Bose–Einstein Distribution Calculator

Bose–Einstein occupation and condensation results

This Bose–Einstein distribution calculator examines a defining property of bosons: multiple particles may occupy the same quantum state. It returns two related quantities. First, it calculates the mean occupation number of one single-particle state with energy E, at the entered temperature T and chemical potential μ. Second, it estimates the critical temperature T_c for condensation in an ideal three-dimensional gas with the entered particle mass and number density. Together, these outputs distinguish strong population of one state from the temperature criterion for condensation of the gas as a whole.

The form is compact, but neither output is a generic score. The occupation number is an expected population for a specified quantum state, while the critical temperature changes when the boson mass or density changes. Reading the values in their physical context is therefore as important as carrying out the calculation.

Bose–Einstein input quantities and SI units

Energy E (J) is the single-particle energy of the state whose population is being evaluated. It can describe a particular mode, trap level, or level in a textbook spectrum. Temperature T (K) is the absolute temperature of the boson gas. Cooling generally raises the occupation of low-energy states because thermal spreading into excited states is reduced.

Chemical potential μ (J) controls the equilibrium particle distribution among states. In an ideal Bose gas, it approaches the lowest state energy from below near condensation. Particle mass m (kg) and number density n (m⁻³) enter only the critical-temperature calculation. Lighter bosons and a denser gas give a higher ideal-gas transition temperature.

Enter energy and chemical potential in joules, temperature in kelvin, mass in kilograms, and density in particles per cubic meter. Convert electronvolts, microkelvin, atomic mass units, or densities quoted per cubic centimeter before using the form; the equations use SI values directly.

Bose–Einstein occupation-number equation

The calculator obtains the mean occupation of the selected bosonic state from the standard Bose–Einstein expression. Its dimensionless result is an expected number of particles in one state, rather than an energy or a temperature.

Here k is Boltzmann’s constant. Lowering T makes kT smaller, and moving μ upward toward E from below makes E − μ smaller. Either change raises the mean occupation. As the chemical potential approaches the state energy from below, the denominator tends toward zero and the predicted occupation can become very large, expressing bosonic bunching into low-energy states.

The condensation threshold uses the ideal homogeneous three-dimensional Bose-gas relation with ζ(3/2) ≈ 2.612.

This separation of inputs is important: state energy, temperature, and chemical potential determine n(E), whereas mass and density determine T_c in this ideal-gas estimate.

Physical conditions enforced for Bose–Einstein statistics

The key Bose–Einstein condition checked by the calculator is μ < E for the selected state at positive temperature. If μ reaches or exceeds E, the occupation expression has a zero or negative denominator, so the page reports a nonphysical input instead of a numerical occupation. This is a restriction of the equilibrium distribution, not merely an input-format rule.

The calculation also requires T > 0, m > 0, and n > 0. A nonpositive mass or density cannot define the displayed ideal-gas condensation temperature. For equilibrium photons the chemical potential is normally zero; for ultracold atoms, the energy reference and the chemical potential must be chosen consistently.

Rubidium-87 Bose gas example

For an ideal-gas scale check, consider an ultracold rubidium-87 cloud with E = 2.0 × 10⁻³⁰ J, T = 1.5 × 10⁻⁶ K, μ = 1.6 × 10⁻³⁰ J, m = 1.443 × 10⁻²⁵ kg, and n = 1.0 × 10²¹ m⁻³. The positive energy gap is small, so the selected state is strongly populated compared with a classical dilute-gas expectation.

For these values, the occupation is about 5.1 × 10¹, and the ideal-gas critical temperature is about 1.8 × 10⁻⁶ K. Since the entered temperature is below that estimate, the calculator identifies the ideal model as being below its condensation threshold.

This example also shows why the two outputs should not be conflated. A large value of n(E) concerns the chosen state; comparing T with T_c addresses whether the whole ideal gas lies below the condensation threshold.

Interpreting Bose–Einstein calculator outputs

A very small occupation number means the chosen state is weakly populated under the supplied conditions, commonly because E − μ is large compared with kT. A moderate or large value signals appreciable bosonic enhancement for that state. Extremely large values occur when a cold gas has a state energy very close to the chemical potential from above.

The critical temperature is an idealized threshold, not a complete prediction for an apparatus. For the homogeneous ideal model, T < T_c indicates condensation. Interactions, finite particle number, trap geometry, dimensionality, and nonequilibrium dynamics can shift or broaden a transition in a physical experiment.

Connecting state occupation to Bose-gas condensation

The two equations on this page describe different levels of the same Bose-gas picture. The occupation equation assigns a mean population to one energy state, while the critical-temperature equation identifies the density-and-mass scale at which an ideal three-dimensional gas can no longer accommodate all particles in excited states. Below that threshold, macroscopic occupation of the lowest available state is expected in the ideal model.

When checking a result, focus on the trends dictated by these expressions. Cooling the gas or narrowing the positive gap E − μ increases the displayed state occupation. Raising the density or reducing the particle mass increases the displayed critical temperature. These directional checks are more meaningful than treating either result as an isolated number.

Ideal Bose-gas assumptions and numerical limits

This Bose–Einstein calculator uses an ideal Bose gas: particles are taken to be in thermal equilibrium, interactions are neglected, and the critical temperature is the standard homogeneous three-dimensional result. Those assumptions are useful for instruction, order-of-magnitude checks, and textbook comparisons, but they do not capture every feature of trapped, interacting, lower-dimensional, or driven bosonic systems.

SI energy scales can also make the occupation exponent very large. When E − μ is much larger than kT, the occupation is effectively zero, and the script avoids an unstable exponential evaluation. Conversely, a chemical potential extremely close to a state energy from below produces a sharply rising occupation; verify the energy reference carefully in that regime.

After each calculation, check that all units are coherent and that the trend is sensible. Lower temperature or a chemical potential closer to the selected energy from below should raise n(E). Greater density or lower mass should raise T_c. Those checks directly reflect the equations evaluated by this page.

Condensate Tuner mini-game and Bose–Einstein intuition

This optional Bose–Einstein mini-game turns the same qualitative relationships into a timing challenge. You move the chemical-potential line and cool the trap while energy packets approach the capture gate. The game rewards the same behavior represented by the distribution: keep the gas cold and hold μ below a packet’s energy. It is a memory aid only and does not alter the calculator result.