Physica A: Statistical Mechanics and its Applications
Quantum gases and polylogs
Introduction
As is well known, ideal quantum gases come in two classes, Bose and Fermi, according to the fundamental statistics that the constituents satisfy. The thermodynamic properties of a Fermi gas are profoundly different from those of a Bose gas at low temperatures where statistical effects are not masked by thermal fluctuations. For example, there is a finite pressure at T=0 in a Fermi gas, called zero point pressure, whereas it vanishes with T in a Bose gas. The entropy per particle vanishes with T in a Fermi gas but it remains constant as T→0 for a Bose gas. It is thus customary that the two statistics are treated separately and regarded independent of each other [1]. Can they possibly be subclasses of a unifying big picture, where these seeming differences may not be so different?
Before exploring this idea, let us briefly examine whether there really are ideal quantum gases. Perhaps the closest to an ideal gas are a photon gas, a Bose gas, and a neutrino gas, a Fermi gas. It is suggestive that these two relativistic gases have zero chemical potential at any temperature. Among the more common non-relativistic Fermi gases is an electron gas in a metal, which is nearly an ideal gas. For the Bose counterpart, one might consider a gas. Liquid has often been regarded a Bose fluid although hardly ideal owing to strong interatomic interactions at short distances. In any case, it is profitable to think of ideal quantum gases as a first approximation to most gases or liquids at low temperatures.
The possibility that these Bose and Fermi gases may be fundamentally connected does not appear to have been considered previously. But if one could unify them, some of the mysteries of different statistical consequences could be more easily understood. One main problem faced in unifying quantum gases is that there are at least three parameters, not two. They are temperature T, statistics and dimensionality d. Evidently, T is necessary since as T rises the statistical differences begin to vanish, eventually all going over into Boltzmann statistics. No need to further elaborate on statistics. The importance of d is underscored by the fact that in 2d (but not in 3d) there is no Bose Einstein condensation at finite values of T. A unifying theory is at its best when there are but two parameters, see e.g. the periodic table of chemical elements. We may combine T and statistics into the chemical potential μ or the fugacityAs we shall see, there are always some underlying mathematics that play an essential role in unification, which are special to specific physical or chemical problems. We can see the structure and unity in the problems through these special mathematics. Our problem illustrate this idea rather richly.
Section snippets
Polylogs
Polylogs are transcendental functions of two complex numbers much like say the Bessel functions. The simplest form, known as the dilogs, was invented by Euler in 1768. The next simplest is the trilogs due to Landen almost twenty years later. Polylogs are a modern generalization of Euler's dilogs and Landen's trilogs. The early developments of these special functions in the 18th century appeared to have lost their impetus until re-discovered in the problems related to Bose statistics in the
Density functions
Let us define the number density ρ=N/V, where N is the total number of particles, Bose or Fermi, in a box of volume V. We assume that N and V are large or macroscopic, i.e., the thermodynamic limit is operative at will. By definition,where is the number operator at the state of momentum or wave vector k (we adopt ℏ=1), 〈⋯〉 an ensemble average such that denotes the single particle momentum (or wave vector) distribution function. The systems of our interest are isotropic,
Thermodynamic functions in unified theory
Using (9) we can easily obtain the grand partition function Q(ζ) from the standard relation [1],with the boundary condition that . Substituting (9) in (10) for the density function we obtainAs is well known, and , where P is the pressure and U the energy. Hence all other thermodynamic functions such as the entropy, specific heat follow from (11). We can recover all the known results in 3d whether Fermi or Bose, whether
Equivalence in 2d
As stated above, the unified grand partition function formula (11) at once recovers all the known thermodynamic functions in 3d. It can furthermore account for the underlying structures of physical phenomena ranging from the ground state Fermi to Bose Einstein condensation. In addition the two statistics are brought together very naturally at the classical limit.
Even more remarkable still is an equivalence that (11) shows between the Fermi and Bose gases in 2d and only in 2d. This equivalence
Discussion
Another unexplored direction is to take null dimension d=0 in (11). The physics of null dimension is potentially quite interesting. It pertains to confinement of particles. Confinement is a quantum concept, not attainable in classical statistical physics. If d→0, what would happen to the particles initially packed in a volume V? How particles respond is found to depend strongly on their statistics.
Consider Fermi particles first. Since one may not ignore the Pauli principle if particles in the
Acknowledgements
We thank Professor H.J. Schmidt for informing us of his results (Ref. [5]) prior to publication. We thank Professor C.W. Kim, director of the Institute for Advanced Study, Seoul for his hospitality and support where a portion of this work was completed. The work of JK has been supported by the Korea Research Foundation (2000-Y00070).
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