Thermodynamic fluctuations in the Gibbs and Einstein approaches

A comparative analysis of the description of fluctuations in statistical mechanics (the Gibbs approach) and in statistical thermodynamics (the Einstein approach) is given. On this basis solutions are obtained for the Gibbs and Einstein problems that arise in equilibrium (or slightly non-equilibrium) pressure fluctuation calculations for a spatially limited macroscopic system. A modern version of the Gibbs approach which allows to calculate pressure fluctuations without making any additional assumptions is presented; to this end the generalized forms of the Bogolyubov-Zubarev and Helman-Feynman theorems are proved for the classical and quantum descriptions of a macrosystem. A statistical version of the Einstein approach is developed which, while similar in form to its Gibbs counterpart, leads to fundamentally different pressure fluctuation results. Both the "genetic" relation and the conceptual differences between the Gibbs and Einstein approaches are demonstrated. To illustrate the results, which are valid for any thermodynamic system an ideal non-degenerate gas of microparticles is considered, both classically and quantum mechanically. Based on the results obtained, the correspondence between the micro- and macroscopic descriptions is considered and prospects of statistical thermodynamics are discussed.