a very difficult undertaking. In both experiments and calculations, however, we are interested in free energy differences, AA, between two systems or states of a system, say 0 and 1, described by the partition functions Qo and (), respectively - the arguments N, V., T have been dropped to simplify the notation ... 

In previous work on enhanced configurational sampling [154, 155] it was conjectured that a method based on the Tsallis generalization of the canonical ensemble [156] would be expected to have faster convergence with P, and an application was produced to prove this conjecture [34], Here we present a pedagogical outline of the ideas behind this approach. 

As we have seen in Sect. 8.4.2, in the Tsallis generalization of the canonical ensemble [31], the probability density that the system is at position x is... 

Because of the relationship between the different canonical ensembles the generalized partition function is best defined as the generalization of the... 

In the preceding section we have set up the canonical ensemble partition function (independent variables N, V, T). This is a necessary step whether one decides to use the canonical ensemble itself or some other ensemble such as the grand canonical ensemble (p, V, T), the constant pressure canonical ensemble (N, P, T), the generalized ensemble of Hill33 (p, P, T), or some form of constant pressure ensemble like those described by Hill34 in which either a system of the ensemble is open with respect to some but not all of the chemical components or the system is open with respect to all components but the total number of atoms is specified as constant for each system of the ensemble. We now consider briefly the selection of the most convenient formalism for the present problem. 

Exercise. The transition from the limited domain (1.1) to the full domain with symmetric Qs is especially convenient, if not indispensable, for generalizing the description to random dots on a plane or in space. Write explicitly the functions Qs for the grand-canonical ensemble of an ideal gas in a fixed volume. 

The argument can easily be generalized to yield Des Cloizeanx s result for h 0, m = 0 the Landau-Ginzburg model is equivalent to a grand canonical ensemble of polymer chains specified by a specific choice of the chemical potential i ip(n) for chains of length n... 

Three of the eight thermodynamic potentials for a system with one species are frequently used in statistical mechanics (McQuarrie, 2000), and there are generally accepted symbols for the corresponding partition functions V[T = A = — RTlnQ, where Q is the canonical ensemble partition function ... 

We have derived a formula for the molecular partition function by considering a system containing many molecules at equilibrium with a heat bath. We can generalize our statistical mechanics by a gedanken experiment of considering a large number of identical systems, each with volume V and number of particles N at equilibrium with the heat bath at temperature T. Such a supersystem is called a canonical ensemble. Our derivation is the same the fraction of systems that are in a state with energy Et is... 

According to (XII) it is first of all plausible that in general the average (Eq. 62) over the canonical ensemble will be very nearly identical with the average value taken over a microcanonical or even ergodic ensemble with E=E0> In fact, in that case also Eq. (57), for example, coincides with a relationship derived by Boltzmann (1871) for ergodic ensembles.182 Furthermore, the micro-canonical ensemble is very nearly equivalent to an ensemble that is distributed (cf. Section 12c) with constant density over the shell in T-space belonging to... 

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