Statistics of mesoscopic ensembles of bosons and fermions
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Statistics of Mesoscopic Ensembles of Bosons and Fermions V. A. Alekseev Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia email: [email protected] Received July 9, 2010
Abstract—Equilibrium distribution functions are obtained for boson and fermion ensembles with a limited number of particles. It is shown that the numberofparticle distribution functions in different quantum states are statistically dependent; this dependence disappears only for a large number of particles in the ensemble. The distributions are transformed into the Boltzmann distribution at a high temperature and into the Bose– Einstein and Fermi–Dirac distributions for a large number of particles in the ensemble. DOI: 10.1134/S1063776111050013
1. INTRODUCTION The Bose–Einstein and Fermi–Dirac distribu tions, which make it possible to calculate the mean values of number of particles nk in quantum states with energy Ek, form the basis of modern statistics. These distributions are obtained by summing the Gibbs dis tributions under the assumption that the total number N of particles in the ensemble is very large (in fact, N ∞), the distributions wk(nk) of the number of particles in states with energy Ek are statistically inde pendent [1, 2]. It was shown in [3, 4] that in the case of a boson gas, the assumption concerning statistical independence of distributions wk(nk) leads to an incor rect distribution w0(n0) of the number of particles in the ground state (condensate) even in the limit N ∞; a method for determining the correct distribution was developed. In greater degree, the assumption about the statistical independence of distributions wk(nk) leads to erroneous results for ensembles with a small number of particles (e.g., for N = 2). The latter case has attracted special attention in view of the high expectations associated with application of entangled states of two quantum atoms captured in a trap or in a quantum dot in quantum cryptography, quantum tele portation, and quantummechanical calculations (see, for example, [5]). Apart from interesting applica tions, analysis of the equilibrium distribution of parti cles constituting a mesoscopic ensemble is a concep tual problem in statistics. In this paper, exact equilibrium distributions are obtained for boson and fermion ensembles with an arbitrary number of particles. Obviously, an arbitrary number of particles cap tured in a trap and interacting with the thermodynam ically equilibrium surroundings obey the Gibbs distri bution –1
W ( n 0, n 1, … ) = S exp ( – ε 0 n 0 – ε 1 n 1 – … ),
(1)
where εk = Ek/T, T is the temperature, and S is the normalizing factor. We can state, however, that distri bution (1) also sets in when particles in such an ensem ble do not interact at all with the surroundings because it is only such a distribution that ensures zero value of the collision integral. In this case, the total energy of particles in the ensemble is not a preset conserved quantity, which actually is not surprising. With an overwhelm
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