Ensembles thermodynamic functions

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... 

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 ... 

Each ensemble has "natural" thermodynamic variables, but the usual relationships between macroscopic thermodynamic functions will allow us to obtain the "other" thermodynamic state functions. 

V, N, and T are held constant. In this way, the configurational integral Z is calculated. This immediately allows one to calculate the partition function Q for the ensemble (equation (2.2.31)) and thus the thermodynamic functions for the system. In addition, the pair correlation function for the fiuid is obtained in this calculation. 

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. 

The conserved thermodynamic function for this ensemble is the generalized enthalpy... 

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... 

The language of Statistical Mechanics evolved over a considerable period of time. For example, the term "ensemble" is used to denote a statistical population of molecules "partition function" is the integral, over phase-space of a system, of the exponential of -E/kT [where E is the energy of the system, k is Boltzmann s eonstant, and T is the temperature in °K]. From this "function", all of the thermodynamic functions can be derived. The definitions that we shall need are given as follows in 2.5.1. on the next page. 

The fundamental thermodynamic function for a canonical ensemble is the Helmholtz free energy... 

The thermodynamic functions, as defined above, can be shown to be given by ensemble averages. For a dynamical variable A = A(, Pj ), the canonical-ensemble average is given by... 

When we want to calculate thermodynamics from MD, the first question which arises is, how are thermodynamic functions to be defined in the MD ensemble It is natural to define thermodynamic functions in the MD ensemble in terms of averages of dynamic variables, exactly as In equations (16)-(21) for the canonical ensemble. However, in different ensembles, the averages of the same dynamical variable are different, the difference being of... 

The next step in preparing to calculate thermodynamic functions from MD Is to invoke the quasiergodic hypothesis. This is well described by Reif, and for a general ensemble composed of systems in contact with a reservoir, it may be stated as follows. 

Express thermodynamic functions as canonical ensemble averages. 

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. 

Expressing other thermodynamic functions and thermodynamic coefficients in the canonical ensemble... 

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