The quasi-chemical equilibrium, or pair correlation, approximation to statistical mechanics is written in terms of second-quantization formalism. This involves an entension of the original theory to include an Ansatz for off-diagonal elements of the density matrix. The formal expressions involve “labelling operators” acting in a purely formal Hilbert space. These labelling operators obey Bose commutation rules for correlations between even numbers particles (e.g. pair correlations), Fermi commutation rules for correlations between odd numbers of particles. The labelling operators provide an algebraic way of formulating the restriction to certain types of “graphs”. We also show how to include higher-order correlations into the Ansatz. The main results of the original theory can be derived more rapidly with the new formalism. However, certain correction terms obtained earlier are shown to be in error, necessitating a reinvestigation of the nature of the condensation phenomenon. This re-investigation is carried out in two following papers, and proves to lead to no essential modifications of the previous results. The correlation matrices which enter into the Ansatz need to be related to the Hamiltonian of the system, and to the thermodynamic variables which define its thermodynamic state. A variational formulation is developed for this purpose, but no explicit calculations are carried out in this paper.
In the pair-correlation, or quasi-chemical equilibrium, approximation to statistical mechanics, there appears the possibility of a condensation phenomenon closely analogous to a Bose-Einstein condensation of correlated pairs of particles. However, the expansion used in earlier work to establish this phenomenon fails to yield an adequate approximation at and below the condensation point. In this paper, we give an alternative development, first formally for an arbitrary number of quantum states of the pair, then carry out in detail for the case of only one, or only two, quantum states of the pair. It is shown that, in spite of the Fermi-Dirac statistics of the particles making up the pairs, it is nevertheless possible to accomodate an arbitrarily large number of particles within a single pair state. Furthermore, if two quantum states of the pair are possible, and conditions are such that either the pair state would be occupied by a macroscopic number of particles if it alone were present, then the Pauli exclusion principle works in such a way that the lower one of the two pair states is occupied by a macroscopic number of particles, whereas the occupation number of the upper pair state remains independent of the volume. This result is not restricted to pairs whose mean separation exceeds their internal size, but applies also to pairs which completely overlap each other. The result is therefore of significance in the theory of superconductivity where the electron pairs have an internal size of the order of the Pippard coherence length, which is much larger than the mean separation of their centres of gravity.
In this work we revisit the concept of chemical potential
μ in both classical and quantum gases from a perspective of Equilibrium Statistical Mechanics (ESM). Two new results regarding the equation of state
μ=μ(n,T), where
n is the particle density and
T the absolute temperature, are given for the classical interacting gas and for the weakly-interacting quantum Bose gas. In order to make this review self-contained and adequate for a general reader we provide all the basic elements in an advanced-undergraduate or graduate statistical mechanics course required to follow all the calculations. We start by presenting a calculation of
μ(n,T) for the classical ideal gas in the canonical ensemble. After this, we consider the interactions between particles and compute the effects of them on
μ(n,T) for the van der Waals gas. For quantum gases we present an alternative approach to calculate the Bose-Einstein (BE) and Fermi-Dirac (FD) statistics. We show that this scheme can be straightforwardly generalized to determine what we have called Intermediate Quantum Statistics (IQS) which deal with ideal quantum systems where a single-particle energy can be occupied by at most
j particles with
0⩽j⩽N with
N the total number of particles. In the final part we address general considerations that underlie the theory of weakly interacting quantum gases. In the case of the weakly interacting Bose gas, we focus our attention to the equation of state
μ=μ(n,T) in the Hartree-Fock mean-field approximation (HF) and the implications of such results in the elucidation of the order of the phase transitions involved in the BEC phase for non-ideal Bose gases.