Bose–Einstein Condensation

Bose–Einstein condensation is the appearance of a one-particle mode whose occupation is proportional to the total number of bosons. In a sequence of systems approaching a thermodynamic limit,

$$N_0 \sim O(N),$$

or, more sharply,

$$\lim_{N\to\infty} \frac{N_0}{N} > 0.$$

For a uniform ideal gas, the macroscopically occupied mode is the zero-momentum ground state. That familiar case is important but not the general definition. In an interacting or inhomogeneous system, the condensate mode is identified as an eigenvector of the one-body density matrix. It need not be a plane wave or the bare one-particle ground orbital.

The ideal-gas mechanism is an excited-state capacity limit. As the chemical potential approaches the lowest one-particle energy, the excited modes can hold only a finite density of particles in suitable dimensions. Any particles beyond that capacity accumulate macroscopically in the lowest mode.

This page owns:

• the basis-independent Penrose–Onsager criterion;
• the excited-state saturation mechanism;
• the critical temperature and condensate fraction of the uniform ideal gas;
• the general density-of-states criterion;
• dimensional and harmonic-trap variants;
• finite-size and ensemble qualifications;
• the distinction among condensation, coherence, symmetry breaking, and superfluidity;
• the boundary between the ideal benchmark and interacting condensates.

The Ideal Bose Gas page owns the full box model, partition function, pressure, energy, and entropy. The historical article owns the path from Einstein’s prediction to dilute-gas experiments.

Off-Diagonal Long-Range Order places the one-body criterion inside the reduced-density-matrix hierarchy, relates its spectrum to spatial coherence, and develops the corresponding fermion-pair criterion.

Evaporative Cooling owns the selective-loss and rethermalization route by which many dilute-gas experiments approach the degenerate regime.

Ultracold Atoms owns the broader experimental criteria for degeneracy, interaction tuning, and dimensional freeze-out.

After establishing condensation, use Superfluidity in Condensed Matter for a neutral material’s stiffness, flow, and evidence claim. Use Superfluidity and Superconductivity to route a charged-superconductor claim; this page stops at the condensation criterion and ideal-gas benchmark.

Three Levels of the Concept

It is useful to separate three statements.

Statistical permission.
Bosonic statistics allows

$$n_i = 0,1,2,\ldots$$

in one mode. This makes macroscopic occupation possible but does not guarantee it.

Thermodynamic mechanism.
For some spectra and thermodynamic limits, the excited states have finite capacity at fixed temperature. A macroscopic occupation then becomes necessary below a critical scale.

General many-body criterion.
Condensation is diagnosed by an extensive eigenvalue of the one-body density matrix. This definition remains meaningful with interactions and without translation invariance.

Confusing these levels leads to statements such as “all bosons condense because they can share a state” or “BEC always means zero momentum.” Neither is generally correct.

Penrose–Onsager Criterion

For field operators $\psi(\mathbf r)$ and $\psi^\dagger(\mathbf r)$, define the equal-time one-body density matrix

$$\gamma(\mathbf r,\mathbf r') = \left\langle \psi^\dagger(\mathbf r') \psi(\mathbf r) \right\rangle.$$

As a positive Hermitian kernel, it has orthonormal eigenfunctions:

$$\int d^d r'\, \gamma(\mathbf r,\mathbf r') \phi_\alpha(\mathbf r') = N_\alpha \phi_\alpha(\mathbf r).$$

The eigenfunctions $\phi_\alpha$ are natural orbitals and the eigenvalues $N_\alpha$ are natural occupations. Their sum is the mean particle number:

$$\sum_\alpha N_\alpha = \langle\hat N\rangle.$$

A conventional, or simple, condensate has one eigenvalue that is extensive:

$$N_0 = O(N),$$

while all other $N_\alpha$ are subextensive. The condensate fraction is

$$f_0 = \frac{N_0}{N}.$$

If two or more eigenvalues are extensive, the condensate is called fragmented. Fragmentation can arise from symmetry, degeneracy, strong correlations, or constraints. It is not captured by saying that one bare orbital is occupied.

The Occupation Numbers page develops the distinction between integer Fock-state counts, mean mode occupations, and natural occupations.

The Ideal Uniform Benchmark

Consider $N$ spinless, noninteracting bosons in a cubic box of volume $V$ with periodic boundary conditions. Measure energies relative to the unique ground mode:

$$\epsilon_{\mathbf k} = \frac{\hbar^2k^2}{2m}, \qquad \epsilon_{\mathbf0} = 0.$$

The mean occupation of mode $\mathbf k$ is

$$\overline n_{\mathbf k} = \frac{1}{ z^{-1}e^{\beta\epsilon_{\mathbf k}}-1 }, \qquad z = e^{\beta\mu}.$$

For a finite system,

$$0 < z < 1, \qquad \mu < 0.$$

Separate the ground occupation:

$$N = N_0 + N_{\mathrm{ex}},$$

where

$$N_0 = \frac{z}{1-z},$$

and

$$N_{\mathrm{ex}} = \sum_{\mathbf k\neq\mathbf0} \frac{1}{ z^{-1}e^{\beta\epsilon_{\mathbf k}}-1 }.$$

The ground mode must remain discrete. Replacing every mode by a continuum integral would remove the very degree of freedom that becomes macroscopic.

Thermal Wavelength and Excited-State Density

Define the thermal de Broglie wavelength

$$\lambda_T = \sqrt{ \frac{2\pi\hbar^2}{ mk_{\mathrm B}T } }.$$

In the three-dimensional thermodynamic limit, the excited-state sum becomes

$$\frac{N_{\mathrm{ex}}}{V} = \int \frac{d^3k}{(2\pi)^3} \frac{1}{ z^{-1}e^{\beta\hbar^2k^2/(2m)}-1 }.$$

Evaluating the integral gives

$$n_{\mathrm{ex}} = \frac{1}{\lambda_T^3} \operatorname{Li}_{3/2}(z).$$

The polylogarithm increases with $z$, but the bosonic constraint prevents $z$ from exceeding one. Therefore

$$n_{\mathrm{ex}} \leq \frac{\zeta(3/2)}{ \lambda_T^3 }.$$

The maximum excited-state density is finite:

$$n_{\mathrm{ex,max}}(T) = \frac{\zeta(3/2)}{ \lambda_T^3 }.$$

This finite capacity is the mathematical core of ideal-gas condensation.

Critical Temperature in a Box

At the critical temperature, the total density just equals the maximum excited-state density:

$$n = \frac{\zeta(3/2)}{ \lambda_{T_c}^3 }.$$

Equivalently,

$$n\lambda_{T_c}^3 = \zeta(3/2) \simeq 2.612.$$

Solving for temperature gives

$$T_c^{\mathrm{box}} = \frac{2\pi\hbar^2}{ mk_{\mathrm B} } \left[ \frac{n}{ \zeta(3/2) } \right]^{2/3}.$$

This formula assumes:

• a uniform three-dimensional gas;
• a quadratic nonrelativistic dispersion;
• one independent internal component;
• a unique ground mode;
• no interactions;
• the thermodynamic limit at fixed $n=N/V$.

Changing any of these ingredients can change the formula or the existence of a sharp transition.

Condensate Fraction

Below $T_c$, the thermodynamic-limit fugacity reaches

$$z \to 1, \qquad \mu \to 0^-.$$

The excited-state population is then

$$N_{\mathrm{ex}}(T) = \frac{V}{\lambda_T^3} \zeta(3/2).$$

Since

$$\lambda_T^{-3} \propto T^{3/2},$$

comparison with the number equation at $T_c$ yields

$$\frac{N_{\mathrm{ex}}}{N} = \left( \frac{T}{T_c} \right)^{3/2}.$$

The ideal uniform condensate fraction is therefore

$$\frac{N_0}{N} = 1 - \left( \frac{T}{T_c} \right)^{3/2}, \qquad 0 \leq T \leq T_c.$$

Above $T_c$, the thermodynamic-limit condensate fraction vanishes and $z<1$ is fixed by

$$n\lambda_T^3 = \operatorname{Li}_{3/2}(z).$$

Uniform three-dimensional ideal gas in the thermodynamic limit. Below $T_c$, the thermal cloud is saturated and its capacity scales as $T^{3/2}$; the remaining particles occupy the ground mode. A finite system rounds the corner near $T_c$.

The formula does not mean that particles are divided into two distinguishable species. “Condensate” and “thermal cloud” refer to contributions to the one-body density matrix and mode occupations.

Why Saturation Occurs

At low momentum and $z=1$,

$$\frac{1}{ e^{\beta\hbar^2k^2/(2m)}-1 } \sim \frac{ 2mk_{\mathrm B}T }{ \hbar^2k^2 }.$$

In three dimensions, the momentum-space measure contributes $k^2\,dk$. The low-momentum excited-number integral therefore behaves as

$$\int_0 dk\, k^2 \frac{1}{k^2} \sim \int_0 dk,$$

which is finite at the lower limit.

The Bose occupation of each low-energy mode diverges as $\mu\to0$, but the phase-space volume of those modes shrinks quickly enough that the total excited density remains finite. Condensation depends on this competition between the occupation factor and the density of states.

General Density-of-States Criterion

Suppose the low-energy density of excited one-particle states is

$$D(\epsilon) \simeq C\epsilon^{\alpha-1}, \qquad \epsilon \to 0^+.$$

At the condensation boundary,

$$\mu \to \epsilon_0,$$

so, measuring $\epsilon$ from $\epsilon_0$,

$$N_{\mathrm{ex,max}} \simeq \int_0^\infty d\epsilon\, \frac{ C\epsilon^{\alpha-1} }{ e^{\beta\epsilon}-1 }.$$

The low-energy integrand behaves as

$$\epsilon^{\alpha-2}.$$

The integral converges at zero exactly when

$$\alpha > 1.$$

When the power law applies over the relevant spectrum,

$$N_{\mathrm{ex,max}} = C \Gamma(\alpha) \zeta(\alpha) (k_{\mathrm B}T)^\alpha.$$

The ideal critical temperature is

$$k_{\mathrm B}T_c = \left[ \frac{N}{ C\Gamma(\alpha)\zeta(\alpha) } \right]^{1/\alpha},$$

and below it,

$$\frac{N_0}{N} = 1 - \left( \frac{T}{T_c} \right)^\alpha.$$

This compact criterion unifies boxes, traps, and other spectra. It also makes clear that unrestricted bosonic mode occupation is not enough: the low-energy density of states must permit finite excited-state capacity.

Dimension and Dispersion

For a uniform $d$-dimensional system with low-momentum dispersion

$$\epsilon_{\mathbf k} \propto k^s,$$

the density of states scales as

$$D(\epsilon) \propto \epsilon^{d/s-1}.$$

Thus

$$\alpha = \frac{d}{s},$$

and the ideal excited-state integral converges when

$$d > s.$$

For ordinary massive particles,

$$s = 2,$$

so a uniform ideal gas has a finite-temperature condensation transition only for

$$d > 2.$$

In three dimensions, $\alpha=3/2$. In two dimensions, the low-momentum integral diverges logarithmically. In one dimension, it diverges more strongly.

This is a thermodynamic-limit statement. A finite one- or two-dimensional box can show a large ground-state population and a pronounced crossover, but no exact finite-temperature nonanalytic transition survives the standard uniform thermodynamic limit.

For interacting homogeneous Bose systems with short-range forces, long-wavelength phase fluctuations also forbid ordinary one-body long-range order at finite temperature in one and two dimensions under the hypotheses of Hohenberg-type results. A two-dimensional interacting fluid can nevertheless undergo a Berezinskii–Kosterlitz–Thouless transition with algebraically decaying order and nonzero superfluid response. That phenomenon is not the ideal-gas condensation derived here.

Low-Dimensional Quantum Gases develops the explicit one- and two-dimensional densities of states, finite-size qualifications, phase-fluctuation argument, dimensional-freezing criteria, and bridges to BKT and Luttinger-liquid physics.

Harmonic Traps

Laboratory atomic gases are often confined by an anisotropic harmonic potential:

$$V(\mathbf r) = \frac{m}{2} \left( \omega_x^2x^2 + \omega_y^2y^2 + \omega_z^2z^2 \right).$$

Define the geometric mean frequency

$$\overline\omega = (\omega_x\omega_y\omega_z)^{1/3}.$$

At excitation energies large compared with the level spacings, the three-dimensional trap density of states is

$$D_{\mathrm{trap}}(\epsilon) \simeq \frac{\epsilon^2}{ 2\hbar^3 \omega_x\omega_y\omega_z }.$$

Here

$$\alpha = 3,$$

so the maximum excited population is

$$N_{\mathrm{ex,max}} = \zeta(3) \left( \frac{ k_{\mathrm B}T }{ \hbar\overline\omega } \right)^3.$$

The leading ideal-gas critical temperature is

$$T_c^{\mathrm{trap}} = \frac{ \hbar\overline\omega }{ k_{\mathrm B} } \left[ \frac{N}{ \zeta(3) } \right]^{1/3}.$$

Below this scale,

$$\frac{N_0}{N} = 1 - \left( \frac{T}{ T_c^{\mathrm{trap}} } \right)^3.$$

The exponent is $3$, not $3/2$, because a harmonic trap has a different density of states from a uniform box.

The approximation requires

$$k_{\mathrm B}T \gg \hbar\omega_i$$

for thermally relevant directions. A sharp trapped-gas thermodynamic limit is constructed by taking

$$N \to \infty, \qquad \overline\omega \to 0,$$

while keeping

$$N\overline\omega^3$$

fixed. At finite $N$, discrete-level corrections shift and round the crossover. Interactions also change the density profile and transition region.

The condensate in a trap occupies a spatial orbital. It is not generally a zero-momentum plane wave.

Example: Box and Trap Scales

Uniform rubidium-87 estimate

Take an ideal uniform gas of rubidium-87 atoms with

$$m \simeq 1.443\times10^{-25}\ {\rm kg},$$

and density

$$n = 10^{20}\ {\rm m}^{-3}.$$

The box formula gives

$$T_c \simeq 3.98\times10^{-7}\ {\rm K} = 398\ {\rm nK}.$$

This is an order-of-magnitude benchmark. A real trapped, interacting cloud must be analyzed with its actual confinement, component populations, and scattering properties.

Harmonic-trap estimate

For

$$N = 10^5$$

atoms in a trap with

$$\overline\omega = 2\pi\times100\ {\rm Hz},$$

the leading ideal result is

$$T_c^{\mathrm{trap}} \simeq 2.10\times10^{-7}\ {\rm K} = 210\ {\rm nK}.$$

At half this temperature, the ideal trap fraction is

$$\frac{N_0}{N} = 1-\left(\frac12\right)^3 = \frac78.$$

For a uniform box at $T/T_c=1/2$, the corresponding ideal fraction is

$$1-\left(\frac12\right)^{3/2} \simeq 0.646.$$

The contrast comes entirely from the different density-of-states exponents.

Real-Space Coherence

For a translation-invariant uniform gas,

$$\gamma(\mathbf r,\mathbf r') = \frac{N_0}{V} + \int \frac{d^3k}{(2\pi)^3} \overline n_{\mathbf k} e^{i\mathbf k\cdot(\mathbf r-\mathbf r')},$$

where the integral contains the noncondensed modes. At large separation, the thermal contribution decays, leaving

$$\lim_{ |\mathbf r-\mathbf r'| \to \infty } \gamma(\mathbf r,\mathbf r') = n_0,$$

with

$$n_0 = \frac{N_0}{V}.$$

This nonzero asymptote is off-diagonal long-range order. It is the real-space form of an extensive eigenvalue of the one-body density matrix.

In an inhomogeneous trap there is no translation-invariant separation limit of this simple form. Diagonalizing $\gamma(\mathbf r,\mathbf r')$ provides the more general criterion.

Long-range one-body coherence is not complete state tomography. Density correlations, phase fluctuations, depletion, and higher-order correlations contain additional information.

Condensation and Symmetry Breaking

The microscopic Hamiltonian of a number-conserving Bose gas is invariant under the global transformation

$$\psi(\mathbf r) \longrightarrow e^{i\theta} \psi(\mathbf r).$$

In an exact fixed-$N$ state,

$$\langle\psi(\mathbf r)\rangle = 0$$

because $\psi$ changes particle number. Nevertheless, the one-body density matrix can have an extensive eigenvalue. The Penrose–Onsager criterion therefore defines condensation without assuming broken symmetry.

In a symmetry-breaking description, one writes the complex order parameter

$$\Phi(\mathbf r) = \langle\psi(\mathbf r)\rangle,$$

with

$$\int d^d r\, |\Phi(\mathbf r)|^2 \simeq N_0.$$

This representation is powerful in the thermodynamic limit and in mean-field theory. It selects a phase and replaces the condensate operator partly by a classical field. A number-conserving treatment and a broken-symmetry treatment can agree on local observables in their common regime while organizing the calculation differently.

The condensate phase is not an absolute observable. Interference experiments reveal relative phases, and measurement can establish a relative phase even between initially number-defined clouds.

The general logic of fixed-number symmetry, source-selected phase states, and their local thermodynamic equivalence is developed in Spontaneous Symmetry Breaking.

Finite Systems

At finite $N$ and finite volume:

• the spectrum is discrete;
• the partition function is analytic;
• $\mu$ remains below the lowest one-particle energy;
• the condensate fraction changes smoothly;
• there is no exact thermodynamic singularity.

For the ideal ground mode,

$$N_0 = \frac{z}{1-z}.$$

Solving for fugacity gives

$$z = \frac{N_0}{ N_0+1 }.$$

Therefore

$$\beta\mu = \ln \left( \frac{N_0}{ N_0+1 } \right) \simeq - \frac{1}{N_0}$$

for macroscopic $N_0$. The statement $\mu=0$ below $T_c$ is a thermodynamic-limit shorthand. A finite ideal system has $\mu<0$.

A finite cloud may still have a clearly dominant natural occupation and experimentally useful crossover temperature. “No exact phase transition” does not mean “no observable condensate.” The general distinction among finite-$N$ rounding, mode resolution, trap scales, and a bulk comparison belongs to Finite-Size Effects.

Ensemble Dependence of Condensate Fluctuations

The leading thermodynamic-limit mean condensate fraction can agree across ensembles, while condensate-number fluctuations do not.

For a single ideal bosonic mode in the grand-canonical ensemble,

$$\operatorname{Var}(N_0) = N_0(1+N_0).$$

If $N_0$ is extensive,

$$\frac{ \sqrt{ \operatorname{Var}(N_0) } }{ N_0 } \to 1.$$

These order-one relative fluctuations are often called the grand-canonical condensate-fluctuation problem. They arise from coupling an exactly zero-energy ideal mode to an unlimited particle reservoir.

In a canonical ensemble, fixed total number couples the condensate and thermal cloud:

$$N_0 = N - N_{\mathrm{ex}}.$$

Total-number fluctuations vanish, and condensate fluctuations are controlled by fluctuations of the excited population. Interactions modify the result again.

Ensemble equivalence must therefore be stated for a specified observable and limit. Bulk pressure can be ensemble equivalent while a zero-mode fluctuation is not.

Thermodynamic Signatures of the Ideal Transition

Below $T_c$ in the uniform ideal gas,

$$P = \frac{ k_{\mathrm B}T }{ \lambda_T^3 } \zeta(5/2),$$

and

$$U = \frac32 PV.$$

The condensate carries no kinetic energy in the convention $\epsilon_0=0$. Hence

$$U \propto T^{5/2}$$

below $T_c$, and

$$\frac{C_V}{ Nk_{\mathrm B} } = \frac{15}{4} \frac{ \zeta(5/2) }{ \zeta(3/2) } \left( \frac{T}{T_c} \right)^{3/2}.$$

There is no latent heat in the ideal transition. Thermodynamic derivatives become nonanalytic only in the thermodynamic limit. The general distinction between first-order and continuous thermal transitions, including finite-size rounding, is developed in Finite-Temperature Phase Transitions.

The ideal uniform gas also becomes anomalously compressible as $z\to1^-$. Its pathological softness below the transition is another sign that interactions are essential for a realistic stable superfluid.

The complete ideal-gas thermodynamics and its assumptions belong to the Ideal Bose Gas.

Interactions Change the State, Not the Definition

For a dilute gas with short-range interactions, a common low-energy Hamiltonian is

$$\hat H = \int d^3r\, \left[ \frac{\hbar^2}{2m} \boldsymbol\nabla\psi^\dagger \cdot \boldsymbol\nabla\psi + V_{\mathrm{ext}} \psi^\dagger\psi + \frac{g}{2} \psi^\dagger\psi^\dagger\psi\psi \right],$$

with

$$g = \frac{4\pi\hbar^2a}{m},$$

where $a$ is the $s$-wave scattering length.

For a weakly repulsive homogeneous gas, the dilute parameter is

$$na^3 \ll 1.$$

Interactions have several effects:

• they make the condensate orbital self-consistent rather than a bare one-particle eigenstate;
• they produce finite compressibility and sound;
• they create quantum depletion even at zero temperature;
• they change collective modes and condensate fluctuations;
• they connect condensation with superfluid response in a nontrivial way;
• they shift and reshape the critical region;
• they provide collisions needed for equilibration.

At zero temperature, the leading homogeneous Bogoliubov depletion is

$$\frac{N-N_0}{N} = \frac{8}{ 3\sqrt\pi } \sqrt{na^3} + \cdots.$$

Thus an interacting condensate need not have $N_0=N$ even at $T=0$.

The Penrose–Onsager criterion still applies: interactions change $\gamma$ and its eigenvalues, not the definition of condensation. The Weakly Interacting Bose Gas Preview develops the dilute parameter, mean-field scales, Bogoliubov sound branch, depletion, and superfluidity boundary. Those results require their own approximations and are not consequences of the ideal saturation derivation.

Condensate Fraction Is Not Superfluid Fraction

The condensate fraction is

$$f_0 = \frac{N_0}{N},$$

defined from the largest eigenvalue of the one-body density matrix.

The superfluid fraction is a response coefficient. It can be defined through the free-energy cost of a phase twist, response to slow rotation, or nondissipative transport under stated conditions.

The two quantities are related in many weakly interacting Bose gases but are not identical:

• the uniform ideal Bose gas condenses, yet its quadratic excitation spectrum gives zero Landau critical velocity;
• strongly interacting liquid helium can have a superfluid fraction much larger than its condensate fraction;
• an infinite homogeneous two-dimensional fluid can have BKT superfluidity with algebraic one-body order but no conventional finite-temperature condensate;
• disorder, lattices, and finite geometry can further separate one-body coherence from transport response.

“BEC” is therefore not a synonym for “superfluid,” and neither term should be inferred from the other without assumptions.

Condensation Is Not Ordinary Spatial Condensation

Bose–Einstein condensation occurs in one-particle state space. The condensate orbital can extend across the entire sample or trap.

It is not the same as:

• forming a liquid droplet;
• freezing into a solid;
• increasing the real-space density without bound;
• every atom sitting at one spatial point;
• every particle following a classical trajectory with zero velocity.

Repulsive interactions, trapping, kinetic energy, and quantum pressure determine the spatial profile of an atomic condensate. The word “condensation” refers to macroscopic mode occupation.

Experimental Diagnostics

No single measurement is universally sufficient in every system. Common diagnostics include:

Momentum-space bimodality.
After release from a trap, a narrow low-momentum component can appear on top of a broad thermal distribution.

One-body coherence.
Interference and correlation measurements probe the spatial coherence encoded in $\gamma(\mathbf r,\mathbf r')$.

Condensate-mode fitting.
In trapped gases, density profiles can be decomposed into a condensate contribution and a thermal cloud using a model appropriate to the interaction and expansion regime.

Macroscopic natural occupation.
When the one-body density matrix can be reconstructed or computed, its leading eigenvalue gives the basis-independent condensate fraction.

Response and collective behavior.
Sound, vortices, collective modes, and superfluid response provide important information about interactions and superfluidity, but they probe more than condensation alone.

The experimental sequence, 1995 dilute-gas observations, and historical interpretation belong to Bose–Einstein Condensation as Historical and Modern Topic and the compact experiment card.

Variants and Boundaries

Internal components

If $g$ internal states are independent, degenerate, and share one chemical potential, the normal-phase excited density acquires a factor $g$. For a uniform ideal gas,

$$n_{\mathrm{ex,max}} = \frac{ g\zeta(3/2) }{ \lambda_T^3 }.$$

Then

$$T_c \propto \left( \frac{n}{g} \right)^{2/3}.$$

But the factor cannot be inserted blindly. If component numbers are separately conserved, if levels are split, or if the ground manifold is degenerate, several chemical potentials or fragmented condensation may be relevant.

Lattices

On a lattice, the low-energy band dispersion and density of states replace the continuum quadratic formulas. Interactions can drive a superfluid-to-Mott transition, where a simple ideal-band condensation picture is inadequate.

Quasiparticles

Photons, magnons, polaritons, and other bosonic quasiparticles may have nonconserved number, driven-dissipative dynamics, effective masses, or finite lifetimes. Their condensation criteria require the actual conservation laws and equilibration mechanism. The atomic fixed-$N$ formulas should not be transplanted automatically.

Degenerate minima

If several one-particle minima are exactly degenerate, the condensate can occupy a superposition, choose one minimum through symmetry breaking, or fragment across modes. The one-body density matrix decides which description applies.

Practical Workflow

To analyze a possible Bose condensate:

1. Identify the bosonic particles or quasiparticles and the conserved quantities.
2. Specify the one-particle spectrum, geometry, boundary conditions, and internal components.
3. Separate any discrete lowest modes from the continuum approximation.
4. Determine the low-energy density-of-states exponent $\alpha$.
5. Check whether the excited-state integral converges as $\mu\to\epsilon_0$.
6. Define the thermodynamic limit appropriate to a box, trap, lattice, or other geometry.
7. Compute $T_c$ and $N_0/N$ only within the assumptions of the chosen spectrum.
8. For interacting or inhomogeneous systems, use the one-body density matrix rather than assuming the condensate orbital.
9. Treat finite-size rounding and ensemble-dependent fluctuations explicitly.
10. Keep condensate fraction, coherence, symmetry breaking, and superfluid response conceptually distinct.

Common Mistakes

Defining BEC only as ground-state occupation.
That is correct for a simple ideal gas with a unique lowest orbital. The general criterion uses the largest eigenvalue of the one-body density matrix.

Assuming every bosonic gas condenses at finite temperature.
The low-energy density of states and thermodynamic limit decide whether the excited-state capacity is finite.

Applying the box formula to a harmonic trap.
The critical-temperature scaling and condensate-fraction exponent differ.

Calling a finite-system crossover an exact phase transition.
Finite partition functions are analytic; sharp nonanalyticity requires a limit.

Setting $\mu=\epsilon_0$ in a finite ideal system.
For finite $N_0$, $\mu$ remains strictly below $\epsilon_0$.

Replacing the ground mode by the continuum density of states.
This can erase the condensate contribution.

Equating condensate fraction with the probability that one particle is condensed.
$N_0/N$ is a natural occupation divided by total number, not a particle-label probability.

Equating BEC with superfluidity.
Condensation is one-body order; superfluidity is a response property.

Assuming all particles occupy the condensate at zero temperature.
Interactions generate quantum depletion.

Treating a narrow momentum peak as a complete many-body diagnosis.
Geometry, expansion dynamics, finite resolution, interactions, and coherence measurements matter.

Ignoring component degeneracy.
Every spin or internal state is a distinct mode unless dynamics imposes a different effective description.

Using broken-symmetry notation as the definition.
An exact fixed-number condensate can have $\langle\psi\rangle=0$ and still satisfy the Penrose–Onsager criterion.

Exercises

Uniform critical temperature

Starting from

$$n = \frac{\zeta(3/2)}{ \lambda_{T_c}^3 },$$

derive $T_c$ for a three-dimensional uniform ideal gas. Check the dimensions.

Solution

Insert

$$\lambda_{T_c} = \left( \frac{ 2\pi\hbar^2 }{ mk_{\mathrm B}T_c } \right)^{1/2}.$$

Then

$$n \left( \frac{ 2\pi\hbar^2 }{ mk_{\mathrm B}T_c } \right)^{3/2} = \zeta(3/2).$$

Raise both sides to the power $2/3$ and solve:

$$T_c = \frac{ 2\pi\hbar^2 }{ mk_{\mathrm B} } \left[ \frac{n}{ \zeta(3/2) } \right]^{2/3}.$$

The factor $\hbar^2/m$ has units of energy times length squared, while $n^{2/3}$ has units of inverse length squared. Their product is an energy, and division by $k_{\mathrm B}$ gives temperature.

General low-energy criterion

Suppose

$$D(\epsilon) = C\epsilon^{\alpha-1}$$

near the lowest energy. Determine when the excited-state capacity is finite and derive its temperature scaling.

Solution

At the saturation boundary,

$$N_{\mathrm{ex,max}} = C \int_0^\infty d\epsilon\, \frac{ \epsilon^{\alpha-1} }{ e^{\beta\epsilon}-1 }.$$

For small $\epsilon$,

$$e^{\beta\epsilon}-1 \simeq \beta\epsilon,$$

so the integrand behaves as

$$\frac{C}{\beta} \epsilon^{\alpha-2}.$$

The lower-limit integral converges when

$$\alpha-2 > -1,$$

or

$$\alpha > 1.$$

With $x=\beta\epsilon$,

$$N_{\mathrm{ex,max}} = C (k_{\mathrm B}T)^\alpha \int_0^\infty dx\, \frac{ x^{\alpha-1} }{ e^x-1 }.$$

The remaining integral is

$$\Gamma(\alpha)\zeta(\alpha),$$

so

$$N_{\mathrm{ex,max}} = C \Gamma(\alpha) \zeta(\alpha) (k_{\mathrm B}T)^\alpha.$$

Infrared divergence in two dimensions

Show directly that a uniform two-dimensional ideal Bose gas with quadratic dispersion has no finite excited-state capacity at nonzero temperature.

Solution

At $\mu=0$,

$$n_{\mathrm{ex}} = \int \frac{d^2k}{(2\pi)^2} \frac{1}{ e^{\beta\hbar^2k^2/(2m)}-1 }.$$

For small $k$,

$$\frac{1}{ e^{\beta\hbar^2k^2/(2m)}-1 } \simeq \frac{ 2mk_{\mathrm B}T }{ \hbar^2k^2 }.$$

The two-dimensional measure is proportional to $k\,dk$, so the infrared contribution is

$$n_{\mathrm{ex}} \sim \int_0 \frac{dk}{k}.$$

This diverges logarithmically. The excited modes can absorb arbitrarily large density as $\mu\to0^-$, so there is no standard finite-temperature ideal-gas BEC in the uniform two-dimensional thermodynamic limit.

Harmonic-trap result

Use

$$D_{\mathrm{trap}}(\epsilon) = \frac{\epsilon^2}{ 2(\hbar\overline\omega)^3 }$$

to derive the ideal three-dimensional harmonic-trap critical temperature and condensate fraction.

Solution

At saturation,

$$\begin{aligned} N_{\mathrm{ex,max}} &= \frac{1}{ 2(\hbar\overline\omega)^3 } \int_0^\infty d\epsilon\, \frac{\epsilon^2}{ e^{\beta\epsilon}-1 } \\ &= \frac{ (k_{\mathrm B}T)^3 }{ 2(\hbar\overline\omega)^3 } \Gamma(3)\zeta(3). \end{aligned}$$

Since

$$\Gamma(3) = 2,$$

this becomes

$$N_{\mathrm{ex,max}} = \zeta(3) \left( \frac{ k_{\mathrm B}T }{ \hbar\overline\omega } \right)^3.$$

Setting this equal to $N$ gives

$$T_c = \frac{ \hbar\overline\omega }{ k_{\mathrm B} } \left[ \frac{N}{ \zeta(3) } \right]^{1/3}.$$

Below $T_c$,

$$\frac{N_{\mathrm{ex}}}{N} = \left( \frac{T}{T_c} \right)^3,$$

so

$$\frac{N_0}{N} = 1 - \left( \frac{T}{T_c} \right)^3.$$

Finite-system chemical potential

For a finite ideal system with ground energy zero, show that

$$\mu \simeq - \frac{ k_{\mathrm B}T }{ N_0 }$$

when $N_0\gg1$.

Solution

The ground occupation is

$$N_0 = \frac{z}{1-z}.$$

Solving gives

$$z = \frac{N_0}{ N_0+1 }.$$

Because $\mu=k_{\mathrm B}T\ln z$,

$$\mu = k_{\mathrm B}T \ln \left( 1 - \frac{1}{ N_0+1 } \right).$$

For $N_0\gg1$, use $\ln(1-x)\simeq-x$:

$$\mu \simeq - \frac{ k_{\mathrm B}T }{ N_0+1 } \simeq - \frac{ k_{\mathrm B}T }{ N_0 }.$$

Thus $\mu$ approaches zero from below as the condensate becomes macroscopic but is not exactly zero at finite $N_0$.

Simple and fragmented condensation

Consider a two-mode one-body density matrix

$$\gamma = N \begin{pmatrix} 1/2 & c\\ c^* & 1/2 \end{pmatrix},$$

with $|c|\leq1/2$. Find its natural occupations. Classify the limits $|c|=1/2$ and $c=0$.

Solution

The eigenvalues are

$$N_\pm = N \left( \frac12 \pm |c| \right).$$

When

$$|c| = \frac12,$$

the occupations are

$$(N_+,N_-) = (N,0).$$

There is one macroscopically occupied natural orbital: simple condensation.

When

$$c = 0,$$

the occupations are

$$(N_+,N_-) = \left( \frac N2, \frac N2 \right).$$

Two orthogonal natural orbitals are macroscopically occupied, so the state is fragmented at the one-body level.

For fixed $0<|c|<1/2$ as $N\to\infty$, both eigenvalues remain proportional to $N$, which is also fragmented condensation.

Condensate and superfluid diagnostics

For each statement, identify whether it concerns condensation, superfluidity, both, or neither:

1. the largest eigenvalue of $\gamma$ is $0.4N$;
2. the free energy increases quadratically under a small phase twist;
3. the Landau critical velocity is zero;
4. $\langle\psi\rangle=0$ in a fixed-number state.

Solution

1. An eigenvalue $0.4N$ is extensive, so it directly diagnoses condensation.
2. A phase-twist stiffness diagnoses superfluid response.
3. Zero Landau critical velocity means the Landau criterion does not support stable superflow; it does not by itself decide whether there is a condensate. The uniform ideal Bose gas is the standard example with BEC but zero Landau critical velocity.
4. This decides neither. Exact particle-number conservation forces $\langle\psi\rangle=0$, but condensation can still be present in the one-body density matrix.

Quantum depletion estimate

For a homogeneous weakly interacting gas with

$$na^3 = 10^{-6},$$

estimate the leading zero-temperature noncondensed fraction.

Solution

The Bogoliubov result is

$$\frac{N-N_0}{N} \simeq \frac{8}{ 3\sqrt\pi } \sqrt{na^3}.$$

Here

$$\sqrt{na^3} = 10^{-3}.$$

Therefore

$$\frac{N-N_0}{N} \simeq \frac{8}{ 3\sqrt\pi } \times 10^{-3} \simeq 1.50\times10^{-3}.$$

The leading depletion is about $0.15\%$. It is small but nonzero, illustrating that an interacting ground state is not a pure occupation of one bare mode.

Cross-Links

• Ideal Bose Gas — box spectrum, partition function, density of states, equation of state, and ideal thermodynamics.
• Weakly Interacting Bose Gas Preview — contact interactions, mean-field stiffness, Bogoliubov phonons, depletion, and the Landau criterion.
• Bose–Einstein Statistics — the one-mode occupation factor and chemical-potential constraint.
• Occupation Numbers — Fock counts, means, fluctuations, one-body density matrices, and natural occupations.
• Thermodynamic Limit — why a finite crossover becomes a sharp phase transition only in a limit.
• Field Operators — field expansions and real-space one-body density matrices.
• Correlation Functions Overview — off-diagonal long-range order and correlation diagnostics.
• Equal-Time Correlations — one-body coherence conventions, positivity checks, and the boundary between static one-body and pair diagnostics.
• Fluctuations and Susceptibilities — ensemble-dependent number fluctuations and response.
• Bose–Einstein Condensation as Historical and Modern Topic — prediction, experimental realization, and historical context.
• Bose–Einstein Condensates Overview — AMO preparation, effective-field scales, collective modes, interference, and experimental evidence.
• Ideal Bose Gas Reference Card — compact formulas for lookup.
• Bose Gas Formula Sheet — ideal occupations, condensation scales, thermodynamics, and dimensional limits.

References

• R. K. Pathria and P. D. Beale, Statistical Mechanics, 4th ed., Academic Press (2021) — ideal Bose gas, condensation, dimensionality, and thermodynamic limits.
• K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987) — standard ideal-gas derivation and thermodynamic behavior.
• C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases, 2nd ed., Cambridge University Press (2008) — dilute trapped gases, interactions, and condensate physics.
• L. Pitaevskii and S. Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016) — condensation criteria, weak interactions, superfluidity, and finite-temperature theory.
• O. Penrose and L. Onsager, “Bose–Einstein Condensation and Liquid Helium”, Physical Review 104, 576–584 (1956) — basis-independent definition through an extensive eigenvalue of the one-body density matrix.
• P. C. Hohenberg, “Existence of Long-Range Order in One and Two Dimensions”, Physical Review 158, 383–386 (1967) — rigorous restrictions on long-range order in low-dimensional equilibrium systems.
• F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose–Einstein Condensation in Trapped Gases”, Reviews of Modern Physics 71, 463–512 (1999) — authoritative review of trapped dilute-gas theory.
• W. Ketterle and N. J. van Druten, “Bose–Einstein Condensation of a Finite Number of Particles Trapped in One or Three Dimensions”, Physical Review A 54, 656–660 (1996) — finite-number and trap-dimensionality analysis.