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.

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 DISTRIBUTION OF PARASTATISTICS FUNCTIONS: AN OVERVIEW OF THERMODYNAMICS PROPERTIES

2016 ◽  
Vol 4 (2) ◽  
pp. 179
Author(s):  
R. Yosi Aprian Sari ◽  
W. S. B. Dwandaru
Keyword(s):  
Partition Function ◽  
Thermodynamic Properties ◽  
Thermodynamic Functions ◽  
Energy Levels ◽  
Canonical Partition Function ◽  
Thermodynamics Properties ◽  
Grand Canonical Partition Function ◽  
Bose Einstein ◽  
Grand Canonical ◽  
Number Of Particles

This study aims to determine the thermodynamic properties of the parastatistics system of order two. The thermodynamic properties to be searched include the Grand Canonical Partition Function (GCPF) Z, and the average number of particles N. These parastatistics systems is in a more general form compared to quantum statistical distribution that has been known previously, i.e.: the Fermi-Dirac (FD) and Bose-Einstein (BE). Starting from the recursion relation of grand canonical partition function for parastatistics system of order two that has been known, recuresion linkages for some simple thermodynamic functions for parastatistics system of order two are derived. The recursion linkages are then used to calculate the thermodynamic functions of the model system of identical particles with limited energy levels which is similar to the harmonic oscillator. From these results we concluded that from the Grand Canonical Partition Function (GCPF), Z, the thermodynamics properties of parastatistics system of order two (paraboson and parafermion) can be derived and have similar shape with parastatistics system of order one (Boson and Fermion). The similarity of the graph shows similar thermodynamic properties. Keywords: parastatistics, thermodynamic properties

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.

scholarly journals STRING PICTURE OF BOSE–EINSTEIN CONDENSATION

1999 ◽  
Vol 13 (11) ◽  
pp. 349-362 ◽  
Author(s):  
S. BUND ◽  
A. M. J. SCHAKEL
Keyword(s):  
Chemical Potential ◽  
Canonical Ensemble ◽  
String Tension ◽  
Einstein Condensation ◽  
Winding Number ◽  
Imaginary Time ◽  
Bose Einstein Condensation ◽  
Bose Einstein ◽  
Grand Canonical ◽  
Number Of Particles

A nonrelativistic Bose gas is represented as a grand-canonical ensemble of fluctuating closed spacetime strings of arbitrary shape and length. The loops are characterized by their string tension and the number of times they wind around the imaginary time axis. At the temperature where Bose–Einstein condensation sets in, the string tension, being determined by the chemical potential, vanishes and the strings proliferate. A comparison with Feynman's description in terms of rings of cyclicly permuted bosons shows that the winding number of a loop corresponds to the number of particles contained in a ring.

λ-Transition to the Bose-Einstein Condensate

1995 ◽  
Vol 50 (10) ◽  
pp. 921-930 ◽  
Author(s):  
Siegfried Grossmann ◽  
Martin Holthaus
Keyword(s):  
Heat Capacity ◽  
Three Dimensional ◽  
Einstein Condensate ◽  
Einstein Condensation ◽  
Condensation Temperature ◽  
Harmonic Oscillator Potential ◽  
Bose Einstein Condensate ◽  
Bose Einstein ◽  
Analytical Expressions ◽  
Grand Canonical

Abstract We study Bose-Einstein condensation of comparatively small numbers of atoms trapped by a three-dimensional harmonic oscillator potential. Under the assumption that grand canonical statis­tics applies, we derive analytical expressions for the condensation temperature, the ground state occupation, and the specific heat capacity. For a gas of TV atoms the condensation temperature is proportional to N1/3, apart from a downward shift of order N-1/3. A signature of the condensation is a pronounced peak of the heat capacity. For not too small N the heat capacity is nearly discon­tinuous at the onset of condensation; the magnitude of the jump is about 6.6 N k. Our continuum approximations are derived with the help of the proper density of states which allows us to calculate finite-AT-corrections, and checked against numerical computations.

Effect of a finite number of particles in the Bose-Einstein condensation of a trapped gas

1997 ◽  
Vol 55 (5) ◽  
pp. 3954-3956 ◽  
Author(s):  
R. Napolitano ◽  
J. De Luca ◽  
V. S. Bagnato ◽  
G. C. Marques
Keyword(s):  
Finite Number ◽  
Einstein Condensation ◽  
Bose Einstein Condensation ◽  
Trapped Gas ◽  
Bose Einstein ◽  
Number Of Particles

scholarly journals GROSS–PITAEVSKII MODEL FOR NONZERO TEMPERATURE BOSE–EINSTEIN CONDENSATES

2006 ◽  
Vol 16 (09) ◽  
pp. 2713-2719 ◽  
Author(s):  
KESTUTIS STALIUNAS
Keyword(s):  
Power Spectra ◽  
Pitaevskii Equation ◽  
Nonzero Temperature ◽  
Ginzburg Landau ◽  
Long Wavelength ◽  
Occupation Numbers ◽  
Ginzburg Landau Equations ◽  
Bose Einstein ◽  
Canonical Ensembles ◽  
Grand Canonical

Momentum distributions and temporal power spectra of nonzero temperature Bose–Einstein condensates are calculated using a Gross–Pitaevskii model. The distributions are obtained for micro-canonical ensembles (conservative Gross–Pitaevskii equation) and for grand-canonical ensembles (Gross–Pitaevskii equation with fluctuations and dissipation terms). Use is made of equivalence between statistics of the solutions of conservative Gross–Pitaevskii and dissipative complex Ginzburg–Landau equations. In all cases the occupation numbers of modes follow a 〈Nk〉 ∝ k-2 dependence, which corresponds in the long wavelength limit (k → 0) to Bose–Einstein distributions. The temporal power spectra are of 1/fα form, where: α = 2 - D/2 with D the dimension of space.

The isotherms of an imperfect gas

1953 ◽  
Vol 49 (1) ◽  
pp. 130-135 ◽  
Author(s):  
D. ter Haar
Keyword(s):  
Equation Of State ◽  
General Model ◽  
General Type ◽  
Liquid Drop ◽  
Liquid Drop Model ◽  
Imperfect Gas ◽  
The One ◽  
Different Parts ◽  
Number Of Particles

ABSTRACTThe liquid drop model of an imperfect gas in the form introduced by Wergeland is discussed by using the method of the grand ensembles and the equation of state of the system is derived. This equation of state is of the same general type as the one derived by Mayer for a more general model. It is shown that in both cases the isotherms consist of two analytically different parts in the limit where the number of particles in the system, N, goes to infinity.

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