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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]}

The Gibbs phenomenon is the persistent overshoot near a jump discontinuity in ordinary truncated Fourier series. For this square wave, retain $M$ nonzero odd modes:As $M$ grows, the first peak approaches the jump while its height tends to approximately $1.17897974$. This is an excess of about $0.17898$ above the upper level $1$, or $8.949\%$ of the jump size $2$. The narrower region does not imply a vanishing maximum error. At the jump itself every $Q_M(0)=0$. These observations are compatible with pointwise convergence and $L^2$ convergence but exclude uniform convergence to the discontinuous square wave. ^[ref].

For this particular sum, its derivative is $Q_M^{\prime }(x)=(2/\pi )\sin (2Mx)/\sin x$ away from multiples of $\pi$, so the first positive peak is exactly at $x=\pi /(2M)$. There,This is a midpoint Riemann sum tending to $2\mathrm{Si}(\pi )/\pi$, where $\mathrm{Si}$ is the sine integral. It gives a short explanation of the nonzero limiting excess.

The phenomenon matters whenever a sharp boundary is represented by a limited set of Fourier harmonics: a visible ripple can be a property of the approximation rather than an oscillation in the original profile. Josiah Willard Gibbs reported the phenomenon in his 1899 letter “Fourier's Series” in Nature, volume 59, page 606. This is a documented historical report, without a claim of first discovery. ^[ref].

The strings in Boltzmann entropy of a two-box occupancy are complete microstates only within the binary toy model. They retain a coarse-grained variable for each particle and omit its exact position within a box and its momentum. An actual classical gas instead occupies continuous phase space; obtaining a dimensionless count from a phase space volume requires a specified cell convention and the appropriate treatment of particle identity. A quantum state count must respect the allowed quantum states and quantum statistics. Thus $2^N$ is not a general count of microstates for a gas obeying quantum ideal-gas statistics. ^[ref]

For a discrete probability distribution with unequal weights, use the existing Gibbs entropy article; its equal-weight case gives Boltzmann's entropy formula. Shannon entropy uses the corresponding information measure, with the logarithm base setting its units. A count alone cannot replace the specified probability distribution, as the Boltzmann distribution examples demonstrate. ^[ref]

Overwhelming multiplicity explains why thermodynamic equilibrium is overwhelmingly probable in macroscopic systems under the appropriate statistical assumptions. It does not make low-entropy thermal fluctuations logically impossible. Nor does counting alone derive irreversible macroscopic evolution: one also needs dynamics, an initial preparation, and a reason the system samples the allowed microstates on the timescale of interest. ^[ref]

Historically, Boltzmann's 1877 paper counted microscopic arrangements, called complexions, and related the natural logarithm of their number to entropy. The Sharp and Matschinsky translation appeared in Entropy 17 (2015), pages 1971–2009, DOI 10.3390/e17041971. Its notation differs from ours: Boltzmann's permutability measure $\Omega$ is already logarithmic, whereas here $\Omega$ is the count. ^[ref] Josiah Willard Gibbs's Elementary Principles in Statistical Mechanics (1902) developed the statistical ensemble framework; Chapter XIV compares several statistical definitions of entropy with thermodynamics. ^{[ref] [ref]}