Indian Journal of Pure & Applied Physics Vol. 42, October 2004, pp. 749-757

Micro-canonical ensemble model of particles obeying Bose-Einstein and Fermi-Dirac statistics

Y K Ayodo1, K M Khanna2 & T W Sakwa1 1Department of Physical Sciences, Western University College, Box 190, Kakamega, Kenya 2Department of Physics, Moi University, Box 1125 Eldoret, Kenya Received 13 February 2004; accepted 12 April 2004

A micro-canonical ensemble for an assembly of bosons and fermions is considered in which the number of particles, and volume are kept constant. A statistical distribution model, which is fermion dominated and where bosons and fermions interact in pairs, is developed. The partition function is derived. Macroscopic thermodynamic quantities such as , internal energy and specific heat are obtained in terms of the partition function. The model equations are applied to a mixture of liquid helium-3 and liquid helium-4 atoms.

[Keywords: Bose-Einstein Statistics, Fermi-Dirac statistics, Partition function, 3He- 4He mixture] IPC Code: C01B 23/00

1 Introduction was made by Gentile1. He proposed statistics in which A micro-canonical ensemble represents a collection up to N particles (N>>1) were allowed to occupy a of configurations of isolated systems that have single quantum state instead of just one particle for reached . A system is isolated Fermi case due to the Pauli exclusion principle, and from its environment if it does not exchange either infinitely many for the Bose case. However, Gentile’s particles or energy with its surroundings. The volume, approach was found to be too much of a internal energy and the number of particles of such a generalisation and contained the violation of the system are constant and are the same for all conventionally accepted Pauli principle. Furthermore, configurations that are part of the same micro- his model did not distinguish which particles were canonical ensemble. In this paper, a configuration of a fermions and which ones were bosons. However, mixture of bosons and fermions is studied and a Gentile’s work laid the emphasis and the foundation partition function is developed for the same. that the of a mixture of bosons Thermodynamic quantities, such as internal energy, and fermions can be worked out. specific heat and entropy can be calculated from the The next attempt was that of Medvedev2. In his knowledge of the statistical distribution and the paper entitled ‘properties of particles obeying partition function. So far most of the studies deal ambiguous statistics’, Medvedev proposed a new either with a system of bosons or with a system of class of , which may exhibit both fermions. In nature, there do exist systems, which are Bose and Fermi statistics with respective probabilities 1 2 mixtures of bosons and fermions such as 1 H, 1 H and P′b and P′f. The model admits only primary Bose- Einstein and Fermi-Dirac statistics as existing. He 3 H, and the most interesting mixture is 4 He and 1 2 assumed that a particle is neither a pure boson nor a 3 2 He. It should be clearly understood that in the pure fermion. He let another particle, which interacts mixture, bosons obey Bose-Einstein statistics and with the first one, play the role of an external fermions obey Fermi-Dirac statistics. What observer. During the interaction it performs a distribution law or what will be the expression for the measurement at the first particle and identifies it as most probable distribution-in-energy in the mixture is either a boson or a fermion with respective the subject matter of study in this paper. probabilities P′b and P′f. According to the result of The first attempt to generalise quantum Bose and this measurement, it interacts with