Bose-Einstein Distribution Simulator — Quantum Statistics of Bosons
Visualize the boson occupation number n_BE(E) from chemical potential, temperature and energy. The Maxwell-Boltzmann limit and the BEC divergence near E→μ are tracked in real time.
Parameters
Energy E
eV
Chemical potential mu
eV
Temperature T
K
Degeneracy g
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For photons and phonons mu = 0 (particle number not conserved). For massive boson gases (^87Rb, ^4He) mu < 0 and approaches zero from below as T is reduced - the onset of BEC.
Results
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n_BE occupation
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n_MB comparison
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Thermal energy kT
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BE-MB relative diff.
Distribution n(E)
x = (E - mu)/(kT) / y = occupation n / blue solid = BE, orange dashed = MB / yellow line = current observation / BE diverges as E -> mu (BEC signature).
Temperature dependence n_BE(T)
Sweep T from 1 to 1000 K with E and mu fixed. Low temperature drives n_BE up sharply, the BEC signature. Yellow dot = current T.
Theory & Key Formulas
The mean number of integer-spin bosons occupying a state of energy $E$ in thermal equilibrium follows the Bose-Einstein distribution.
Occupation number ($\mu$ is the chemical potential, $k$ Boltzmann's constant, $T$ absolute temperature, $g$ the degeneracy):
How does the Bose-Einstein distribution actually differ from Fermi-Dirac?
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Both are quantum statistics, but they apply to different particles. Fermi-Dirac describes half-integer-spin "fermions" like electrons, where the Pauli exclusion principle limits each state to a single particle. Bose-Einstein covers integer-spin "bosons" - photons, phonons, ^4He, cold ^87Rb atoms - which can pile up many particles in the same state. The formula is $n_{BE}(E) = 1/(\exp((E-\mu)/kT)-1)$; the minus one in the denominator (versus plus one for Fermi-Dirac) is the fundamental distinction.
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The defaults give n_BE = 0.109 with E = 0.001 eV, mu = -0.001 eV, T = 10 K. The 10.9 % difference from MB is bigger than I expected.
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Yes, (E - mu)/kT = 0.002/0.000862 ≈ 2.32 sits right in the regime where quantum effects start to appear. With exp(2.32) ≈ 10.18, BE = 1/(10.18 - 1) = 0.109 while MB = exp(-2.32) = 0.0982 - a 10.9 % gap. At (E - mu)/kT = 5 the gap shrinks to 0.7 %, and below 0.5 it exceeds 30 %, with BE diverging as E -> mu. The "bunching" of bosons - their preference for shared states - shows up most strongly at low temperature.
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Pressing "Sweep temperature" makes n_BE jump up rapidly at low T!
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That is the signature of Bose-Einstein condensation (BEC). For a real conserved-particle gas, lowering T pushes mu upward, and at the critical temperature mu reaches the ground-state energy E_0 - at which point a macroscopic number of bosons collapses into a single quantum state. In this simulator you set mu directly, so when T is lowered with mu fixed the bunching grows but does not formally diverge. The 1995 ^87Rb experiment by Cornell, Wieman and Ketterle won the 2001 Nobel Prize for the first realisation of BEC.
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Pulling mu close to zero (say -0.0001 eV) makes n_BE explode...
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That is precisely the mathematical signature of BEC: x = (E - mu)/kT becomes tiny and the small-x expansion gives $n_{BE} \approx kT/(E - \mu)$, a power-law blow-up rather than exponential decay. In real cold-atom experiments thousands to millions of atoms below T_c "condense" into the ground state, forming a macroscopic matter wave. Standard cooling pipelines combine laser cooling, evaporative cooling and magnetic trapping to reach the nK regime.
FAQ
The chemical potential mu is the Lagrange multiplier conjugate to particle number conservation, $\mu = \partial F/\partial N$. Photons and phonons are created and destroyed freely by black-body emission or lattice vibration so N is not conserved, and minimising the free energy F demands mu = 0. That is why Planck's black-body formula $u(\nu) \propto \nu^3/(\exp(h\nu/kT)-1)$ has no extra term. For massive particles such as ^4He or ^87Rb atoms N is conserved, mu varies with temperature and density, and BEC sets in as mu approaches the ground-state energy at low T.
The sign in the denominator is decisive. FD is $1/(e^x+1)$, always between 0 and 1 because of the Pauli principle. BE is $1/(e^x-1)$, ranging from 0 to infinity due to the bosonic bunching effect. With x = (E - mu)/kT, both reduce to the MB form when x >> 1, but for x << 1 they diverge: FD saturates at 1, while BE behaves like 1/x. Physically fermions "avoid" each other while bosons "prefer" sharing the same state - a direct consequence of their wavefunction symmetry under particle exchange.
For a 3D free boson gas of particle density n and mass m, $T_c = \dfrac{2\pi\hbar^2}{m k}\left(\dfrac{n}{\zeta(3/2)}\right)^{2/3}$, where $\zeta(3/2) \approx 2.612$ is the Riemann zeta function. For ^87Rb at n ~ 10^14 cm^-3 this gives T_c ~ 100 nK; for superfluid ^4He at n ~ 2.2×10^22 cm^-3 it predicts T_c ~ 3.13 K (versus the measured lambda transition at 2.17 K). Trap geometry and interactions add corrections; this simulator focuses on the distribution function itself.
g counts the number of distinct states sharing energy E. Photons have two polarisations (g = 2), spin-zero ^4He has g = 1, the ^87Rb hyperfine F = 1 manifold is three-fold degenerate while F = 2 is five-fold. Use g = 1 to compute the average occupation of a single quantum state, or the actual degeneracy when treating an entire energy shell. The simulator multiplies n_BE by g - a simple scaling that does not change the qualitative shape of the distribution.
Real-world applications
Bose-Einstein condensation (BEC): In 1995 Cornell and Wieman cooled ^87Rb atoms to 170 nK and produced the first atomic BEC; later that year Ketterle achieved it in ^23Na, and the trio shared the 2001 Nobel Prize. BEC is a macroscopic quantum state that powers atom lasers, matter-wave interferometry, and quantum simulation. BEC in optical lattices reproduces the superfluid-to-Mott-insulator quantum phase transition, providing a clean experimental platform for strongly correlated electronic systems with light and atoms.
Planck radiation and the cosmic microwave background: Photons are mu = 0 bosons, with energy density $u(\nu) \propto h\nu^3/(\exp(h\nu/kT)-1)$ - Planck's law that launched quantum theory in 1900. The cosmic microwave background follows a near-perfect Planck spectrum at T ≈ 2.725 K, the smoking gun for Big Bang cosmology. COBE's 1989 measurement bounded deviations from a black-body to less than 10^-5, earning the 2006 Nobel Prize.
Superfluid ^4He and superconductivity: Liquid ^4He becomes a frictionless superfluid below 2.17 K (the lambda point), interpreted as the manifestation of bosonic ^4He BEC and described by Landau's quasi-particle picture. Superconductivity is a condensate of Cooper pairs (electron pairs that act as bosons), captured by BCS theory. From MgB_2 (T_c = 39 K) to cuprates (T_c > 100 K) and most recently hydrides (LaH_10, T_c ~ 250 K under pressure), bosonic condensation underpins active research in superconductivity.
Lasers and optical communication: Laser light is a coherent state with a macroscopic number of photons in a single mode, fueled by stimulated emission - a direct manifestation of bosonic bunching. When Einstein derived his A and B coefficients in 1917 he implicitly used Bose statistics for photons. Semiconductor lasers, optical communications, LIDAR, and optical-lattice atomic clocks - core technologies of the modern world - all rely on the statistics of bosons.
Common misconceptions and caveats
The most common misconception is to treat BEC as ordinary condensation (liquefaction). The two are entirely different - BEC is a condensation in momentum space (or in the manifold of energy eigenstates), not in real space. Atoms continue floating as a dilute gas, but their quantum state is described by a single wavefunction - a purely quantum-mechanical phenomenon distinct from classical liquefaction driven by intermolecular attraction. The divergence of n_BE as E -> mu shown in this simulator is the mathematical fingerprint of that condensation into a single state.
A second pitfall is to think you can input mu >= E. The physical domain of the BE distribution is strictly E > mu. With E <= mu the denominator exp((E - mu)/kT) - 1 vanishes or becomes negative, giving an unphysical infinite or negative occupation. The simulator constrains mu < 0 and E > 0 so that E - mu is always positive. For massive boson gases, lowering T pushes mu toward the ground-state energy E_0 - that is the mathematical hallmark of BEC.
Finally, do not assume that all bosons undergo BEC. BEC occurs in 3D free boson gases, but the Mermin-Wagner theorem rules it out at finite temperature in 2D (a different phenomenon, the Berezinskii-Kosterlitz-Thouless transition, can occur instead). It generally does not occur in 1D either. With interactions, Bogoliubov quasi-particles emerge and the picture differs slightly from pure BEC. This simulator visualises the distribution of a non-interacting free boson gas; real-material applications must add the appropriate corrections.