PhD thesis in English

3. Rotating ideal BECBy rearranging the summation order in the previous equation, we finally obtain:N = B 0 (µ, T) += B 0 (µ, T) +∞∑ ∑ ∞e jβµ e −jβEn ,j=1n=1∞∑e ( jβµ Z 1 (jβ) − e ) −jβE 0, (3.7)j=1where the summation on the right-hand side of the previous equation corresponds tothe cumulant expansion. In order to avoid any double-counting, we have subtractedthe contribution of the ground state within the single-particle partition functionbecause a possible macroscopic occupation of the ground state is separately takeninto account.The BEC phase transition is achieved only in the thermodynamic limit of aninfinite number of atoms, thus making numerical studies of the condensation increasinglydifficult. Usually, the problem is solved by fixing the chemical potential µat the low temperatures of the condensate phase to the ground-state energy, i.e. bysetting µ = E 0 . This requires that the ground state is treated separately by associatinga macroscopic value N 0 to the ground-state occupation number B 0 (µ, T), forT < T c . Thus, from Eq. (3.7), the total number of particles in the condensate phasefollows to be:∞∑ (N = N 0 + ejβE 0Z 1 (jβ) − 1 ) . (3.8)j=1The equation (3.8) yields the temperature dependence of N 0 . Within the gas phase,where the macroscopic occupation of the ground state vanishes, i.e. we have N 0 =0, Eq. (3.7) determines the temperature dependence of the chemical potential µ.Therefore, the value of β c = 1/k B T c , which characterizes the boundary betweenboth phases, follows from Eq. (3.8) by setting N 0 = 0 and µ = E 0 :N =∞∑ [ejβ cE 0Z 1 (jβ c ) − 1 ] . (3.9)j=1We conclude that, for a given number N of ideal bosons, the condensation temperaturecan be exactly calculated only if both the single-particle ground-state energyE 0 and the full temperature dependence of the one-particle partition function (3.5)are known.61