Abstract
Within the theory of statistical ensemble, the so-called mu PT ensemble describes equilibrium systems that exchange energy, particles, and volume with the surrounding. General, model-independent features of volume and particle number statistics are derived. Non-analytic points of the partition function are discussed in connection with divergent fluctuations and ensemble equivalence. Quantum and classical ideal gases, and a model of Bose gas with mean-field interactions are discussed as examples of the above considerations.
Highlights
Within the theory of statistical ensemble, the so-called μPT ensemble describes equilibrium systems that exchange energy, particles, and volume with the surrounding
We focus on the Legendre transform of the NPT ensemble for the classical ideal homogeneous gas
The statistics of the volume and of the particle number do not depend on the specific model, i.e. on the Hamiltonian, as a consequence of the Legendre transforms performed on all the extensive quantities
Summary
Within the theory of statistical ensemble, the so-called μPT ensemble describes equilibrium systems that exchange energy, particles, and volume with the surrounding. Each ensemble is derived from the maximisation of the Shannon entropy with constraints that fix the average of the fluctuating extensive quantities, following the Jaynes’ approach[7,8] These two constructions are equivalent, resulting in the well-known Boltzmann weight of exponential form. This paper concerns the statistical ensemble, missing in the above picture, which represents a system exchanging energy, particles, and volume with the surrounding, and is parametrised by the intensive variables β , μ , and P, called μPT ensemble[9,10,11]. The μPT ensemble is the extension of the μVT ensemble when the pressure instead of the volume is fixed, or the extension of the NPT ensemble when the chemical potential instead of the particle number is fixed The latter conditions are met in several physical and chemical processes, e.g. naturally arise in systems confined within a porous and elastic membranes.
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