The Density Operator and the Boltzmann Distribution

Roger Balian

Shigeji Fujita, Roman Schubert, Chi‐Ting Ho +1 more

In the classical mechanical limit in which quantum cells of volume (2πħ)2 are reduced to points in phase space, the Bose and Fermi distributions are shown to approach the Boltzmann distribution. This limit is a continuous limit and therefore does not alter the indistinguishability of bosons (fermions) which is defined in terms of a discontinuous permutation symmetry of the density operator and Hamiltonian for the system. This means that the classical particles which correspond to the identical boson (fermions) remain indistinguishable. In the same limit, the partition function for N identical bosons approaches (2πħ)−3N (N!)−1 f. f d3r1 d3p1…d3rN d3pN exp (–βH)  Zc. The two factors, (2πħ)−3N and (N!)−1 which are often added in an ad hoc manner in many books on statistical mechanics, are thus derived from first principles. In the classical statistical limit in which the number density n and the reciprocal temperature β become small, the quantum statistical distributions also approach the Boltzmann distribution. This is shown by examining the chemical potential μ as a function of n and β. The classical approximation is valid if the thermal de Broglie wavelength is much shorter than the interparticle distance irrespective of any interparticle interaction.

Christian B. Mendl

We devise a lattice Boltzmann method (LBM) for a matrix-valued quantum Boltzmann equation, with the classical Maxwell distribution replaced by Fermi–Dirac functions. To accommodate the spin density matrix, the distribution functions become 2 × 2 matrix-valued. From an analytic perspective, the efficient, commonly used BGK approximation of the collision operator is valid in the present setting. The numerical scheme could leverage the principles of LBM for simulating complex spin systems, with applications to spintronics.

Philip L. Bowers

Lev I. Plimak, Daniel F. Walls

The spatial and momentum distributions of the uncondensed cloud associated with a confined Bose-condensed gas are found to have a universal form which differs from the classical Maxwell-Boltzmann distribution. This universality is a direct consequence of the condensation which stabilizes the density of the cloud on the threshold of condensation. The deviation of the distributions from the Maxwell-Boltzmann distribution is a macroscopic signature of the quantum statistics. \textcopyright{} 1996 The American Physical Society.