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International Journal of Modern Physics B

THERMODYNAMICS OF A BOSE GAS TRAPPED IN A BOUNDED HARMONIC POTENTIAL

Shalini Lumb, Satish Kumar Muthu

Published Dec 30, 2003• 4 citations
Publisher

Abstract

Some thermodynamic features of an assembly of a finite number of bosons trapped in a bounded harmonic potential are investigated. P–V isotherms are drawn for both the degenerate and the non-degenerate phases. At any temperature, the pressure is a decreasing function of volume, unlike the free Bose gas for which the pressure becomes independent of volume in the degenerate phase. At the absolute zero of temperature the quantum pressure for a spherical enclosure of radius r0 equal to aho, aho being the characteristic harmonic oscillator length, is found to be of order 10-10 Torr while for [Formula: see text] it is of order 10-12 Torr. The isothermal compressibility has a sharp drop near the critical point and becomes negligibly small for temperatures above the critical temperature, irrespective of the size of the trap. The coefficient of thermal expansion also shows a sudden drop at the critical temperature. The specific heat at constant pressure shows a peak and is well-defined in the degenerate phase, in contradistinction to the ideal Bose gas. Results recently derived by Singh on the basis of local equilibrium theory are found to be in good agreement with our numerical computations.

DOI
10.1142/s0217979203023379

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Some thermodynamic features of an assembly of a finite number of bosons trapped in a bounded harmonic potential are investigated. P–V isotherms are drawn for both the degenerate and the non-degenerate phases. At any temperature, the pressure is a decreasing function of volume, unlike the free Bose gas for which the pressure becomes independent of volume in the degenerate phase. At the absolute zero of temperature the quantum pressure for a spherical enclosure of radius r0 equal to aho, aho being the characteristic harmonic oscillator length, is found to be of order 10-10 Torr while for [Formula: see text] it is of order 10-12 Torr. The isothermal compressibility has a sharp drop near the critical point and becomes negligibly small for temperatures above the critical temperature, irrespective of the size of the trap. The coefficient of thermal expansion also shows a sudden drop at the critical temperature. The specific heat at constant pressure shows a peak and is well-defined in the degenerate phase, in contradistinction to the ideal Bose gas. Results recently derived by Singh on the basis of local equilibrium theory are found to be in good agreement with our numerical computations.
We present theoretical aspects concerning the thermodynamics of an ideal bosonic gas trapped by a harmonic potential. Working in the Grand Canonical ensemble we are able to properly identify the extensive thermodynamic variable equivalent to the volume and the intensive thermodynamic variable equivalent to the pressure. These are called the "harmonic volume" and the "harmonic pressure" and their physical meaning is discussed. With these variables, the problem of Bose-Einstein condensation is studied in terms of the behavior of the corresponding equation of state and in terms of measurable susceptibilities such as the heat capacities, the isothermal compressibility and the coefficient of thermal expansion. From the analysis, an interesting analogy with Black-Body radiation emerges, showing that at and below the critical temperature, the non-condensate fraction of atoms behaves thermodynamically like a gas of massless particles.
The behavior of a finite number of bosons trapped in a bounded harmonic potential is investigated. The eigenvalue spectrum is worked out numerically for three different sizes of the trap. The condensate fraction is determined and is found to increase suddenly below a certain temperature which is a characteristic signature of BEC. The specific heat-temperature curve also shows a peak, with the maximum shifting to lower values and occurring at higher temperatures, as the size of the assembly is reduced.
We employ the q-digamma function to calculate the internal energy E, the entropy S, the Helmholtz free energy F, and the heat capacity at constant number of ideal Bose atoms in a one-dimensional harmonic trap. The first main finding in this paper is that in a one-dimensional harmonic trap and when the temperature approaches zero, the entropy of a finite number of ideal Bose atoms is nonzero but the entropy of an infinite number of ideal Bose atoms is zero. The second main finding is that the heat capacity at constant number has a classic limit as . In the thermodynamic limit, we find that at critical temperature the heat capacity at constant number is discontinuous. Hence, at there is a jump in the heat capacity and this jump is calculated. Consequently, the transition between normal and condensed states is a second-order phase transition.
Effects of finite spatial size of boson assemblies in traps are studied in a self-consistent lattice theory by modeling the trap as a bounded harmonic potential of size R0. The thermodynamic quantities exhibit scaling and crossover from ideal gas behaviour at small (R0/a0) to that appropriate to an unbounded harmonic potential at large (R0/a0) with a crossover parameter [Formula: see text], a0 being the harmonic oscillator length, and τ denoting the dimensionless thermal energy. The numerical results obtained earlier by computing the energy levels of the bounded harmonic oscillator fit the general structure predicted by the theory very well. For a1>10, the spatial size effects are negligible but for a1<10 they become appreciable and experimentally measurable in suitably designed traps. At low temperatures the self consistent cell size is found to be about 2.5a0 implying that the condensate is essentially a single coherent state contained in the central cell.
We have investigated the thermodynamic behaviour of ideal Bose gases with an arbitrary number of particles confined in a harmonic potential. By taking into account the conservation of the total number N of particles and using a saddle-point approximation, we derive analytically the simple explicit expression of mean occupation number in any state of the finite system. The temperature dependence of the chemical potential, specific heat, and condensate fraction for the trapped finite-size Bose system is obtained numerically. We compare our results with the usual treatment which is based on the grand canonical ensemble. It is shown that there exists a considerable difference between them at sufficiently low temperatures, especially for the relative small numbers of Bose atoms. The finite-size scaling at the transition temperature for the harmonically trapped systems is also discussed. We find that the scaled condensate fractions for various system sizes and temperatures collapse onto a single scaled form.
The chemical potential and its differential coefficients as a function of temperature of an ideal system trapped in three dimensions isotropic harmonic traps with an finite number of particles were calcu- lated from the statistical theory.The results show that,there is no phase transition in a system with an finite number of particles,but the Bose-Einstein condensation occurs at a finite temperature.The critical temperatures were well defined by the minimum temperatures of the second derivative of chemical potential, which were in good agreement with the experimental results.