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arXiv:1208.4888

Bose - Einstein condensation in arbitrary dimensions

Ajanta Bhowal Acharyya, Muktish Acharyya

Published Aug 24, 2012• 0 citations
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Abstract

The density of bosonic states are calculated for spinless free massive bosons in generalised d dimensions. The number of bosons are calculated in the lowest energy state. The Bose Einstein condensation was found in generalised d dimensions (on and above d = 3) and the condensation temperature is calculated which is observed to drop abruptly above three dimensions and decreases monotonically as the dimensionalities of the system increases. The rate of fall of the condensation temperature decreases as the dimensionality increases. Interestingly, in the limit of infinite dimensions, the condensation temperature is observed to approach a nonzero finite value.

DOI
10.48550/arxiv.1208.4888

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The density of bosonic states are calculated for spinless free massive bosons in generalised d dimensions. The number of bosons are calculated in the lowest energy state. The Bose Einstein condensation was found in generalised d dimensions (on and above d = 3) and the condensation temperature is calculated which is observed to drop abruptly above three dimensions and decreases monotonically as the dimensionalities of the system increases. The rate of fall of the condensation temperature decreases as the dimensionality increases. Interestingly, in the limit of infinite dimensions, the condensation temperature is observed to approach a nonzero finite value.
The density of bosonic states are calculated for spinless free massive bosons in generalised d dimensions. The number of bosons are calculated in the lowest energy state. The Bose Einstein condensation was found in generalised dimensions (on and above d = 3) and the condensation temperature was observed to drop abruptly above three dimensions and decreases monotonically as the dimensionalities of the system increases. The rate of fall of the condensation temperature decreases as the dimensionality increases. This article is aimed to the University graduate students.
The phenomenon of Bose-like condensation, the continuous change of the dimensionality of the particle distribution as a consequence of freezing out of one or more degrees of freedom in the limit of low particle density, is investigated theoretically in the case of closed systems of massive bosons and fermions, described by general single-particle Hamiltonians. This phenomenon is similar for both types of particles and, for some energy spectra, exhibits features specific to multiple-step Bose-Einstein condensation, for instance, the appearance of maxima in the specific heat. In the case of fermions, as the particle density increases, another phenomenon is also observed. For certain types of single particle Hamiltonians, the specific heat is approaching asymptotically a divergent behavior at zero temperature, as the Fermi energy epsilon(F) is converging towards any value from an infinite discrete set of energies epsilon(i)(i>/=1). If epsilon(F)=epsilon(i), for any i, the specific heat is divergent at T=0 just in infinite systems, whereas for any finite system the specific heat approaches zero at low enough temperatures. The results are particularized for particles trapped inside parallelepipedic boxes and harmonic potentials.
Standard arguments state that Bose Einstein condensation (BEC) cannot occur in dimensions lower than three in the thermodynamic limit as the expressions for the number of bosons in the excited states are unbounded. These arguments imply that the number density is infinite, which is an extraordinary condition. As an alternative to this pedagogy, we explore the use of regularization techniques to show that the number density of bosons in the excited state is finite in two and one dimensions at low temperatures, making it possible to have a BEC transition in the thermodynamic limit. We suggest creating a two-dimensional optical traps of increasing sizes to test this hypothesis as the thermodynamic limit is approached.
An exact expression for the single-particle reduced density matrix of a translationally invariant system of bosons is obtained, as an average of a functional of the density and current operators. An approximation scheme, based on small fluctuations in density and current, is used in its lowest order, to approximate the single-particle reduced density matrix and the condensate density by a functional of the structure factor, which is a measurable quantity. This is done for the ground state and for finite temperatures. The resulting expressions for the condensate density are discussed. For the ideal gas the result is correct below the condensation temperature. For the case of an interacting liquid, Bose-Einstein condensation exists, in the ground states for two- and three-dimensional systems, and for finite temperatures it does exist only in three dimensions. The present experimental knowledge of the structure factor enables only a crude estimate of the condensed fraction which is found to be 0.1-0.2.
Low-dimensional Bose-Einstein condensations (BEC) for g(e) (the density of states) = Cde(superscript d-1) with d ≤ 1, corresponding to a dimensionality of not larger than 2, are theoretically analyzed. When the energy quantization of the first excited state is taken into account, the boson number in the excited states is finite even for d ≤ 1 with the density of states remaining proportional toe(superscript d-1). The condensation temperature increases with the particle number faster for d ≤ 1 than for d ＞ 1. The condensation depends highly on the quantized energye1 of the first excited state. Whene1 just deviates slightly from zero, the condensation temperature quickly increases withe1. Theoretical estimation shows that a low-dimensional BEC can occur at a temperature above 1 K for excitons in semiconductor quantum wells even without the harmonic-oscillator potential.
Bose-Einstein condensation (BEC) of an ideal gas is investigated for a finite number of particles. In three dimensions, we find a transition temperature which is lower than in the thermodynamic limit. Lowering the dimension increases the transition temperature and is therefore favorable for BEC. This is in contrast to the standard result obtained in the thermodynamic limit which states that BEC is not possible in, e.g., a one-dimensional (1D) harmonic potential. As a result, 1D atom traps, such as radially tightly confining magnetic traps or optical dipole traps, are promising for studying BEC. \textcopyright{} 1996 The American Physical Society.