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The Boltzmann distribution assigns probabilities to individual microstates in a canonical ensemble, a statistical ensemble for thermal contact. The system exchanges energy with a thermal reservoir at fixed positive absolute temperature $T$, while its particle number $N$ and external parameters, such as volume, stay fixed. The thermal reservoir is large enough that these exchanges barely change its temperature; the coupling permits thermal equilibrium but contributes negligible interaction energy. All the microstates considered have the same $N$. No chemical potential term is needed in their weights; fixed $N$ does not imply that the chemical potential is zero.

For a finite discrete set of microstates with energies $E_i$, writeHere $k_B$ is the Boltzmann constant, $\beta$ is the inverse temperature, $e^{-\beta E_i}$ is a Boltzmann factor, and $Z$ is the partition function. The Boltzmann factor is a weight; dividing by the partition function makes it a probability. ^[ref]

The useful surprise is that a fixed temperature still permits thermal fluctuations in the system's energy. The Boltzmann distribution predicts their probabilities and expected values without tracking every microscopic exchange. Its exponential function also makes rare excited microstates sensitive to temperature: an energy gap much larger than $k_{B}T$ gives strong Boltzmann suppression, while warming increases their relative populations. A population ratio can therefore reveal a gap or serve as a thermometer, provided thermal equilibrium and the relevant state degeneracies are known.

Historically, Josiah Willard Gibbs's Elementary Principles in Statistical Mechanics (1902) presents the canonical ensemble in Chapter IV. His equations 90–92 give an exponential function of energy and its normalization, in classical phase space. This classical formulation exhibits the same weighting principle. ^{[ref] [ref]}

Boltzmann suppression is the exponential reduction of an equilibrium population when creating it costs energy much greater than the temperature. For a nonrelativistic species with negligible chemical potential, the leading factor is $e^{-m/T}$, multiplied by its momentum phase-space volume. It concerns the equilibrium abundance; freeze-out can preserve a larger population than the instantaneous equilibrium value at later times.

The expanded feature resolves the Q branch: each line has $\Delta J=0$ but starts from a different lower rotational level. In the simple rigid rotor band model,Because the upper rotational constant is larger, the lines extend towards higher wavenumber. Their adjacent spacings areso they crowd together at the low-frequency edge and become increasingly separated. A strictly resolved bending-band analysis may distinguish the upper parity component sampled by Q lines from that sampled by P/R lines; it is not necessary to impose identical high-resolution constants on every branch in this sketch.

The intensity envelope reflects rotational state populations and the rotational transition strength. For this perpendicular Q branch, the usual rotational line-strength factor gives the approximate dependenceAt small $J$, the increasing magnetic degeneracy makes the line intensity rise; at large $J$, Boltzmann suppression dominates. The maximum is therefore at an intermediate $J$. This is a population envelope across distinct transitions, rather than one intrinsically broad vibrational line.

At nucleosynthesis temperatures, $50\mathrm{GeV}$ particles are nonrelativistic. Their equilibrium abundance is negligible because of Boltzmann suppression, so they do not behave as an extra relativistic species contributing its full degree count to $g_{\ast }$. A stable thermal weakly interacting massive particle generally freezes out much earlier; a surviving relic population with the usual dark-matter abundance remains negligible compared with radiation during nucleosynthesis. In that standard relic scenario the helium abundance is essentially unchanged.

The mass and weak coupling alone do not specify the relic density. If an unusually abundant stable population contributes appreciably to the expansion density, it raises $H$ at fixed radiation temperature and increases helium by the same freeze-out and survival mechanism as part (g). Decays or energetic annihilation products would require their own specified history. Thus a universal numerical change cannot be inferred from the mass alone.

A thermal relic is a surviving particle population whose earlier abundance was set by chemical thermal equilibrium with the cosmic plasma. After number-changing reactions cease and cosmic entropy is conserved, its comoving abundance is described by the yield $Y=n/s$. A relativistically decoupled species can retain an appreciable yield, whereas a species remaining in thermal equilibrium after becoming nonrelativistic is subject to Boltzmann suppression. Its later mass density, lifetime and momentum distribution determine its cosmological effects.

A weakly interacting massive particle is a massive dark-matter candidate whose interaction strength permits thermal production and later cosmological particle freeze-out. Its relic abundance depends on the annihilation rate and cosmological history, not on its mass alone. At nucleosynthesis temperatures a 50 GeV example is nonrelativistic: its equilibrium density is reduced by Boltzmann suppression, while an already frozen relic population need not follow the equilibrium density. Its effect on nucleosynthesis depends on whether the surviving energy density is appreciable or whether it injects reaction-altering products.