Ensemble thermodynamic potential

The chemical potential p, = 6J f6p enters the respective Euler-Lagrange equation obtained by minimizing the grand ensemble thermodynamic potential — p J pd a , which defines the equilibrium particle density distribution... 

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 ... 

In this chapter, the most common procedures for augmenting electronic-structure calculations in order to convert single-molecule potential energies to ensemble thermodynamic variables will be detailed, and key potential ambiguities and pitfalls described. Within the context of certain assumptions, this connection can be established in a rigorous way. 

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 ... 

If a system contains two types of species, but the membrane is permeable only to species number 1, the natural variables for the system are T, K //, and N2, where N2 is the number of molecules of type 2 in the system. The thermodynamic potential for this system containing two species is represented by U[T, //,]. The corresponding ensemble is referred to as a semigrand ensemble, and the semigrand partition function can be represented by P(71 K /q, N2). The thermodynamic potential of the system is related to the partition function by... 

In order to determine a system thermodynamically, one has to specify some independent parameters (e.g. N, T, P or V) besides the composition of the system. The most common choice in MC simulation is to specify N, V and T resulting in the canonical ensemble, where the Helmholtz free energy A is the natural thermodynamical potential. However, MC calculations can be performed in any ensemble, where the suitable choice depends on the application. It is straightforward to apply the Metropolis MC algorithm to a simple electric double layer in the iVFT ensemble. It is however, not so efficient for polymers composed of more than a few tens of monomers. For long polymers other algorithms should be considered and the Pivot algorithm [21] offers an efficient alternative. MC simulations provide thermodynamic and structural information, but time-dependent properties are not accessible. If kinetic or time-dependent properties are of interest one has to use molecular dynamic or brownian dynamic simulations. 

Our perspective is that the PDT should be recognized as directly analogous to the partition functions which express the Gibbsian ensemble formulation of statistical mechanics. From this perspective, the PDT is a formula for a thermodynamic potential in terms of a partition function. Merely the identification of a... 

The general mathematical formulation of the equilibrium statistical mechanics based on the generalized statistical entropy for the first and second thermodynamic potentials was given. The Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles were investigated as an example. It was shown that the statistical mechanics based on the Tsallis statistical entropy satisfies the requirements of equilibrium thermodynamics in the thermodynamic limit if the entropic index z=l/(q-l) is an extensive variable of state of the system. 

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... 

Such an ensemble generalized ground-state energy functional, E = E[N, v] = E[p[N. i l- represents the thermodynamic potential of the N, v -representation, with the corresponding generalized Hellmann-Feynman expression for its differential (see equations (17), (22) and (27)) ... 

In Section 4 we focused on the implementation of the MD method for the microcanonical (N, V, E) ensemble. Other ensembles are distinguished by the use of different independent variables which function as the control parameters during the simulation. In the canonical ensemble the independent variables are (N, V, T). The calculated value of the energy E is the same for both ensembles in the thermodynamic limit N - oo at constant Nj V. However, different formulae may apply for thermodynamic properties that are derivatives of a thermodynamic potential (Allen and Tildesley, 1987). 

The canonical ensemble corresponds to a system of fixed N and V, able to exchange energy with a thermal bath at temperature T, which represents the effects of the surroundings. The thermodynamic potential is the Helmholtz free energy, and it is related to the partition function Q ryT as follows ... 

All the quantities on the left-hand side of the expressions are second derivatives of the Gibbs free energy, while all the quantities on the right-hand side involve second derivatives of pF—the characteristic thermodynamic potential of the grand canonical ensemble. Application of the stability conditions for stable (miscible) solutions indicates that we must have rju > 0 and C2 > 0 (Prigogine and Dufay 1954). 

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