Entropy canonical ensemble

Parvan A S. Extensive statistical mechanics based on nonadditive entropy Canonical ensemble. Phys. Lett. A. 2006 360(l) 26-34. 

The statistical analogue of entropy for the canonical ensemble follows as S = —kfj. Since fj = trace rje 1, the entropy expression becomes... 

The definition of entropy in the grand canonical ensemble follows from the definition of — S/k according to (50), combined with (46) and (49), as... 

Figure 2. Entropy per site, 5(0), versus lattice coverage, 0. The curves denoted (a) to (d), corres-pond one-to-one to the cases displayed in fig. 1 (e) k = 2, w / kBT = +2 (f) k = 10, w / kBT = -2. Results of thermodynamic integration by Monte Carlo Simulation in the canonical ensemble are shown in symbols for the respective cases.

Standard procedures permit the evaluation of the entropy of a Fermi gas under the conditions of a grand canonical ensemble, which we will have to adjust to obtain the microcanonical entropy. For low excitation energies, E, the entropy is... 

The above realization of the abstract mesoscopic equilibrium thermodynamics is called a Canonical-Ensemble Statistical Mechanics. We shall now briefly present also another realization, called a Microcanonical-Ensemble Statistical Mechanics since it offers a useful physical interpretation of entropy. 

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... 

Substituting Eq. (144) into Eq. (130) for the entropy of the microcanonical ensemble, we obtain the entropy of the canonical ensemble (Eq. (90)). Equation (134) for the pressure of the microcanonical ensemble is identical to Eq. (92) for the pressure of the canonical ensemble. Substituting Eqs. (144) and (86) into Eq. (135) for the chemical potential of the microcanonical ensemble, we obtain Eq. (94) for the chemical potential of the canonical ensemble. Moreover, substituting Eqs. (144) and (86) into Eq. (136) for the variable E of the microcanonical ensemble, we obtain Eq. (96) for the variable E of the canonical ensemble. Thus, for the Tsallis statistics, the canonical and microcanonical ensembles are equivalent in the thermodynamic limit when the entropic parameter z is considered to be an extensive variable of state. 

Section II briefly reviews some arguments about the applicability of one or another ensemble in studying various aspects of small systems. For example, negative heat capacities can be detected in microcanonical ensembles [21-24] if the entropy has a convex dip. The canonical ensemble of the same system does not show any negative heat capacity [25], which is consistent with the general theory for example, the heat capacity is proportional to the energy variance in the canonical ensemble and can never be negative. 

From statistical mechanics the second law as a general statement of the inevitable approach to equilibrium in an isolated system appears next to impossible to obtain. There are so many different kinds of systems one might imagine, and each one needs to be treated differently by an extremely complicated nonequilibrium theory. The final equilibrium relations however involving the entropy are straightforward to obtain. This is not done from the microcanonical ensemble, which is virtually impossible to work with. Instead, the system is placed in thermal equilibrium with a heat bath at temperature T and represented by a canonical ensemble. The presence of the heat bath introduces the property of temperature, which is tricky in a microscopic discipline, and relaxes the restriction that all quantum states the system could be in must have the same energy. Fluctuations in energy become possible when a heat bath is connected to the... 

For strongly asymmetric mixtures (e.g., mixtures where the A-chains are stiff while the B-chains are flexible) the semi-grandcanonical approach is clearly not feasible, and one must work in a canonical ensemble where both the number of A-chains nA and the number of B-chains nB are fixed. However, the finite size scaling ideas for PL(M) as exposed above still can be exploited if one considers the order parameter M in L x L subsystems of a much larger system [267]. The usefulness of this concept was demonstrated earlier for Ising models and Len-nard-Jones fluids [268-271]. Gauger and Pakula [267] find an entropy-driven phase separation without any intermolecular interactions. 

In this section, we focus on a relation between entropy and the probability distribution function P,. If P, obeys Boltzmann statistics for the canonical ensemble, then one arrives at a correspondence between entropy S and the partition function Z. Equation (28-26) is interpreted within the context of the first law of thermodynamics in differential form ... 

Integration of (28-34) allows one to express entropy in terms of Pi for a canonical ensemble of molecules that obey Boltzmann statistics. The result is exact to within an integration constant that comprises the third law of thermodynamics at absolute zero, where Pi is unity for the state of lowest energy and zero for all higher-energy states. Hence,... 

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