Canonical ensembles thermodynamic functions

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 ... 

Statistical thermodynamics has defined, in addition to the particle partition function z, the canonical ensemble partition function Zas follows ... 

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q, which turns out to be linked to matrix elements of the Hamiltonian operator H in Eq. (2.39). However, the systems treated below exist in a region of thermodjniamic state space where the exact quantum mechanical treatment may be abandoned in favor of a classic dc.scription. The transition from quantum to classic statistics was worked out by Kirkwood [22, 23] and Wigner [24] and is rarely discussed in standard texts on statistical physics. For the sake of completeness, self-containment, and as background information for the interested readers we summarize the key considerations in this section. 

Thermodynamics of ideal quantum gases is typically obtained using a grand canonical ensemble. In principle this can also be done using a canonical ensemble partition function, Q =, exp(-p E ). For the photon and... 

Statistical mechanics may be used to derive practical microscopic formulae for thermodynamic quantities. A well-known example is the virial expression for the pressure, easily derived by scaling the atomic coordinates in the canonical ensemble partition function... 

This quantity q is called the partition function. It plays a central role in statistical thermodynamics. Because we defined our system as a canonical ensemble, q is commonly called the canonical ensemble partition function. 

A system of N particles under thermodynamic constraints of constant volume and temperature is described by the canonical ensemble partition function Q. In the classical limit for a three-dimensional system Q is given by... 

The derivation of this relationship from the canonical ensemble partition function is straightforward. It is given here to illustrate the type of partition function manipulations commonly used in developing simulation expressions for thermodynamic quantities. The excess chemical potential is defined as the Helmholtz free energy difference between two (N + l)-particle systems, one... 

The ideal gas law and the thermodynamic properties of an ideal gas are completely derived from the canonical ensemble partition function. This is a remarkable illustration of how statistical mechanics explains macroscopic observables in terms of microscopic properties. 

Thus far, we have only introduced systems of monoatomic, or point mass, particles. In this chapter we present a statistical mechanical derivation of polyatomic system thermodynamics. We focus on diatomic molecules in an ideal gas phase and we use the canonical ensemble partition function. The reason this ensemble is chosen is the ease with which the integration of the Boltzmann factor can be performed over the entire phase space. All the ensembles of molecular systems are again equivalent at the thermodynamic limit. 

To obtain thermodynamic averages over a canonical ensemble, which is characterized by the macroscopic variables (N, V, T), it is necessary to know the probability of finding the system at each and every point (= state) in phase space. This probability distribution, p(r, p), is given by the Boltzmann distribution function. 

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]. 

The Helmholtz free energy, A, which is the thermodynamic potential, the natural independent variables of which are those of the canonical ensemble, can be expressed in terms of the partition function ... 

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 ... 

We can, therefore, let /cx be the subject of our calculations (which we approximate via an array in the computer). Post-simulation, we desire to examine the joint probability distribution p(N, U) at normal thermodynamic conditions. The reweighting ensemble which is appropriate to fluctuations in N and U is the grand-canonical ensemble consequently, we must specify a chemical potential and temperature to determine p. Assuming -7CX has converged upon the true function In f2ex, the state probabilities are given by... 

Equation (51) again has the form of a thermodynamic differential equation for a characteristic function and may be used to identify the thermodynamic analogues of the grand canonical ensemble. 

Use of the Grand Ensemble has the disadvantage that all the calculated thermodynamic functions are dependent on z, V, and T. However, Mayer s programme can be adapted to the canonical ensemble (N, V, T). 

Many thermodynamic functions can be derived from the partition function of the canonical ensemble by a weighted average, or by differentiation of the partition function. For instance, the average energy of the ensemble can be given by a weighted... 

Monte Carlo simulations are performed within a statistical ensemble. In the canonical ensemble (with the number of molecules, volume, and temperature fixed), the average value of a thermodynamic quantity, (T(x)), as a function of the states of system, x, is given by... 

The organization of this chapter is as follows. In Sect. 5.1 we present the basic formalism and work out the Feynman rules for the grand canonical ensemble. Diagrammatic representations valid in the thermodynamic limit are derived for both thermodjmamic quantities and correlation functions. The proof of the Linked Cluster Theorem is given in Appendix A 5.1. Section 5.2... 

Briefly, we recall some basic definitions involving the short-order structural functions typical of the liquid state and their relationships with thermodynamic quantities. Considering a homogenous fluid of N particles, enclosed in a definite volume V at a given temperature T (canonical ensemble), the two-particles distribution function [7, 9, 17, 18] is defined as... 

This statistical mechanical expression for surface tension depends explicitly on the potentials of inteimolecular force and molecular distribution functions. Upon recognition that the two-phase system under consideration is thermodynamically open, it follows that the distribution functions must be represented in the grand canonical ensemble. Thus, the dependence of y on temperature, T, and chemical potentials, enters through the implicit dependence of the distribution func-... 

One also finds that fixing the director generates a new equilibrium ensemble where the Green-Kubo relations for the viscosities are considerably simpler compared to the conventional canonical ensemble. They become linear functions of time correlation function integrals instead of rational functions. The reason for this is that all the thermodynamic forces are constants of motion and all the thermodynamic fluxes are zero mean fluctuating phase functions in the constrained ensemble. 

The microcanonical ensemble is characterized by fixed values of the thermodynamic variables N, the total particle number, V, the volume of the system, and E, the total energy. The phase space distribution function for the micro-canonical ensemble is... 

In real situations surface and volume changes are often made with systems that are at equilibrium with their environment, characterized by a set of chemical potentials p, rather than keeping In ] fixed, as in [2.2.7 and 8j. In other words, area changes in open systems are considered. In statistical thermodynamics the conversion from closed to open implies the transition from the canonical to the grand canonical ensemble. The characteristic function of the latter is nothing other than the sum of the bulk and surface mechemical work terms (see [1.3.3.12] and [I.A6.23D which are the quantities of interest ... 

The Kirkwood—Buff (KB) theory of solution (often called fluctuation theory) employs the grand canonical ensemble to relate macroscopic properties, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volnmes, to microscopic properties in the form of spatial integrals involving the radial distribution function. This theory allows one to obtain information regarding some microscopic characteristics of mnlti-component mixtures from measurable macroscopic thermodynamic quantities. However, despite its attractiveness, the KB theory was rarely used in the first three decades after its publication for two main reasons (1) the lack of precise data (in particular regarding the composition dependence of the chemical potentials) and (2) the difficulty to interpret the results obtained. Only after Ben-Naim indicated how to calculate numerically the Kirkwood—Buff integrals (KBIs) for binary systems was this theory used more frequently. 

Each individual pore has a fixed geometry, and is open and in contact with bulk gas at a fixed temperature. For this system, the grand canonical ensemble provides the appropriate description of the thermodynamics. In this ensemble, the chemical potential temperature T, and pore volume V are specified. In the presence of a spatially varying external potential Vea, the grand potential functional Qoithe fluid is [11]... 

The thermodynamic potential of the canonical ensemble, the Helmholtz free energy, is the first thermodynamic potential g=F, which is a function of the variables of state u 1 = T, x2=V, x3=N, and x4=z. It is obtained from the fundamental thermodynamic potential / =E (the energy) by the Legendre transform (Eq. (7)), exchanging the variable of state x1 =S of the fundamental thermodynamic potential with its conjugate variable u 1 = / . In the canonical ensemble, the first partial derivatives (Eq. (1)) of the fundamental thermodynamic potential are defined asu2=-p, u3=p, and u 4 = - S. The entropy (Eq. (46)) for the Tsallis and Boltzmann-Gibbs statistics in the canonical ensemble can be rewritten as... 

A system that is constrained to have a constant number of molecules, volume, and temperature constitutes a canonical ensemble. The thermodynamic properties of the system can be calculated from the corresponding partition function, Q N, V, T) [24]. For the adsorbed phase the partition function can be written as... 

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