Thermal phonon contribution to the reduced entropy s_th = S_th/N_ik_BT as a function of log    (T/T_p) at β = ħω_ci/k_BT_p = 0, 0.1, 1, 10, 100, and 10⁴ (numbers near the lines). The analytical approximations in Eq. (77) (dotted lines) and in Eq. (79) (dashed lines) are compared with the numerical results of Baiko (2009) (solid lines for ). The inset shows the same approximations in the linear scale for s_th; here, the solid line corresponds to the numerical data at β = 100.