Gibbs-Boltzmann statistical ensemble

When g = 1 the extensivity of the entropy can be used to derive the Boltzmann entropy equation 5 = fc In W in the microcanonical ensemble. When g 1, it is the odd property that the generalization of the entropy Sq is not extensive that leads to the peculiar form of the probability distribution. The non-extensivity of Sq has led to speculation that Tsallis statistics may be applicable to gravitational systems where interaction length scales comparable to the system size violate the assumptions underlying Gibbs-Boltzmann statistics. [4]... 

Tet us consider the nonrelativistic ideal gas of N identical particles governed by the classical Maxwell-Boltzmann statistics in the framework of the Tsallis and Boltzmann-Gibbs statistics in the microcanonical ensemble. For this special model, the statistical weight (111) can be written as (see [6] and reference therein)... 

There have apparently been two parallel developments of the statistical mechanics theory, which are typified by the work of Boltzmann [6] and Gibbs [33]. The main difference between the two approaches lies in the choice of statistical unit [15]. In the method of Boltzmann the statistical unit is the molecule and the statistical ensemble is a large number of molecules constituting a system, thus the system properties are said to be calculated as... 

In the following, we first describe (Section 13.3.1) a statistical mechanical formulation of Mayer and co-workers that anticipated certain features of thermodynamic geometry. We then outline (Section 13.3.2) the standard quantum statistical thermodynamic treatment of chemical equilibrium in the Gibbs canonical ensemble in order to trace the statistical origins of metric geometry in Boltzmann s probabilistic assumptions. In the concluding two sections, we illustrate how modem ab initio molecular calculations can be enlisted to predict thermodynamic properties of chemical reaction (Sections 13.3.3) and cluster equilibrium mixtures (Section 13.3.4), thereby relating chemical and phase thermodynamics to a modem ab initio electronic stmcture picture of molecular and supramolecular interactions. 

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... 

In the thermodynamic limit (82), the chemical potential (80) and the variable (81) for the ideal gas in the canonical ensemble for the Tsallis and Boltzmann-Gibbs statistics are [7]... 

The entropy S for the Tsallis and Boltzmann-Gibbs statistics in the microcanonical ensemble can be written as... 

F is thus expressed as the statistical average of the random variable exp[-l-E(x)/fe3T] with the Boltzmann PD. Notice that F (like S) is defined up to an additive constant. < E>, S, and F are extensive variables that is, they are proportional to the number of particles N. One can also define the Gibbs free energy G = F + PV of the (NPT) ensemble. Even though G is calculated in simulations more often than F, for the sake of simplicity (and without a loss of generality) we shall describe the general theory and the various methods with respect to F. 

All macroscopic observables are obtainable from the distribution of microscopic states (henceforth, microstates) of elements that obey mechanics. Strictly speaking, this is the most basic assumption of statistical thermodynamics. However, to elucidate macroscopic phenomena, it is not necessary to know the true distribution of microstates of the system. Boltzmann introduced the concept of orthodic ensembles that are compatible with thermodynamics. In practice, an orthodic ensemble is established hy demonstrating that it is compatible with the laws of mechanics and with the laws of thermodynamics. Gibbs demonstrated that the canonical distribution fimction... 

In order to get expressions for Gibbs energy G and the Helmholtz energy A, we will need an expression for the entropy, S. The statistical thermodynamic approach for S is somewhat different. Rather than derive a statistical thermodynamic expression for S (which can be done but will not be given here ), we present Ludwig Boltzmanns 1877 seminal contribution relating entropy S and the distribution of particles in an ensemble il ... 

Gibbs ensemble formulation of statistical mechanics avoids specific reference to atoms and molecules. In many chemical applications, however, Gibbs and Boltzmann s equation become identical. 

Looking at the phase space not as a succession in time of microscopic states that follow Newtonian mechanics, but as an ensemble of microscopic states with probabilities that depend on the macroscopic state, Gibbs and Boltzmann set the foundation of statistical thermodynanucs, which provides a direct connection between classical thermodynamics and microscopic properties. 

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