Equilibrium statistical mechanics canonical thermodynamic functions

In the present work, the general mathematical scheme of construction of the equilibrium statistical mechanics on the basis of an arbitrary definition of statistical entropy for two types of thermodynamic potential, the first and the second thermodynamic potentials, was proposed. As an example, we investigated the Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles. On the example of a nonrelativistic ideal gas, it was proven that the statistical mechanics based on the Tsallis entropy satisfies the requirements of the equilibrium thermodynamics only in the thermodynamic limit when the entropic index z is an extensive variable of state of the system. In this case the thermodynamic quantities of the Tsallis statistics belong to one of the classes of homogeneous functions of the first or zero orders. 

The fundamental problem in classical equilibrium statistical mechanics is to evaluate the partition function. Once this is done, we can calculate all the thermodynamic quantities, as these are typically first and second partial derivatives of the partition function. Except for very simple model systems, this is an unsolved problem. In the theory of gases and liquids, the partition function is rarely mentioned. The reason for this is that the evaluation of the partition function can be replaced by the evaluation of the grand canonical correlation functions. Using this approach, and the assumption that the potential energy of the system can be written as a sum of pair potentials, the evaluation of the partition function is equivalent to the calculation of... 

In order to theoretically understand the Monte Carlo method, we need to use the concepts of statistical mechanics. These concepts are developed in detail in books of fundamental statistical mechanics [16]. This formalism guides us in the task of calculating various quantities and/or properties that describe the behavior of the system. This is accomplished from the partition function that specifies the statistical properties of the system in thermodynamic equilibrium. There are different classes of partition functions, each corresponding to a determined statistical ensemble. The canonical ensemble is applied for a system of N particles in a volume V and temperature T. These magnitudes have fixed values and heat exchange is allowed. For this ensemble the corresponding partition function is ... 

The energy and entropy, which are related in this equation to the molecular partition function, are statistical entities, defined by particles which do not interact except to maintain the equilibrium conditions. To obtain similar relationships for real systems, it is necessary to apply statistical mechanics to the calculation of the thermodynamic entities, which correspond to the molar quantities of particles, or that is N approaches 1 this treatment it is convenient to use the canonical ensemble already discussed and presented in Figure n.l. This ensemble consists of a very large number of systems, N, each containing 1 mol of molecules and separated from the others by diathermic walls, which allow heat conduction but do not allow particles to pass. The set of all the systans is isolated from the outside and has a fixed energy E, which is the energy of the canonical ensemble. 

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