PhD thesis in English

Summations in the last expression can be exactly performed only numerically, so inorder to proceed further in an analytic way, we use a semiclassical approximation [19,5]. Practically, this means that we neglect the discretness of energy levels and replacesummations by the integrals over now continuous n x , n y and n z . Mathematicallyspeaking, the approximation is justified if the values of the functions which wesum over do not vary significantly over the summation step. For the Eq. (1.7), thementioned condition translates into the high-temperature limit k B T ≫ (E n+1 −E n ),where the thermal energy is larger than the typical spacing of energy levels. Withthis simplification, we obtain:N ≈===∫ ∞ ∫ ∞ ∫ ∞0∞∑0∏m=1 j=x,y,z0∫ ∞0dn x dn y dn ze β0 c (ωxnx+ωyny+ωznz) − 1dn j e −β0 c mω jn j1 ∑ ∞1(βc 0)3 ω x ω y ω z m 3m=11(β 0 c) 3 ω x ω y ω zζ 3 , (1.8)where we introduce the Bose function (the polylogarithm function) ζ α (x) = ∑ ∞ x nn=1 n αand the abbreviation ζ α ≡ ζ α (1) for the Riemann zeta function. From the lastexpression, we find that the condensation sets in for:k B Tc 0 = ¯ω N 1/3 , (1.9)ζ 1/33where ¯ω is a geometric mean ¯ω = (ω x ω y ω z ) 1/3 . Consequently, we distinguish thephase with a macroscopic occupation of the ground state and denote it as the condensatephase, and the phase without a macroscopic value of N 0 that is designatedas the (normal) gas phase. However, the phase transition is well defined only in thethermodynamic limit [5, 20] and hence in the case of finite-size systems we refer toT 0 cas the condensation temperature instead of the critical temperature.The BEC phase transition is usually depicted in the phase diagram showing thecondensate fraction N 0 /N versus temperature. From Eq. (1.8), we infer that in thecondensate phase, the number of atoms in the thermal component is proportional5