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The dilogarithm is the order-two polylogarithm, defined for $|z|<1$ by ${\mathrm{Li}}_2(z)={\sum }_{n\ge 1}{z}^n/n^2$. Its derivative is $-\mathrm{Log}(1-z)/z$, with a removable value at zero; a chosen complex logarithm branch gives its analytic continuation.

For $0<p<1$, $\beta >0$ and $x>0$, this probability density function has cumulative distribution function $F(x)=1-\log (1-(1-p)e^{-\beta x})/\log p$. Differentiating gives the density, while the endpoint limits are zero and one. Its quantile function is $F^{-1}(u)={\beta }^{-1}[\log (1-p)-\log (1-p^{1-u})]$ for $0<u<1$, enabling inverse transform sampling.

Expanding the denominator as a geometric series givesTermwise integration, justified by the monotone convergence theorem, and the gamma function integral yield raw moments $\mathbb{E}[X^j]=-j!{\mathrm{Li}}_{j+1}(1-p)/({\beta }^j\log p)$. Thus the polylogarithm expresses its mean, variance and higher moments. Here $p$ is the complement of the series parameter $1-p$; parameter conventions must be checked when comparing formulas.

For fixed $s$, the factor $n^{-s}$ grows at most polynomially in $n$, whereas $|z{|}^n$ decays exponentially when $|z|<1$. The defining series for the polylogarithm therefore converges absolutely and locally uniformly for every complex $s$ and $|z|<1$. It is holomorphic in $z$ on the unit disc and entire in $s$ there.

Let $m=\mathbb{E}X$ and $v=\mathrm{Var}(X)$, evaluated from the given polylogarithm formulas. Generate a large sample of independent and identically distributed random variables with r_sample_f, and compute its sample mean ${\overline{X}}_n$ and sample variance $S_n^2$ using mean and var. The central limit theorem gives the first standardized check:For the second check, the correct asymptotic variance of the sample variance involves the fourth central moment ${\mu }_4=\mathbb{E}[(X-m{)}^4]$:The denominator is not $\sqrt{2v^2}$ unless the probability distribution has the requisite fourth central moment, as does the normal distribution. Here the probability density function decays exponentially, so all required moments are finite.

To compute ${\mu }_4$, expand the denominator of the probability density function as a geometric series and integrate each nonnegative term. The gamma function integral gives the raw momentsThus $v=m_2-m_1^2$ and ${\mu }_4=m_4-4m_1{m}_3+6m_1^2{m}_2-3m_1^4$. The following R code implements both checks for chosen interior parameters and large n:

p <- 0.4
beta <- 1.3
n <- 100000
raw_m <- vapply(1:4, function(j) {
  -factorial(j) * polylog(j + 1, 1 - p) / (beta^j * log(p))
}, numeric(1))
m <- raw_m[1]
v <- raw_m[2] - m^2
mu4 <- raw_m[4] - 4 * m * raw_m[3] +
  6 * m^2 * raw_m[2] - 3 * m^4
x <- r_sample_f(n, p, beta)
c(mean_check = sqrt(n) * (mean(x) - m) / sqrt(v),
  variance_check = sqrt(n) * (var(x) - v) / sqrt(mu4 - v^2))

For each check separately, an absolute value below $1.96$ is consistent with its approximate $95\%$ confidence interval. Repeat with many independent samples: the two standardized statistics should each resemble a standard normal distribution, with about $5\%$ outside that range. Also vary $p$, $\beta$ and n; a systematic discrepancy that grows with $\sqrt{n}$ suggests a bias. The two checks from the same sample are generally dependent, so passing both is not itself a joint $95\%$ statement, nor can a finite test prove that a generator is correct.

The second central limit theorem follows from the central limit theorem for the sample variance: $\sqrt{n}(S_n^2-v)$ differs by a quantity tending to zero in probability from $n^{-1/2}{\sum }_i[(X_i-m{)}^2-v]$.

Measure energies from the single-particle ground state, and assume the three-dimensional ideal Bose gas in the large-particle-number limit. The harmonic-trap chemical-potential pinning discussed here uses a chemical potential satisfying ${\mu }_{\mathrm{ch}}\le 0$. On cooling, it approaches zero; below the condensation temperature it remains at zero to leading thermodynamic order. Above that temperature it is negative. The detailed negative-chemical-potential branch shape is determined by the density of states; the figure uses the harmonic trap analysed below. The finite-size condensate chemical potential stays slightly below zero rather than being exactly pinned there.

A Bose-Einstein condensate has macroscopic occupation of a single quantum state. A dilute atomic gas needs low temperature and high phase-space density while remaining metastable against ordinary liquid or solid formation. Laser cooling followed by evaporative cooling of atoms in a weak trap can achieve the required regime. For the trapped ideal gas the quantitative threshold is derived below; a uniform ideal gas has $n{\lambda }_T^3=\zeta (3/2)$ at onset. An experimental signature is a narrow low-momentum component appearing within the broader thermal distribution after time-of-flight expansion, as observed in magnetically trapped rubidium-87.^[ref]

Spinless cubic-box wave-vector counting. Impose periodic boundary conditions on a cube of side $L$. The allowed vectors are $\mathbf{k}=(2\pi /L)(n_x,n_y,n_z)$ with integer components. Each state occupies $(2\pi /L{)}^3$ in wave-vector space. For a spinless particle, counting the spherical shell givesThere is no spin multiplicity. This is continuum shell counting; a finite box has discrete levels. Using $ϵ={\hslash }^2{k}^2/(2m)$ gives the free-particle density of states per unit energy and volume

Semiclassical density of states in a central potential. At position $\mathbf{r}$, the available kinetic energy is $ϵ-U(r)$. The local-density approximation integrates the preceding free-particle result over the classically allowed region:This form assumes the allowed radial region is the connected interval $0\le r\le r_{ϵ}$. A nonmonotonic potential would require integrating all allowed shells.

For the isotropic harmonic trap, $r_{ϵ}=\sqrt{2ϵ/(m{\omega }^2)}$. Substituting $r=r_{ϵ}\sin \vartheta$ givesConsequentlyThe exact oscillator ground energy is $3\hslash \omega /2$ above the potential minimum. The harmonic-trap energy-zero convention subtracts that constant, while the displayed density of states is the leading semiclassical continuum expression, neglecting the finite-level corrections produced by that shift. The discrete ground state must be counted separately.

Condensate population. Below the transition, use ${\mu }_{\mathrm{ch}}\simeq 0$ for the excited states. The saturation of harmonic-trap excited states follows from the supplied integral:where $\zeta$ is the Riemann zeta function and $\zeta (3)\simeq 1.202$. Setting $N_{\mathrm{ex}}(T_0)=N$ gives the harmonic-trap critical-temperature prefactor:Above $T_0$ there is no macroscopic ground-state fraction in this approximation. For completeness, above the transition the quantum gas fugacity $z=e^{{\mu }_{\mathrm{ch}}/(k_{B}T)}<1$ satisfies $N=(k_{B}T/\hslash \omega {)}^3{\mathrm{Li}}_3(z)$; the polylogarithm ${\mathrm{Li}}_3$ therefore defines the plotted negative branch. A finite trapped gas has rounded behaviour, weak interactions shift the threshold, and the continuum thermal population is inaccurate when $k_{B}T$ is comparable to or below $\hslash \omega$. The power law is intended for the large-$N$ condensed regime, not an exact discrete-spectrum formula at arbitrarily small temperature.^[ref]

Excited-state saturation. Consider first a uniform three-dimensional noninteracting gas of identical spinless bosons in a large box. Set the one-particle ground energy to zero. The Bose-Einstein distribution is $\bar{n}(ϵ)=[z^{-1}{e}^{\beta ϵ}-1{]}^{-1}$, with quantum gas fugacity $z=e^{\beta \mu }\le 1$. For dispersion $ϵ=p^2/(2m)$, the excited density of states isUsing the thermal de Broglie wavelength ${\lambda }_T=\sqrt{2\pi {\hslash }^2/(m{k}_{B}T)}$, integration givesThe polylogarithm is ${\mathrm{Li}}_s(z)={\sum }_{\ell \ge 1}{z}^{\ell }/{\ell }^s$. Its finite value at $z=1$ implies a maximum excited population $V\zeta (3/2)/{\lambda }_T^3$, with $\zeta (3/2)\simeq 2.612$. Once this is less than $N$, the excess particles occupy the ground state macroscopically. This is Bose-Einstein condensation, rather than occupation of a state by merely one or a few atoms.

Transition and condensate fraction. Setting the saturation number equal to $N$ gives the homogeneous ideal Bose gas condensation temperatureBelow this temperature, $\mu$ approaches zero from below in the thermodynamic limit, and particle conservation givesAbove $T_c$ there is no macroscopic condensate and $\mu <0$ adjusts to keep $N$ fixed. At finite $N$, the chemical potential remains slightly below the ground energy and the transition is rounded.

Energy and the role of geometry. The excited-state internal energy isThe zero-energy condensate contributes no term to this energy in the ideal model. Below $T_c$,Condensation depends on low-energy state counting: for a uniform quadratic-dispersion gas, a finite-temperature bulk condensate requires spatial dimension greater than two. A trap changes the density of states and therefore the temperature law; a three-dimensional harmonic trap gives a cubic, rather than $3/2$, condensate-fraction exponent. Finally, an ideal-gas condensate should not automatically be called a superfluid: its free quadratic spectrum has zero critical speed by the Landau criterion. Interactions matter for the robust collective-flow properties of real dilute condensates.

For $|z|<1$, $a\ge 0$ and integer $N\ge 0$, truncating the defining polylogarithm power series after $N$ terms has remainderThe inequality uses monotonicity of $k^a$ and then the geometric series. The bound applies to negative real $z$ as well as positive real $z$, and gives a rigorous absolute truncation tolerance independently of cancellation. It does not bound floating-point rounding error.