# Boltzmann distribution
The **Boltzmann distribution** is the probability distribution that governs a system in thermal equilibrium with a heat bath at temperature \( T \): the probability of finding the system in a microstate \( n \) of energy \( E_n \) is
\[ P(n) = \frac{1}{Z} e^{-\frac{E_n}{k_\mathrm{B} T}} \]
where \( k_\mathrm{B} \) is the [Boltzmann constant](https://www.edgepedia.org/boltzmann-constant) and the normalizing factor
\[ Z = \sum_n e^{-\frac{E_n}{k_\mathrm{B} T}} \]
is the partition function. The distribution applies to a system of fixed volume and particle number in contact with a reservoir, the setting called the canonical ensemble, and it is the equilibrium distribution of that ensemble[1](https://interactivetextbooks.tudelft.nl/NB2220/content/Exercise_set_4.html). The exponential factor \( e^{-E_n/k_\mathrm{B}T} \) is called the Boltzmann factor[2](https://farside.ph.utexas.edu/teaching/sm1/lectures/node61.html).
| Key fact | Detail |
|---|---|
| Equilibrium probability | \( P_j = \frac{1}{Q} \exp\left(-\frac{E_j}{k_\mathrm{B}T}\right) \) for the energy eigenstates of a canonical-ensemble system[3](https://fb15.pages.uni-marburg.de/ag-von-domaros/teaching/fecomp/canonical-ensemble.html) |
| Partition function | \( Z = \sum_n e^{-E_n/k_\mathrm{B}T} \) over all microstates; with level degeneracy \( g_i \), \( Z = \sum_i g_i e^{-\beta \varepsilon_i} \)[1](https://interactivetextbooks.tudelft.nl/NB2220/content/Exercise_set_4.html) • [4](https://physics.umd.edu/courses/Phys603/kelly/Notes/Ensembles.pdf) |
| Temperature parameter | \( \beta = (k_\mathrm{B}T)^{-1} \), identified with the thermodynamic temperature of the reservoir[2](https://farside.ph.utexas.edu/teaching/sm1/lectures/node61.html) |
| Thermal energy scale | \( kT \) at 298.15 K equals 207.2 cm\(^{-1}\), 2.479 kJ/mol, or 0.0257 eV; at 1000 K it is 695.0 cm\(^{-1}\), 8.314 kJ/mol, or 0.0862 eV[5](https://cccbdb.nist.gov/thermox.asp) |
| Thermodynamics from \( Z \) | Helmholtz energy \( A = -k_\mathrm{B}T \ln Q \); internal energy, entropy, and pressure all follow from derivatives of \( \ln Q \)[3](https://fb15.pages.uni-marburg.de/ag-von-domaros/teaching/fecomp/canonical-ensemble.html) |
| Two-level ratio | \( n_2/n_1 = (g_2/g_1)\, e^{-\Delta E/kT} \) for populations of two levels separated by \( \Delta E \)[6](https://casper.astro.berkeley.edu/astrobaki/index.php/Boltzmann_distribution) |
| Classical limit | Fermi-Dirac and Bose-Einstein statistics both reduce to the Boltzmann result when occupation numbers are far below 1[7](https://ocw.mit.edu/courses/5-62-physical-chemistry-ii-spring-2008/2351f20e4727ae0a7e03ccaca02452d7_08_562ln08.pdf) |
## Derivation from a reservoir
The standard derivation considers a small system \( A \) weakly coupled to a much larger reservoir \( A' \), with the total energy \( E^{(0)} \) of the isolated combined system fixed. If \( A \) is in a particular microstate of energy \( E_r \), the reservoir must hold the remainder, \( E' = E^{(0)} - E_r \), and the probability of the situation is proportional to the number of microstates then accessible to the reservoir, \( \Omega'(E^{(0)} - E_r) \)[2](https://farside.ph.utexas.edu/teaching/sm1/lectures/node61.html).
Because \( E_r \ll E^{(0)} \), the slowly varying logarithm \( \ln \Omega' \) can be Taylor expanded to first order. The derivative \( [\partial \ln \Omega'/\partial E']_0 \) is the temperature parameter \( \beta \) of the reservoir, so \( \ln P_r = \text{const} - \beta E_r \), giving \( P_r = C \exp(-\beta E_r) \). Normalization fixes \( C^{-1} = \sum_r \exp(-\beta E_r) \), which is the partition function[2](https://farside.ph.utexas.edu/teaching/sm1/lectures/node61.html). Matching the statistical definition to the pre-existing thermodynamic one identifies \( \beta = (k_\mathrm{B}T)^{-1} \)[3](https://fb15.pages.uni-marburg.de/ag-von-domaros/teaching/fecomp/canonical-ensemble.html).
The same distribution follows from a different starting point: maximizing the information entropy \( S = -k_\mathrm{B} \sum_j P_j \ln P_j \) subject to fixed normalization and fixed mean energy \( U = \sum_j P_j E_j \). The maximum-entropy distribution for a fixed expectation value is the exponential distribution \( P_j = \exp(-\alpha - \beta E_j) \), and the thermodynamic definition of temperature again fixes \( \beta = (k_\mathrm{B}T)^{-1} \)[3](https://fb15.pages.uni-marburg.de/ag-von-domaros/teaching/fecomp/canonical-ensemble.html).
The exponential form can also be read as a consistency requirement. For independent subsystems with energies \( \varepsilon_i \) and \( \varepsilon_j \), the joint probability of the combined state must factor into a product \( P_i(\varepsilon_i) P_j(\varepsilon_j) \), and the only function with this property is the exponential[8](https://ocw.mit.edu/courses/5-60-thermodynamics-kinetics-spring-2008/e4de766625026dad311c5b410c0e839c_5_60_lecture24.pdf).
## Degeneracy and the density of states
The sum in \( Z \) runs over microstates, not over energy levels. If a level \( \varepsilon_i \) has degeneracy \( g_i \), meaning \( g_i \) distinct states share that energy, it must be counted \( g_i \) times; summing over levels instead gives \( Z = \sum_i g_i e^{-\beta \varepsilon_i} \)[4](https://physics.umd.edu/courses/Phys603/kelly/Notes/Ensembles.pdf). The population ratio of two levels then carries the degeneracy ratio explicitly:
\[ \frac{n_2}{n_1} = \frac{g_2}{g_1} e^{-\frac{\Delta E}{kT}} \]
with \( \Delta E = E_2 - E_1 \)[6](https://casper.astro.berkeley.edu/astrobaki/index.php/Boltzmann_distribution). When the level spacing is small enough that the number of states per energy interval is a smooth function, the sum becomes an integral over the density of states \( g(\varepsilon) \), \( Z = \int d\varepsilon\, g(\varepsilon) e^{-\beta \varepsilon} \)[4](https://physics.umd.edu/courses/Phys603/kelly/Notes/Ensembles.pdf).
Degeneracy can dominate the outcome. Rotational levels of a molecule carry a degeneracy \( 2J+1 \), so the probability of level \( J \) is \( p_J = (2J+1) e^{-E_J/k_\mathrm{B}T} \) divided by the corresponding weighted sum[9](https://chemdata.r.umn.edu/pchem_with_jupyter/M3.D1._Boltzmann_factor_and_partition_functions.html). In hydrogen at 5800 K, the second excited level lies 12.1 eV above the ground state, so a single state of that level is about \( 3 \times 10^{-11} \) as populated as a ground-state state; with its ninefold degeneracy, roughly 270 to 280 atoms per trillion sit in the second excited level[10](https://www.eg.bucknell.edu/physics/PHYS317/2005/rQ/rq22.html).
## What the partition function gives
Once \( Z \) is known, the equilibrium thermodynamics of the system follows. The average energy is the thermal average \( \langle E \rangle = \sum_i \langle i|\hat{H}|i\rangle \rho_i \), and differentiating \( \ln Z \) with respect to \( \beta \) at fixed volume gives \( \langle E \rangle = -\partial \ln Z / \partial \beta \); the second derivative gives the energy fluctuations \( (\Delta E)^2 \)[11](https://www.damtp.cam.ac.uk/user/wingate/Notes/StatPhys/canonical.pdf). In thermodynamic variables, the Helmholtz energy is \( A = -k_\mathrm{B}T \ln Q \), the internal energy is \( U = k_\mathrm{B}T^2 (\partial \ln Q / \partial T)_{V,N} \), the entropy is \( S = k_\mathrm{B} \ln Q + k_\mathrm{B}T (\partial \ln Q / \partial T)_{V,N} \), and the pressure is \( P = k_\mathrm{B}T (\partial \ln Q / \partial V)_{T,N} \)[3](https://fb15.pages.uni-marburg.de/ag-von-domaros/teaching/fecomp/canonical-ensemble.html).
In molecular statistical thermodynamics, the energy levels \( \varepsilon_i \) come from the molecule's translation, rotation, vibration, and electronic excitation, obtained from spectroscopy or calculation. With the motions treated as uncoupled, the molecular partition function factors as \( Q = Q_\mathrm{trans} Q_\mathrm{rot} Q_\mathrm{vib} Q_\mathrm{elec} \), and the molar enthalpies, entropies, and heat capacities follow from \( Q \)[5](https://cccbdb.nist.gov/thermox.asp). The lowest level is usually chosen as the zero of energy, so levels far above \( kT \) contribute little to \( Q \) and its derivative quantities[5](https://cccbdb.nist.gov/thermox.asp).
## Temperature as the control parameter
The distribution depends on energy only through the ratio \( E/kT \). At \( T \to 0 \), the Boltzmann distribution forces the system into its ground state, since all higher states have vanishing probability[12](https://www.damtp.cam.ac.uk/user/tong/statphys/statmechhtml/S1.html). As \( T \) rises, higher-energy states become more probable, and the population shifts upward[8](https://ocw.mit.edu/courses/5-60-thermodynamics-kinetics-spring-2008/e4de766625026dad311c5b410c0e839c_5_60_lecture24.pdf).
The size of \( kT \) sets what counts as "high" energy. At room temperature \( kT \) is about 0.0257 eV, so an electronic excitation of a few electron volts is essentially unpopulated, while rotational spacings of a few hundred wavenumbers are comparable to \( kT \) and substantially populated[5](https://cccbdb.nist.gov/thermox.asp).
## Applications
**Spectroscopy and plasmas.** The NIST Atomic Spectra Database computes level populations in a modeled plasma with the Boltzmann formula \( N_k = (N_0/g_0)\, g_k \exp(-E_k/T_e) \), where \( E_k \) is the level energy above the ground state and \( T_e \) the electron temperature; combined with the Saha formula for the ionization balance, this underlies synthetic spectra of optically thin plasmas[13](https://physics.nist.gov/PhysRefData/ASD/Html/lineshelp.html). In stellar astrophysics, the two-level ratio together with the Saha equation gives the populations needed to interpret stellar spectra[6](https://casper.astro.berkeley.edu/astrobaki/index.php/Boltzmann_distribution).
**Nuclear magnetic resonance.** The net magnetization measured in an NMR experiment comes from a tiny Boltzmann imbalance between spin states. For a proton in the 11.76 T field of a 500 MHz spectrometer, the level splitting is about \( 2.1 \times 10^{-6} \) eV against \( kT \approx 2.57 \times 10^{-2} \) eV at 298 K, so the population ratio is \( e^{\Delta E/kT} = 1.0000817 \) and the net alignment is about 40 per million, or 0.004%[14](https://nmr.chem.ucsb.edu/education/part0.html).
**Molecular speeds.** Applying the Boltzmann weight to translational states of an ideal gas, and counting the \( v^2 \) states available at each speed in three-dimensional velocity space, yields the Maxwell-Boltzmann speed distribution \( f(v) = \frac{4}{\sqrt{\pi}} \left(\frac{m}{2k_\mathrm{B}T}\right)^{3/2} v^2 e^{-mv^2/2k_\mathrm{B}T} \)[15](https://openstax.org/books/university-physics-volume-2/pages/2-4-distribution-of-molecular-speeds). The \( v^2 \) factor reflects the growing number of velocity directions at larger speed, while the exponential suppresses large kinetic energies; the two factors combine into a single-peaked curve[15](https://openstax.org/books/university-physics-volume-2/pages/2-4-distribution-of-molecular-speeds).
## Relation to quantum statistics
The Boltzmann distribution is the classical limit of the two quantum distributions. For indistinguishable particles, the occupation of a level is \( n_i = 1 / (e^{(\varepsilon_i - \mu)/kT} \pm 1) \), with \( +1 \) for fermions (Fermi-Dirac) and \( -1 \) for bosons (Bose-Einstein). When \( e^{(\varepsilon_i - \mu)/kT} \gg 1 \) for all levels, equivalently when occupation numbers are far below 1, the \( \pm 1 \) is negligible and both reduce exactly to the Boltzmann result \( n_i = N e^{-\varepsilon_i/kT}/q \)[7](https://ocw.mit.edu/courses/5-62-physical-chemistry-ii-spring-2008/2351f20e4727ae0a7e03ccaca02452d7_08_562ln08.pdf).
Most atoms and molecules at ordinary temperatures satisfy this condition, so Boltzmann statistics applies regardless of whether the particle is a fermion or a boson. The exceptions are light particles at low temperature, such as \( ^4 \)He and \( \mathrm{H_2} \) (bosons) and \( ^3 \)He (a fermion), which must be treated with their quantum statistics near 0 K; electrons, whose occupation numbers are of order 1, must be treated as fermions at all normal temperatures below about 3000 K[7](https://ocw.mit.edu/courses/5-62-physical-chemistry-ii-spring-2008/2351f20e4727ae0a7e03ccaca02452d7_08_562ln08.pdf).
## References
1. [Boltzmann statistics and entropy, NB2220 Statistical Physics, TU Delft](https://interactivetextbooks.tudelft.nl/NB2220/content/Exercise_set_4.html)
2. [Boltzmann distributions, Statistical Mechanics lecture notes, R. Fitzpatrick, University of Texas at Austin](https://farside.ph.utexas.edu/teaching/sm1/lectures/node61.html)
3. [Canonical Ensemble, Statistical Thermodynamics and Free Energy Computation, M. von Domaros, University of Marburg](https://fb15.pages.uni-marburg.de/ag-von-domaros/teaching/fecomp/canonical-ensemble.html)
4. [Ensembles notes, Phys 603, University of Maryland](https://physics.umd.edu/courses/Phys603/kelly/Notes/Ensembles.pdf)
5. [CCCBDB Essential Statistical Thermodynamics, NIST](https://cccbdb.nist.gov/thermox.asp)
6. [Boltzmann distribution, AstroBaki, UC Berkeley](https://casper.astro.berkeley.edu/astrobaki/index.php/Boltzmann_distribution)
7. [Boltzmann, Fermi-Dirac, and Bose-Einstein Statistics, MIT 5.62 Physical Chemistry II, Spring 2008](https://ocw.mit.edu/courses/5-62-physical-chemistry-ii-spring-2008/2351f20e4727ae0a7e03ccaca02452d7_08_562ln08.pdf)
8. [Boltzmann probability distribution, MIT 5.60 Thermodynamics & Kinetics, Lecture 24](https://ocw.mit.edu/courses/5-60-thermodynamics-kinetics-spring-2008/e4de766625026dad311c5b410c0e839c_5_60_lecture24.pdf)
9. [Boltzmann factor and partition functions, Physical Chemistry with Python, University of Minnesota](https://chemdata.r.umn.edu/pchem_with_jupyter/M3.D1._Boltzmann_factor_and_partition_functions.html)
10. [Reading Quiz #22, PHYS 317, Bucknell University](https://www.eg.bucknell.edu/physics/PHYS317/2005/rQ/rq22.html)
11. [Canonical Ensemble, Statistical Physics notes, University of Cambridge](https://www.damtp.cam.ac.uk/user/wingate/Notes/StatPhys/canonical.pdf)
12. [The Fundamentals of Statistical Mechanics, Statistical Physics, D. Tong, University of Cambridge](https://www.damtp.cam.ac.uk/user/tong/statphys/statmechhtml/S1.html)
13. [Atomic Spectra Database, Spectral Lines Help, NIST](https://physics.nist.gov/PhysRefData/ASD/Html/lineshelp.html)
14. [NMR Theory, UCSB NMR Facility](https://nmr.chem.ucsb.edu/education/part0.html)
15. [Distribution of Molecular Speeds, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/2-4-distribution-of-molecular-speeds)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Thermodynamics › Statistical mechanics and kinetic theory › Ensembles and partition functions*
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