# Boltzmann distribution The **Boltzmann distribution** gives the probability that a system in thermal contact with a reservoir at [[Temperature|temperature]] T is found in a particular state of [[Energy_level|energy]] `E_n`: `p_n = exp(−E_n/(k_B·T))/Z`, where the normalizing sum `Z = sum(exp(−E_n/(k_B·T)))` runs over every state and is called the partition function.[^likharev-sm-ch2] Only differences of energy matter, since the ratio of any two probabilities is `p_i/p_j = exp(−(E_i − E_j)/(k_B·T))` and the zero of energy cancels. The whole of equilibrium [[Statistical_mechanics|statistical mechanics]] is contained in that ratio: the higher the level, the rarer it is, and `k_B·T` is the yardstick that decides how much rarer. In the microsim below the reader slides the reduced temperature T/θ, where `theta = hbar·w0/k_B` is the level spacing expressed as a temperature, and the populations redraw as a column of bars. A two-level preset shows the simplest case: the excited fraction is `1/(1 + exp(theta/T))`, which is 0.119 at T = θ/2, 0.269 at T = θ and 0.475 at T = 10·θ, approaching but never passing one half (derived). A second preset loads the [[Quantum_harmonic_oscillator|harmonic-oscillator]] ladder `E_n = hbar·w0·(n + 1/2)`, whose partition function has the closed form `Z = 1/(1 − exp(−theta/T))` when energies are measured from the ground state, and whose mean occupation is `<n> = 1/(exp(theta/T) − 1)`.[^likharev-ladder] Below T = θ the bars collapse onto the ground state and the mean energy stops following the temperature — it freezes out, sitting at the [[Zero-point_energy|zero-point]] value `hbar·w0/2` while T keeps falling. That freeze-out is the microscopic reason for the heat-capacity steps of the [[Maxwell–Boltzmann_distribution|Maxwell–Boltzmann]] page; the heat-capacity curve C(T) itself belongs to the [[Einstein_solid|Einstein solid]] and is linked from here rather than drawn twice. On the [[Physics]] flagship this page serves Part II — Core theories, in the section *The Boltzmann factor and the partition function* (row P26), which Physics owns. It sits directly beneath the [[Maxwell–Boltzmann_distribution|Maxwell–Boltzmann distribution]], of which it is the general case: that page's speed law is this exponential applied to a continuum of translational states. ## The distribution The distribution describes a system whose energy is *not* fixed, because it can exchange heat with a much larger reservoir. What is fixed is the reservoir's temperature, and the resulting family of probabilities is the [[Canonical_ensemble|canonical ensemble]].[^likharev-sm-ch2] The exponential form follows from one requirement: that the probability of a compound state factorize into the probabilities of its parts while the energies add. Only an exponential turns a sum into a product, and the constant in the exponent must be an energy, which is `k_B·T` with `k_B` = 1.38×10⁻²³ J/K.[^likharev-thermo] Everything measurable follows from Z. The mean energy is `<E> = −d(ln Z)/d(beta)` with `beta = 1/(k_B·T)`; the [[Thermodynamic_free_energy|Helmholtz free energy]] is `F = −k_B·T·ln(Z)`; the [[Entropy|entropy]] and the [[Heat_capacity|heat capacity]] follow by differentiating F.[^likharev-sm-ch2] The partition function is therefore not an accounting convenience but the generating function of the thermodynamics, which is why the [[Partition_function_(statistical_mechanics)|partition function]] gets a name of its own. The microsim's ladder makes the machinery visible on a single closed form. With `x = theta/T`, the mean occupation `<n> = 1/(exp(x) − 1)` is 0.156 at T = θ/2, 0.582 at T = θ and 1.541 at T = 2·θ (derived). Expanding for small x gives `<n> ≈ T/theta − 1/2`, so the mean energy `<E> = hbar·w0·(<n> + 1/2)` approaches `k_B·T` — exactly the classical [[Equipartition_theorem|equipartition]] result of `(1/2)·k_B·T` for each of the oscillator's two quadratic terms (derived). At the other end, when T falls to θ/4 the mean occupation is 0.0187 and the mean energy has flattened at 0.519·`k_B·theta` against a classical prediction of 0.25 (derived): the ladder has frozen. Nothing in the formula changes at the crossover; the exponential simply stops being able to reach the first rung. ## Generalized Boltzmann distribution The same construction handles systems that exchange more than heat. If a system can trade particles as well as energy with its reservoir, the probability of a state becomes `p_n = exp(−(E_n − mu·N_n)/(k_B·T))/Z`, with one multiplier for each conserved quantity: `1/(k_B·T)` for the energy and `mu/(k_B·T)` for the particle number, where μ is the [[Chemical_potential|chemical potential]].[^likharev-sm-ch2] Adding a term for volume, or for magnetization in a field, extends the pattern without changing its shape. Likharev's treatment writes temperature in energy units throughout, so that the ideal-gas law reads `P·V = N·T` and the Gibbs energy `G = F + P·V = N·[T·ln(V/N) + f(T)] + N·T`; the factors of `k_B` reappear only when a temperature is quoted in kelvin.[^likharev-thermo] There is a deeper reading of the multipliers. The Boltzmann distribution is the one that maximizes [[Entropy_(information_theory)|entropy]] subject to a fixed mean energy, and each multiplier is the price of one constraint. That derivation makes no reference to mechanics, which is why the same exponential reappears in [[Information_theory|information theory]], in inference and in the sections below — and why [[Boltzmann's_entropy_formula|Boltzmann's entropy formula]] `S = k_B·ln(W)`, counting [[Microstate_(statistical_mechanics)|microstates]] at fixed energy, and the canonical distribution at fixed temperature are two views of one statement. A generalization that is *not* available is worth naming. The derivation assumes the reservoir is large enough that its temperature does not move when the system takes energy from it. For a small bath, or for a system with long-range interactions whose energy is not additive, the factorization argument fails and the exponential is not guaranteed. ## In statistical mechanics The distribution's reach in physics comes from the fact that it says nothing about what the levels are. Give it any spectrum and it returns the populations. Applied to the translational states of a gas it produces the [[Maxwell–Boltzmann_distribution|Maxwell–Boltzmann distribution]] of speeds, the exponential in `m·v^2/(2·k_B·T)` being the Boltzmann factor for kinetic energy alone. Applied to the rotational and vibrational levels of a molecule it produces the heat-capacity staircase: measured molar heat capacities give `C_V/R` = 1.50 for helium, neon and [[Argon|argon]] and 2.50 for carbon monoxide, while [[Diatomic_molecule|diatomic]] hydrogen climbs from d = 3 below about 60 K to d = 5 between roughly 300 K and 600 K and to d = 7 above about 3,000 K.[^os-cv] Those plateau temperatures are level spacings in disguise: setting `k_B·T` equal to the gap puts the rotational spacing near 5.2 meV and the vibrational spacing near 0.26 eV (derived from the book's plateau temperatures). Applied to a two-level system it gives the physics of magnetic resonance, of [[Spin_(physics)|spin]] populations and of the [[Laser|laser]]. Because `p_1/p_0 = exp(−ΔE/(k_B·T))` is always less than one for positive T, a thermal system can never have more population above than below; [[Population_inversion|population inversion]] is by definition non-thermal, and pumping a medium into that state is what makes stimulated emission outrun absorption. Applied to an energy barrier rather than a level it gives the exponential of [[Chemical_kinetics|reaction kinetics]]: the fraction of molecules carrying at least an [[Activation_energy|activation energy]] `E_a` is `exp(−E_a/(k_B·T))`, which is the temperature dependence the [[Arrhenius_equation|Arrhenius equation]] fits, and the same factor sets carrier concentrations in a [[Doping_(semiconductor)|doped]] [[Semiconductor|semiconductor]] and the equilibrium potentials of the [[Nernst_equation|Nernst equation]]. The distribution also underwrites the standard models of collective behaviour. An [[Ising_model|Ising]] lattice with energy `E = −J·sum(s_i·s_j)` is simulated by sampling exactly this distribution, and the resulting [[Ferromagnetism|ferromagnetic]] transition at a [[Curie_temperature|critical temperature]] is a competition between the energy term, which prefers alignment, and the entropy of the many disordered configurations the Boltzmann weights admit.[^anagnostopoulos-ising] Quantized lattice vibrations — [[Phonon|phonons]] in an [[Einstein_solid|Einstein]] or [[Debye_model|Debye]] solid — carry the same `<n> = 1/(exp(x) − 1)` occupation as the microsim's ladder, which is also the factor in [[Planck's_law|Planck's law]] for [[Bose–Einstein_statistics|photons]]. ## In mathematics Stripped of physics, `exp(−E_n/(k_B·T))/Z` is a way of turning any list of real numbers into a probability distribution: subtract, exponentiate, normalize. The parameter `1/(k_B·T)` controls how sharply the largest term dominates, running from a uniform distribution as T → ∞ to a point mass on the minimum as T → 0. This is the Gibbs measure, and it is the unique distribution of maximum entropy at fixed mean, which is why it turns up wherever a probability has to be assigned from an incomplete constraint — in inference, in [[Machine_learning|machine learning]] classifiers, and in the stochastic units of an energy-based [[Neural_network|neural network]].[^shannon1948] Its most important algorithmic use is sampling. The Metropolis rule accepts a proposed change with probability `A = min(1, exp(−beta·dE))`, a choice constructed from detailed balance so that the long-run visiting frequency of each configuration is exactly its Boltzmann weight — which lets a [[Monte_Carlo_method|Monte Carlo]] simulation compute averages over a space far too large to enumerate.[^anagnostopoulos-ising] A two-dimensional Ising lattice of only a thousand spins already has more configurations than can be summed in the age of the universe, so sampling is not an optimization but the only route.[^likharev-ising] The implementation detail that makes it fast is that only a few distinct values of ΔE occur on a lattice, so the exponentials can be tabulated once per temperature: at β = 0.21 the two entries are 0.4317 and 0.1864 (derived), and the exact critical point of the model is `beta_c = ln(1 + sqrt(2))/2` = 0.4406867935.[^anagnostopoulos-ising] Lowering the temperature on such a sampler while it runs is the basis of annealing methods for [[Metropolis–Hastings_algorithm|optimization]]. ## In economics An exponential distribution appears in economics for a reason that is formally the same and physically quite different. If a conserved quantity — money, in the simplest models — is repeatedly redistributed among agents by random pairwise exchanges that conserve the total, the stationary distribution of holdings is exponential, `p(m) ∝ exp(−m/T)`, with an effective "temperature" equal to the mean holding. Nothing in the argument requires the agents to be molecules: it requires only conservation, randomness and a large number of exchanges, the same three ingredients that give the Boltzmann distribution in a gas.[^dragulescu2000] The fit to real data is partial, and the failure is the interesting part. The exponential describes the lower part of income and wealth distributions reasonably well, but the upper tail follows a power law of the kind Vilfredo Pareto identified in 1897, which no conservative random-exchange model produces.[^pareto1897] The gap is diagnostic: a power-law tail signals multiplicative dynamics — returns proportional to holdings — rather than the additive, conserved exchanges the Boltzmann argument assumes. Read that way the distribution serves as a null model. Where an economy matches it, random exchange of a conserved quantity suffices to explain the spread; where it departs, some other mechanism is at work, and the departure measures it. ## See also - [[Partition_function_(statistical_mechanics)]] - [[Canonical_ensemble]] - [[Einstein_solid]] - [[Statistical_mechanics]] - [[Maxwell–Boltzmann_distribution]] - [[Boltzmann's_entropy_formula]] - [[Equipartition_theorem]] - [[Ludwig_Boltzmann]] ## References [^likharev-sm-ch2]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Chapter 2, pp. 29–72 (page to pin) — the Gibbs (canonical) distribution `p_n = exp(−E_n/T)/Z` with `Z = sum(exp(−E_n/T))`, the free energy `F = −T·ln(Z)`, the mean energy as a derivative of `ln(Z)`, and the grand distribution with a chemical-potential term for exchanged particles. The chapter's page range is taken from the Portal Books chapter index; the sub-manual's deep-read covered Chapters 1 and 4 of this book but not Chapter 2, so the individual pages are unpinned. Portal Book 075, https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^likharev-thermo]: Likharev (2013), *Essential Graduate Physics, Part SM*, Chapter 1, p. 10 (`k_B` = 1.38×10⁻²³ J/K; note that the extract prints the calorie as 4.148 J where the thermochemical value is 4.184 J), pp. 10–12 (`dE = T·dS − P·dV` and `T = (dE/dS)_V`, restored in standard form after extraction damage), p. 19 (`P·V = N·T` with temperature in energy units, and `G = F + P·V = N·[T·ln(V/N) + f(T)] + N·T`) and p. 20 (`C_P − C_V = N`). Portal Book 075. [^likharev-ladder]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, §2.9, pp. ~96–99 (page to pin; the sub-manual records that the extract's own page labels for this subsection are inconsistent) — the oscillator ladder `E_n = hbar·w0·(n + 1/2)`, its variational ground state of exactly `hbar·w0/2`, and the normalized eigenfunctions. The closed forms `Z = 1/(1 − exp(−theta/T))` and `<n> = 1/(exp(theta/T) − 1)` are derived by summing the geometric series over that ladder, with energies measured from the ground state; including the zero-point term multiplies Z by `exp(−theta/(2·T))` and leaves every probability ratio unchanged. Portal Book 047, https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^os-cv]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2*. OpenStax. Chapter 2, p. 97 (`C_V = (d/2)·R`; hydrogen's d = 3 below about 60 K, d = 5 from just under 300 K to about 600 K, and d = 7 above about 3,000 K) and p. 98, Table 2.3 (measured `C_V/R` of 1.50 for He, Ne and Ar and 2.50 for CO). The level spacings of about 5.2 meV and 0.26 eV are derived by setting `k_B·T` equal to the book's plateau temperatures; the book gives the plateaus but not the curve between them. Portal Book 078, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^anagnostopoulos-ising]: Anagnostopoulos, Konstantinos (2016). *Computational Physics*, 2nd edition (C++). Chapter 13, pp. 522–537 (the Ising energy `H = −J·sum(s_i·s_j) − B·sum(s_i)`; the Metropolis acceptance `A = exp(−beta·dE)` for `dE > 0` and 1 otherwise, constructed from detailed balance; `dE = 2·s_k·sum(s_nn)` with `abs(dE)` at most 8; the lookup table `prob[i] = exp(−2·beta·i)` for i = 2 and 4; `beta_c = ln(1 + sqrt(2))/2` = 0.4406867935 for J = 1 and B = 0). The table entries 0.4317 and 0.1864 at β = 0.21 are derived. Portal Book 061. [^likharev-ising]: Likharev (2013), *Essential Graduate Physics, Part SM*, Chapter 4, pp. 134–136 (a brute-force sum over a 1,000-spin lattice would need about 10³⁰⁰ terms, some 10²⁷⁴ years at 10¹⁸ floating-point operations per second; only 2d + 1 terms change when one spin flips, which is what makes local sampling cheap; a lattice of that size already smears the transition by about 3 %). Portal Book 075. [^shannon1948]: Shannon, Claude E. (1948). "A Mathematical Theory of Communication." *Bell System Technical Journal* 27 (3): 379–423 and 27 (4): 623–656. [^dragulescu2000]: Drăgulescu, Adrian; Yakovenko, Victor M. (2000). "Statistical Mechanics of Money." *European Physical Journal B*, volume 17. [^pareto1897]: Pareto, Vilfredo (1897). *Cours d'économie politique*, volume 2. Lausanne: F. 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