Distribution of quantum states in enclosures of finite size: I

1991 ◽  
Vol 69 (7) ◽  
pp. 813-821
Author(s):  
J. Hugo Souto ◽  
A. N. Chaba
Keyword(s):  
Density Of States ◽  
Three Dimensional ◽  
Finite Size ◽  
Summation Formula ◽  
Einstein Condensation ◽  
Euler Formula ◽  
Bose Einstein Condensation ◽  
Rectangular Slab ◽  
Poisson’S Summation Formula ◽  
Bose Einstein

We show that the expression for the density of states of a particle in a three-dimensional rectangular box of finite size can be obtained by using directly the Poisson's summation formula instead of using the Walfisz formula or the generalized Euler formula both of which can be derived from the former. We also derive the expression for the density of states in the case of an enclosure in the form of an infinite rectangular slab and apply it to the problem of the Bose–Einstein condensation of a Bose gas of noninteracting particles confined to a thin-film geometry.

Phase transitions in finite systems

1983 ◽  
Vol 61 (2) ◽  
pp. 228-238 ◽  
Author(s):  
R. K. Pathria
Keyword(s):  
Phase Transitions ◽  
Thermodynamic Functions ◽  
Finite Size ◽  
Summation Formula ◽  
Einstein Condensation ◽  
Restricted Geometry ◽  
Finite Size Effects ◽  
Mathematical Techniques ◽  
Bose Einstein ◽  
The Given

Mathematical singularities, which are known to be at the heart of phase transitions and are responsible for making the thermodynamic functions of the given system nonanalytic, are a consequence of the thermodynamic limit, viz. N and V → ∞ with N/V staying constant. When a system containing a finite number of particles and confined to a restricted geometry undergoes a phase transition, these singularities get rounded off, with the result that all thermodynamic functions become analytic and vary smoothly with the relevant parameters of the problem. Theoretical analysis of such situations requires the use of special mathematical techniques which may vary drastically from case to case.In the present communication we report the results of a rigorous analysis of the problem of "Bose–Einstein condensation in restricted geometries", which has been carried out by making an extensive use of the Poisson summation formula. Particular emphasis is laid on the growth of the condensate fraction [Formula: see text] as the temperature of the system is lowered, and on the influence of the boundary conditions imposed on the wave functions of the particles. The relevance of these results, in relation to the scaling theory of finite size effects, is also discussed.

Thermodynamic Properties of Bosons in Symmetric Double-well Potentials

1999 ◽  
Vol 54 (3-4) ◽  
pp. 204-212
Author(s):  
J. Choy ◽  
K. L. Liu ◽  
C. F. Lo ◽  
F. So
Keyword(s):  
Thermodynamic Properties ◽  
Excited States ◽  
Low Temperatures ◽  
Three Dimensional ◽  
Energy Difference ◽  
Einstein Condensation ◽  
Thermal Variation ◽  
Bose Einstein Condensation ◽  
Bose Einstein ◽  
Double Well Potential

We study the thermodynamic properties and the Bose-Einstein condensation (BEC) for a finite num-ber N of identical non-interacting bosons in the field of a deep symmetric double-well potential (SDWP). The temperature dependence of the heat capacity C(T) at low temperatures is analyzed, and we derive several generic results which are valid when the energy difference between the first two excited states is sufficiently large. We also investigate numerically the properties of non-interacting bosons in three-dimensional superpositions of deep quartic SDWP's. At low temperatures, we find that C(T) displays microstructures which are sensitive to the value of N and the thermal variation of the condensate frac-tion shows a characteristic plateau. The origin of these features is discussed, and some general conclu-sions are drawn.

FRACTIONAL MATHEMATICAL INVESTIGATION OF BOSE–EINSTEIN CONDENSATION IN DILUTE 87Rb, 23Na AND 7Li ATOMIC GASES

2012 ◽  
Vol 26 (17) ◽  
pp. 1250096 ◽  
Author(s):  
HÜSEYİN ERTİK ◽  
HÜSEYİN ŞİRİN ◽  
DOǦAN DEMİRHAN ◽  
FEVZİ BÜYÜKKİLİÇ
Keyword(s):  
Three Dimensional ◽  
Einstein Condensate ◽  
Einstein Condensation ◽  
Atomic Gases ◽  
Transition Temperatures ◽  
Interparticle Interactions ◽  
Bose Einstein Condensation ◽  
Bose Einstein Condensate ◽  
Einstein Distribution ◽  
Bose Einstein

Although atomic Bose gases are experimentally investigated in the dilute regime, interparticle interactions play an important role on the transition temperatures of Bose–Einstein condensation. In this study, Bose–Einstein condensation is handled using fractional calculus for a Bose gas consisting of interacting bosons which are trapped in a three-dimensional harmonic oscillator. In this frame, in order to introduce the nonextensive effect, fractionally generalized Bose–Einstein distribution function which features Mittag–Leffler function is adopted. The dependence of the transition temperature of Bose–Einstein condensation on α (a measure of fractality of space) has been established. The transition temperatures for the dilute 87 Rb , 23 Na and 7 Li atomic gases have been obtained in consistent with experimental data and the nature of the interactions in the Bose–Einstein condensate has been enlightened. In the course of our investigations, we have arrived to the conclusion that for α < 1 attractive interactions and for α > 1 repulsive interactions are predominant.

On the Bose-Einstein condensation

1950 ◽  
Vol 203 (1073) ◽  
pp. 266-286 ◽  
Keyword(s):  
Specific Heat ◽  
Transition Point ◽  
Energy Levels ◽  
Three Dimensional ◽  
Einstein Condensation ◽  
Bose Einstein Condensation ◽  
Quantum Numbers ◽  
The Mean ◽  
Bose Einstein ◽  
Λ Point

The Bose-Einstein condensation of a gas is investigated. Starting from the well-known formulae for Bose statistics, the problem has been generalized to include a variety of potential fields in which the particles of the gas move, and the number w of dimensions has not been restricted to three. The energy levels are taken to be ε i ≡ ε s 1 , . . . . , s 10 = constant h 2 m s 1 α − 1 a 1 2 + . . . + s w α a w 2 ( 1 ≤ α ≤ 2 ) the quantum numbers being s 1 , w = 1, 2, ..., and a 1 , ..., a w being certain characteristic lengths. (For α = 2, the potential field is that of the w -dimensional rectangular box; for α = 1, we obtain the w -dimensional harmonic oscillator field.) A direct rigorous method is used similar to that proposed by Fowler & Jones (1938). It is shown that the number q = w /α determines the appearance of an Einstein transition temperature T 0 ·For q≤ 1 there is no such point, while for q > 1 a transition point exists. For 1 < q≤ 2, the mean energy ϵ - per particle and the specific heat dϵ - /dT are continuous at T = T 0 · For q > 2, the specific heat is discontinuous at T = T 0 , giving rise to a A λ-point. A well-defined transition point only appears for a very large (theoretically infinite) number N of particles. T 0 is finite only if the quantity v = N/(a 1 .... a w )2/ α ¯ is finite. For a rectangular box, v is equal to the mean density of the gas. If v tends to zero or infinity as N→ ∞, then T 0 likewise tends to zero or infinity. In the case q > 1, and at temperatures T < T 0 ' there is a finite fraction N 0 /N of the particles, given by N 0 /N = 1-(T/T 0 ) q , in the lowest state. London’s formula (1938 b ) for the three-dimensional box is an example of this equation. Some further results are also compared with those given by London’s continuous spectrum approximation.

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