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, internal energy 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 entropy, 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 thermal equilibrium. 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 statistical mechanics 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 identical particles, 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 the first particle as Fermi statistics for a mixture of Bosons and Fermions if the last is a fermion or a boson, respectively. The 750 INDIAN J PURE & APPL PHYS, VOL. 42, OCTOBER 2004 first particle is the observer for the second particle and (i) Statistical count for an ensemble that is a mixture so the process is symmetric. Note that (P′b + P′f) is of the bosons and fermions assuming a pair not necessarily equal to one, and, if not, it means that interaction between the bosons and fermions. the second particle (observer) does not detect the first (ii) The most probable distribution in energy for a particle. The probability of this is (1−P′b−P′f). The mixture of the bosons and the fermions in the statistical uncertainty introduced here may be either ensemble. the intrinsic property of a particle itself or the (iii) The partition function for such an ensemble. experimental uncertainty of the measurement process. (iv) Using the partition function, the calculations were Another attempt in this direction is the so-called done for internal energy, specific heat and ‘statistical independence model’ of Landau and entropy. Lifshitz 3 in which two weakly interacting subsystems (bosons and fermions) are together regarded as one The above expressions are used to study the composite system, and the subsystems are assumed to thermodynamic properties of a mixture of liquid be quasi-closed. The statistical distribution or count helium-3 (fermions) and liquid helium-4 (bosons) for such a mixture is the product of the individual with different concentrations. For our model probabilities for two subsystems, one corresponding calculations a fermion concentration of 0.70 was used. to bosons and the other corresponding to fermions. The theoretical results obtained are compared with the With these assumptions, the statistical independence experimental observations on the properties of a model will hold only for an ideal gas assembly of mixture of liquid helium-3 and liquid helium-4. bosons and fermions. In reality, such an assembly does not exist and hence the statistical independence 2 Theory model cannot be used for real mixtures of bosons and Consider a micro-canonical assembly of N particles fermions like 3He and 4He mixtures. in which there are Nb bosons and Nf fermions such Chan et al.4 studied the ‘effect of disorder on that 3 4 superfluid He- He mixtures, and the ‘thermodyna- 3 4 mics of He- He mixtures in aerogel’. However, the N= Nb+ Nf … (1) studies related to more of an ideal system rather than a real system. Let ε1, ε2, ε3… εj… be the energy states of the In our earlier paper5 entitled “statistical mechanics assembly, and in the statistical equilibrium the and thermodynamics for a mixture of bosons and number of particles assigned to these energy levels be fermions”, the statistical distribution model for a n1, n2, n3, nj, respectively, such that the numbers nj mixture that was dominated by bosons was must satisfy the conditions requiring the conservation developed. The partition function that was derived of particles, N, and conservation of energy, E, i.e., worked well for a liquid helium-3 and liquid helium-4 mixture. In this paper, an assembly that is fermion ∞ ∑ nNj = … (2) dominated is studied. j=1 Therefore, the properties of a mixture of bosons and fermions assuming there exists a pair interaction and between the bosons and fermions; and that the concentrations of the bosons and fermions are ∞ different from each other are studied. Since the ∑ nEjjε = … (3) j=1 concentrations of the bosons and fermions will not be the same, and considering only pair interaction, in a such that, given state of equilibrium some fermions will be left unpaired. The value of the occupation number of nn= + n … (4) fermions in a given state will not exceed, rather, will j jb jf be much less, than the degeneracy of that state so that Pauli exclusion principle is not violated. With these where, basic assumptions, the expressions for the following were derived: njb = number of bosons in the energy level j … (5) AYODO et al.: MICRO-CANONICAL ENSEMBLE FOR BOSONS & FERMIONS 751 and njb (ωωjj) ( )! Pjbf = … (9) njf =number of fermions in the energy level j … (6) ()(nnnnjb!!!ω j−− jf )( jf jb ) Let ω j be the number of states in the j-th level, i.e, The statistical count, Cbf, for such a distribution ω j is the degeneracy of the j-th level. Then the among all the levels (j = 1,2,3…) available to the assembly is the product of such expressions as given number of ways Pjb in which njb bosons can be assigned to ω states in the j-th level is given by, in Eq. (9) since every arrangement in a given energy j level can be considered independently of the other energy levels. Thus we can write, n P = ω jb … (7) jb( j ) ∞ CPbf=∏ jbf j=1 Similarly, the number of ways, Pjf, in which njf njb ∞ ⎧ ⎫ … (10) fermions can be assigned to the ω j states is given by, ⎪ ()()ωωjj! ⎪ = ∏⎨ ⎬ j=1 ⎪()(nnnnjb!!!ω j−− jf )( jf jb )⎪ ω j ! ⎩⎭ P = … (8) jf ω − n ! ()jjf Equation (10) will now be used to calculate the most probable distribution in energy for the ensemble. To satisfy Pauli exclusion principle, it is necessary to assume that ω j >> njf. 3 Most probable distribution in energy for the Once the particles are placed in the ω sub-levels ensemble j The objective is to calculate for what values of n in the j-th level, we shall further assume that bosons jb and njf, the statistical count Cbf, is maximum under the and fermions may interact in pairs. However, not all conditions of N and E being fixed. The distribution the fermions may form pairs with bosons since the numbers and the corresponding energies must satisfy bosons and the fermions will not be in equal the following relations: proportions in the mixture.