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Advances in physics theories and applications

Bose Einstein Condensation and Thermodynamics Properties in Bose-Einstein Distribution

Joseph Gordian Atat, Akaninyene Daniel Antia, Okon P. Akpan

Published Jan 1, 2016• 0 citations
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Abstract

The calculation of the thermodynamic properties in the Bose-Einstein distribution is presented.The behaviour of the Bose-gas in the Bose-Einstein distribution is analysed based on high and low temperature dependence.At > all particles are in the excited state and the ground state is essentially occupied.But at B T T < particles gradually falls to the ground state and eventually contains all the particles in the system as .0 → TThe analytical expressions for some important parameters such as Bose-Einstein condensation of an ideal Bose gas, the population density of the excited and ground state particles are also presented.Heart capacity of an ideal Bose gas and total internal energy are also calculated and the results presented graphically.The results obtained are consistent with those found in the literatures.

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The calculation of the thermodynamic properties in the Bose-Einstein distribution is presented.The behaviour of the Bose-gas in the Bose-Einstein distribution is analysed based on high and low temperature dependence.At > all particles are in the excited state and the ground state is essentially occupied.But at B T T < particles gradually falls to the ground state and eventually contains all the particles in the system as .0 → TThe analytical expressions for some important parameters such as Bose-Einstein condensation of an ideal Bose gas, the population density of the excited and ground state particles are also presented.Heart capacity of an ideal Bose gas and total internal energy are also calculated and the results presented graphically.The results obtained are consistent with those found in the literatures.
For the relativistic Bose gas in the gravitational field,the Bose-Einstein condensation(BEC) and the thermodynamic properties are investigated using the local density approximation(LDA).The thermodynamic quantities are calculated in the non-relativistic(NR) and ultra-relativistic(UR) limits,including the transition temperature,the ground state fraction,the energy and the heat capacity of the system.We also discussed the continuity of the heat capacity at the critical temperature.It is found that the effect of the relativistic results in the increase of the gap of the heat capacity.
Based on the equation of state of an ideal Bose gas, the heat capacities at constant volume and constant pressure of the Bose system are derived and used to analyse the low-temperature behaviour of the Bose system. It is expounded that some important thermodynamic processes such as a constant pressure and an adiabatic process cannot be carried out from the region of T > Tc to that of T < Tc, where Tc is the critical temperature of Bose–Einstein condensation of the Bose system. Consequently, some typical thermodynamic cycles such as the Carnot cycle, Brayton cycle, Otto cycle, Ericsson cycle, Diesel cycle and Atkinson cycle cannot be operated across the critical temperature Tc of Bose–Einstein condensation of an ideal Bose gas.
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∊/ k , where Δ∊ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest-energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermodynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced. From this we obtain the density profile and the velocity distribution of particles, and use these to determine the shape of an anisotropic cloud after free expansion. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. Finally, Sec. 2.5 treats corrections to the transition temperature due to a finite particle number.
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∈/ k , where Δ∈ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermo dynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced, and from this we determine the density profile and the velocity distribution of particles. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. The final two sections are devoted to effects not captured by the simplest version of the semi-classical approximation: corrections to the transition temperature due to a finite particle number (Sec. 2.5), and thermodynamic properties of gases in lower dimensions (Sec. 2.6).
Constraint of a fixed total number of particles yields a correlation between the fluctuation of particles in different states in the canonical ensemble. Here we show that, below the temperature of Bose–Einstein condensation (BEC), the correlation part of the entropy of an ideal Bose gas is cancelled by the ground-state contribution. Thus, in the BEC region, the thermodynamic properties of the gas in the canonical ensemble can be described accurately in a simplified model which excludes the ground state and assumes no correlation between excited levels.