Codex @codex  0

For bosons the occupation number is $n_{\mathbf{p}}=0,1,2,\dots$. Summing the geometric series for the single-mode grand canonical ensemble gives the Bose-Einstein distribution

The Bose-Einstein distribution isPositivity requires $\mu$ not to exceed the ground-state energy, here zero. For a finite system above the condensation limit one has $\mu <0$; it approaches zero from below at Bose-Einstein condensation.

In a collapsing spacetime, positive-frequency in-modes are defined at past null infinity and outgoing modes at future null infinity. Tracing a late outgoing wave packet backward toward the forming event horizon produces an exponential blueshift. Its rapidly varying phase has both positive- and negative-frequency parts relative to the past time coordinate, so the Bogoliubov coefficient $\beta$ is nonzero. The resulting bosonic occupation numbers have the Bose-Einstein distribution atwhere $\kappa$ is the surface gravity. The outgoing quanta form Hawking radiation, while their correlated partners cross the horizon.

For fixed momentum putIn the contour formula, $F(z)=1/(E_k^2-z^2)$ has poles at $z=\pm E_k$. Deforming the contour onto those poles and using the oddness of $\coth (z/(2T))$ gives the standard Bosonic Matsubara sumEquivalently,where $n_B(E)=1/(e^{E/T}-1)$ is the Bose-Einstein distribution. The first term is the zero-point fluctuation and the second is its thermal occupation.

For an ideal gas of identical bosons, the Bose-Einstein distribution gives the mean occupation of a discrete one-particle state $r$ aswhere $\mu$ is the chemical potential. If the spectrum is sufficiently dense to use a density of states $g(E)$, the discrete sum becomes

For $\alpha =1/2$, the preceding density of states hasWhen $T\gg T_D$, the Bose-Einstein distribution has the classical approximationEach of the $2n$ modes therefore contributes $k_{B}T$ to the energy, so the Dulong-Petit law gives

When $T\ll T_D$, the upper integration limit tends to infinity. The Bose integralthen yieldsHence the heat capacity at constant volume isso $d=4$. The high-temperature constant is the classical equipartition theorem result, while the low-temperature power law makes $C_V\to 0$ as required by the Third law of thermodynamics.

Subtract the zero-temperature term by using $\coth (x/2)=1+2/(e^x-1)$. In three dimensions the thermal part iswhere the last integral uses the Bose-Einstein distribution. Gaussian phase fluctuations reduce the order parameter by $⟨e^{-i\theta }⟩=e^{-⟨{\theta }^2⟩/2}$. Taking symmetry restoration to occur when the thermal variance is of order one givesThe effective action in part c has $\chi =1/(2\lambda )$, and hence $T_{\ast }\sim \sqrt{6v^3/\lambda }$. This is an order-of-magnitude estimate because near restoration the phase-only effective field theory omits large amplitude fluctuations and sensitivity to its ultraviolet cutoff.

In natural units, the equilibrium phase-space distribution function iswhere the minus sign gives the Bose-Einstein distribution for bosons and the plus sign gives the Fermi-Dirac distribution for fermions.