Bose gas

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

From equation (A2.2.145). the average occupation number of an ideal Bose gas is... 

The chemical potential for an ideal Bose gas has to be lower than the ground-state energy. Otherwise the occupancy (n.p of some state j would become negative. 

The total number of particles in an ideal Bose gas at low temperaPires needs to be written such that the ground-state occupancy is separated from the excited-state occupancies ... 

S. Masuda. Acceleration of adiabatic transport of interacting particles and rapid manipulations of a dilute Bose gas in the ground state. Phys. Rev. A, 86(6) 063624-063630(2012). 

Fig. 1.16. The intermediate order in ideal gas (i) ideal classical gas (full line) (ii) ideal Fermi gas (broken line) (iii) ideal Bose gas (dotted line).

A procedure similar to the condensate separation in the imperfect Bose gas was employed by Lifshitz and Pitaevski [78]. The diagrammatic technique allows us to calculate the reaction rate and steady-state joint correlation functions. A separation of a condensate from terms with k = 0 cannot be done without particle production (p = 0), in which case nA, tiq —> 0 as t —> oo. In this respect the formalism presented by Lushnikov [111] for the non-stationary processes is of certain interest. 

Now from (2) it can be shown that for sufficiently large s, the total number of quanta N=2njc increases proportionally to the total rate of supply s. Equation (4) then shows that a critical rate of supply Sq, exists, such that for s>Sq something analogous to Einstein condensation of a Bose gas must occur. This implies that p, which acts as chemical potential, must approach very closely such that the occupation, n, of becomes very large. Since represents a normal mode, it follows that this excitation is coherent. 

We recall that Bose-Einstein condensation is the macroscopic occupation of the ground state of a system at finite temperature. For a weakly interacting gas, this phase transition occurs when the inter-particle spacing becomes comparable to the thermal de Broglie wavelength A = /2nh /mkBT, where ks is the Boltzmann constant and T is the temperature. A rigorous treatment for the ideal Bose gas yields n > 2.61221 , where n is the density [35]. At a temperature of 50 yuK, for instance, the critical density for hydrogen is 1.8 x 10 cm. ... 

A number of studies on size-sjjecific effects in phase transitions were reported and summarized by Pathria and coworkers. Here we mention only two of them. The first deals with the problem of Bose-Einstein condensation in restricted geometries for an ideal Bose gas. While the general conclusion is that in a finite system the transition is spread over a fractional range of temjjerature AT/7 (oo) L<, where 7 (oo) is the critical temperature of the infinite system and L< is the shortest side of the cuboidal container Li X L2 X L3) enclosing the gas, the actual fraction of the condensate de-jjends on the shajje of the container and on the boundary conditions. Thus in partially infinite geometries (such as L - ao or Li °°) Bose-Einstein... 

Ground state of a weakly interacting Bose gas at zero temperature the Bogoliubov transformation and two-mode squeezing... 

The Josephson current is obtained by multiplying Eq. 50 by the conjugated wave function and integrating by parts. This yields the flow of the charged Bose gas with the current... 

It is worthwhile to note that the charged Bose gas trapped in the double well potential Ua of Eq. 50 behaves as an inverted Josephson junction (N-S-S-N). The super-current, which accompanies the matter wave coherence, is induced between the degenerate resonance states of the adjacent wells at the frequency of the tunnel splitting A response time, as is typical of tunnel junctions (whose frequency cutoff is much smaller than the vibrational frequency even for nano junctions). The coherent oscillations of the Josephson current can be observed by virtue of their slow frequency A V which is robustly controlled by the bias voltage. 

The origin of the nonvauishing Joule-Thomson effect is the effective repulsive (Fermions) and attractive (Bosons) potential exerted on the gas molecules, which arises from the different ways in which quantum states can be occupied in sy.stems obeying Fermi-Dirac and Boso-Einstein statistics, respectively [17]. In other words, the effective fields are a consequence of whether Pauli s antisymmetry principle, which is relativistic in nature [207], is applicable. Thus, a weakly degenerate Fermi gas will always heat up ((5 < 0), whereas a weakly degenerate Bose gas will cool down (5 > 0) during a Joule-Thomson expansion. These conclusions remain valid even if the ideal quantum gas is treated relativistically, which is required to understand... 

The properties of an ideal Bose gas are entirely controlled by permutation symmetry, and the resulting Bose-Einstein statistics are obeyed by the particles. All complicating effects of interparticle interactions, which play a dominant role in determining the properties of bulk liquid He and of ( He)jy clusters, are... 

The basic concept of the existence of a critical temperature for the onset of macroscopic occupation of a single quantum ground state of a boson system is applicable both for liquid He and for weakly interacting low-density atomic vapors. The phenomenon of Bose-Einstein condensation is not limited to an ideal Bose gas and prevails also in a strongly interacting boson system. The bridging between Bose-Einstein condensation in the low-density, weak... 

At T = 0 all the particles reside in the lowest state Go= (3/2) oc)ho- The critical temperature [Eq. (9)] depends on the total number of particles (with a finite value of cohoA ) and not on the density, as is the case for the homogeneous system [Eq. (6)]. The different temperature dependence for the condensate fraction for the confined boson gas [Eq. (10)] and for the uniform Bose gas [Eq. (8)] can be traced to the higher density of states for the harmonic oscillator relative to that for a particle in a box [14, 24]. Theoretical studies for finite size effects in an ideal finite Bose gas [80, 126] and for a Bose gas trapped in a harmonic potential [14, 127] provided novel information on finite boson systems. These issues will be addressed in Section I.E. 

Finite size effects on the critical temperature for Bose-Einstein condensation of a noninteracting Bose gas conhned in a harmonic trap manifests the reduction of the condensate fraction and the lowering of the transition temperature, as compared to the infinite system [14, 127]. Eor an N particle condensate, the shift of the critical temperature Tc, relative to that for the N by the cluster size scaling relation [14, 127]... 

Here A — d—d, and kd and are the wave vectors in d and A dimensions, respectively. We notice the analogy between Equ. (27) and the equation (5) determining the transition temperature of the free anisotropic Bose gas with aGL - 0, S tends towards the constant a /n which here plays a role which is analogous to the total number of free bosons in the boson equation. For Tc < Fmf, Equ. (27) can conveniently be rewritten as... 

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