When the chemical potential μ limits to zero, the steady increase of particles in the ground state with e=0 under Bose-Einstein gas in equilbirium distribution. This is called Bose-Einstein condensation. Though Monte Carlo simulation in quantum ensenble, when the number of particles in the system in small we find: First, the fluctuation of the number of particles in the ground state is very quick and is analogous to energy level distribution. Second, when TT c , the system appears Bose-Einstein condensation, but whether T c is congruous with the theoretical value should be further verified.
When the chemical potential μ limits to zero, the steady increase of particles in the ground state with e=0 under Bose-Einstein gas in equilbirium distribution. This is called Bose-Einstein condensation. Though Monte Carlo simulation in quantum ensenble, when the number of particles in the system in small we find: First, the fluctuation of the number of particles in the ground state is very quick and is analogous to energy level distribution. Second, when TT c , the system appears Bose-Einstein condensation, but whether T c is congruous with the theoretical value should be further verified.
In this paper, first of all, we proved if the ideal Bose gas with a finite volume and number of particles has a non-degenerate single-particle energy level n, the chemical potential μ can take the value μn = n and there is a phase transition temperature Tp,n, where n = 0,1,2 Taking 0 ≤ n Tp,n+1. When the temperature T > Tp,n or T ≤ Tp,n+1, μ # n and the most probable occupation number Nn = 0. In the temperature interval Tp,n ≥ T > Tp,n+1, μ = n and 0 ≤ Nn = N - NjsupNn, where Nj is the most probable occupation number in the degenerate level j. Thus, if the finite ideal Bose gas has some non-degenerate single-particle levels, there exists a characteristic temperature Tp = Tp,0. The chemical potential μ is quantized when T ≤ Tp, and this leads to the creation of a macroscopic quantum state (pure state) or Bose-Einstein condensation phase. Tp = Tp,0 is a first-order phase transition point, Tp,n#0 is a zero-order phase transition point. Next, we obtained a new expression of the most probable distribution of the finite ideal Bose gas. In this expression Nj is directly proportional to gj - 1, where gj and Nj are, respectively, the degeneracy and the most probable occupation number in the degenerate level j. This property agrees with what chemical potential can be quantized if there is a non-degenerate level for the finite ideal Bose gas. Finally, using this expression, we defined a micro-partition function M, obtained the statistical expressions of some thermodynamical quantities.
Based on grand canonical assembles statistics,the Bose Einstein Condensation (BEC) in non interacting bound atomic system is studied.Some basic concepts of BEC in conventional infinite Bose system are extended.The condition of chemical potential μ ≤0 will not be guarantted for the present finite bound system,and the vapour with μ 0,the state without condensation at any eigenstate,might be possible in principle even below a critical temperature T C .Gibbs functions for both the gas and the BEC state at temperature TT C have been calculated.The condition for BEC to appear is presented.The relation of BEC temperature T C with the density of particles is estimated and the result is qualitatively in good agreement with the experimental results.
We investigate the zero-temperature properties of a diluted homogeneous Bose gas made of N particles interacting via a two-body square-well potential by performing Monte Carlo simulations. We tune the interaction strength to achieve arbitrary positive values of the scattering length and compute by Monte Carlo quadrature the energy per particle E/N and the condensate fraction N0/N of this system by using a Jastrow ansatz for the many-body wave function, which avoids the formation of the self-bound ground state and describes instead a (metastable) gaseous state with uniform density. In the unitarity limit, where the scattering length diverges while the range of the interatomic potential is much smaller than the average distance between atoms, we find a finite energy per particle [$E/N=0.70\phantom{\rule{4pt}{0ex}}{\ensuremath{\hbar}}^{2}{(6{\ensuremath{\pi}}^{2}n)}^{2/3}/2m$, with n the number density] and a quite large condensate fraction (N0/N=0.83).
Bose-Einstein condensation is known to occur in one-dimensional power-law potentials $V(x)\ensuremath{\propto}|x{|}^{\ensuremath{\eta}}$ as a true phase transition for \ensuremath{\eta}2, but only if there are no interactions between the particles. We show, via a path-integral quantum Monte Carlo scheme, that for both \ensuremath{\eta}2 and \ensuremath{\eta}=2 the spatial distribution of finite numbers of hard-core bosons suddenly becomes bimodal below a certain temperature, with a central condensate of particles distributed according to the lowest single-particle eigenstate. At still lower temperatures, the hard-core interactions cause the single-particle ground state to become a poor description for the interacting gas. If \ensuremath{\eta}2, the energy per particle undergoes a sudden decrease near the same temperature at which the bimodal distribution appears. It is found that this drop in energy disappears if the range of the interaction potential is sufficiently large.
Bose–Einstein condensation (BEC) is discussed at the level of an advanced course of statistical thermodynamics, clarifying some formal and physical aspects that are usually not covered by the standard pedagogical literature. The non-conventional approach adopted starts by showing that the continuum limit, in certain cases, cancels out the crucial role of the bosonic ground level. If so, a correct treatment of the problem, including the ground level population N 0 by construction, leads to BEC in a straightforward way. For a density of states of the form G ( )∝ γ , the chemical potential µ is explicitly calculated as a function of the temperature T and of the number N of bosons, for various significant values of the positive exponent γ. In the thermodynamic limit, in which the boson number N diverges and BEC is a sharp process, the chemical potential µ is a singular function of T at the critical temperature T B , determined by an appropriate critical exponent. The condensate population N 0 is studied analytically and numerically as a function of the temperature, for various values of N and for different γ. This provides an accurate description of the way BEC approaches the character of a sharp phase transition. Some aspects of the real experiments on BEC, involving a finite number of bosons, are also illustrated.