Bose statistics and classical fields

Classical fields counterpart of the ideal Bose gas statistics in a trap is investigated by performing calculations in the canonical ensemble. There exists the optimal cutoff which allows to match the full probability distribution of the condensate population by its classical counterpart. Universal scaling of that cutoff with temperature and dimensionality is derived.

Figure 1

(Color online) Probability distribution of having $N_{\text{ex}}$ bosons outside the ground state of a 1D trap. Points represent the exact quantum distribution while lines are its classical counterparts with $\hslash \omega \beta K_{\text{max}}=0.9$ (dotted-dashed line), $\hslash \omega \beta K_{\text{max}}=1$ (solid line), and $\hslash \omega \beta K_{\text{max}}=1.1$ (dashed line). Total number of bosons is $N=1000$ and at $\hslash \omega \beta =0.04$.

Figure 2

(Color online) Solution of the Eq. (10) as a function of $T/T_c$ for 3D harmonic trap and total number of bosons $N={10}^3$ (▼), $N=2\times {10}^4$ (◻), and $N=6\times {10}^5$ (●). Dashed line represents a value of $\hslash \omega K_{\text{max}}\beta$ given by the expression (11).

Figure 3

(Color online) Scaling of the cutoff parameter $\hslash \omega K_{\text{max}}\beta$ with dimensionality $D$. Points are given by formula (11); line is the best linear fit: $\hslash \omega K_{\text{max}}\beta =0.477+0.562D$.

Figure 4

(Color online) Probability distribution of having $N_{\text{ex}}$ bosons outside the ground state of a (a) 3D trap and (b) 3D box potentials. Points are exact distribution while solid lines represent its optimal classical counterparts. Total number of bosons is $N=1000$. Notice, for 3D box potential probability distributions are wider as compared to 3D harmonic trap.

Figure 5

(Color online) Average populations $⟨n_{{\mathbf{K}}_{\text{max}}}⟩$ of the final mode $K_{\text{max}}$ (13) (●), and for the classical distribution (14) (◼), for (a) 3D harmonic trap and (b) 3D box potentials. Lines represent expected results in the thermodynamical limit. Number of atoms $N=1000$.