Statistical mechanics and the partition of numbers I. The transition of liquid helium

1949 ◽  
Vol 199 (1058) ◽  
pp. 361-375 ◽  
Keyword(s):  
Low Temperatures ◽  
Energy Levels ◽  
The Other ◽  
Einstein Condensation ◽  
Perfect Gas ◽  
Bose Einstein Condensation ◽  
Condensation Phenomenon ◽  
Orthodox Theory ◽  
Bose Einstein ◽  
Very High

The existing theory of ‘Bose-Einstein condensation’ is compared with some results obtained from the theory of partition of numbers. Two models are examined, one in which the energy levels are all equally spaced, the other being the perfect gas model. It is concluded that orthodox theory can be relied upon at very high and at very low temperatures, also that the condensation phenomenon is a real one, but that it is not correctly described by orthodox theory, the position of the transition temperature and the form of the specific heat anomaly both being given wrongly.

On Bose-Einstein Condensation

1954 ◽  
Vol 50 (1) ◽  
pp. 65-76 ◽  
Author(s):  
P. T. Landsberg
Keyword(s):  
Closed System ◽  
Canonical Ensemble ◽  
Energy Levels ◽  
Volume Density ◽  
Einstein Condensation ◽  
Bose Einstein Condensation ◽  
Quantum Numbers ◽  
Finite Integer ◽  
Bose Einstein ◽  
Grand Canonical

ABSTRACTThis paper contains a proof that the description of the phenomenon of Bose-Einstein condensation is the same whether (1) an open system is contemplated and treated on the basis of the grand canonical ensemble, or (2) a closed system is contemplated and treated on the basis of the canonical ensemble without recourse to the method of steepest descents, or (3) a closed system is contemplated and treated on the basis of the canonical ensemble using the method of steepest descents. Contrary to what is usually believed, it is shown that the crucial factor governing the incidence of the condensation phenomenon of a system (open or closed) having an infinity of energy levels is the density of states N(E) ∝ En for high quantum numbers, a condition for condensation being n > 0. These results are obtained on the basis of the following assumptions: (i) For large volumes V (a) all energy levels behave like V−θ, and (b) there exists a finite integer M such that it is justifiable to put for the jth energy level Ej= c V−θand to use the continuous spectrum approximation, whenever j ≥ M c θ τ are positive constants, (ii) All results are evaluated in the limit in which the volume of the gas is allowed to tend to infinity, keeping the volume density of particles a finite and non-zero constant. The present paper also serves to coordinate much of previously published work, and corrects a current misconception regarding the method of steepest descents.

scholarly journals Bose-Einstein Condensation of Confined Atomic Gases at Ultra Low Temperatures

2017 ◽  
Vol 9 (5) ◽  
pp. 96
Author(s):  
M. Serhan
Keyword(s):  
Low Temperatures ◽  
Chemical Potential ◽  
Scattering Length ◽  
Einstein Condensation ◽  
Pitaevskii Equation ◽  
Atomic Gases ◽  
Bose Einstein Condensation ◽  
Gaussian Type ◽  
Negative Scattering Length ◽  
Bose Einstein

In this work I solve the Gross-Pitaevskii equation describing an atomic gas confined in an isotropic harmonic trap by introducing a variational wavefunction of Gaussian type. The chemical potential of the system is calculated and the solutions are discussed in the weakly and strongly interacting regimes. For the attractive system with negative scattering length the maximum number of atoms that can be put in the condensate without collapse begins is calculated.

Possible Bose-Einstein Condensation of Polygonal Clusters in 2D-Materials

2019 ◽  
Vol 297 ◽  
pp. 204-208
Author(s):  
Abid Boudiar
Keyword(s):  
Free Energy ◽  
Critical Temperature ◽  
Ground State ◽  
Low Temperatures ◽  
2D Materials ◽  
Equilibrium Structure ◽  
Einstein Condensation ◽  
Bose Einstein Condensation ◽  
Exchange Approximation ◽  
Bose Einstein

This study investigates the possibility of Bose-Einstein condensation (BEC) in 2D-nanoclusters. A ground state equilibrium structure involves the single phonon exchange approximation. At critical temperature, the specific heat, entropy, and free energy of the system can be determined. The results support the existence of BEC in nanoclusters, and they lead to predictions of the behaviour of 2Dmaterials at low temperatures.

The perfect Bose-Einstein gas in the theory of the quantum-mechanical grand canonical ensembles

1952 ◽  
Vol 212 (1111) ◽  
pp. 552-558 ◽  
Keyword(s):  
Equation Of State ◽  
Energy Levels ◽  
Einstein Condensation ◽  
Quantum Mechanical ◽  
Condensation Phenomenon ◽  
Bose Einstein ◽  
Canonical Ensembles ◽  
Grand Canonical ◽  
Different Parts ◽  
Number Of Particles

The theory of quantum-mechanical grand canonical ensembles is used to derive for the case of a perfect Bose-Einstein gas the average number of particles in the different energy levels, the fluctuations in these numbers and the equation of state. The Einstein condensation phenomenon is then discussed, and it is shown that in a p-v diagram (v being the specific volume) the isotherm consists of two analytically different parts in the limit where the number of particles in the system, JV, goes to infinity. It is also shown that for finite N at the critical volume ∂ n p/∂v n is of the order N1/3 (n-2) in accordance with a result obtained by Wergeland & Hove-Storhoug.

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.

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.

On the Bose-Éinstein condensation of a perfect gas

Physica ◽  
1949 ◽  
Vol 15 (7) ◽  
pp. 671-672 ◽  
Author(s):  
S.R. De Groot ◽  
C.A. Ten Seldam
Keyword(s):  
Einstein Condensation ◽  
Perfect Gas ◽  
Bose Einstein Condensation ◽  
Bose Einstein

scholarly journals INEQUIVALENT REPRESENTATIONS OF A q-OSCILLATOR ALGEBRA IN A QUANTUM q-GAS

1995 ◽  
Vol 09 (14) ◽  
pp. 883-887 ◽  
Author(s):  
M. R-MONTEIRO ◽  
L. M. C. S. RODRIGUES
Keyword(s):  
Critical Temperature ◽  
Virial Expansion ◽  
Einstein Condensation ◽  
Oscillator Algebra ◽  
Fundamental Representation ◽  
Bose Einstein Condensation ◽  
Condensation Phenomenon ◽  
Bose Einstein ◽  
Inequivalent Representations ◽  
Λ Point

We study the consequences of inequivalent representations of a q-oscillator algebra on a quantum q-gas. As in the "fundamental" representation of the algebra, the q-gas presents the Bose-Einstein condensation phenomenon and a λ-point transition. The virial expansion and the critical temperature of condensation are very sensible to the representation chosen; instead, the discontinuity in the λ-point transition is unaffected.

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