Condensation in an interacting Bose gas—and the particles left outside it

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Family 267 · Mathematical physics
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

Bose–Einstein condensation means that a macroscopic fraction of particles occupies the same one-particle state. For an ideal gas this follows from particle statistics. Interactions make the problem much harder: repulsion changes the many-particle state, and even at zero temperature some particles remain outside the condensate.

Can one establish condensation for an exact interacting gas while keeping its density and interaction fixed as the sample grows without bound? And in a dilute gas, can one determine the fraction left outside the condensate, rather than merely show that the condensate exists?

Family 267 contains five manuscripts addressing these questions in three dimensions. They treat hard spheres and fixed bounded repulsive potentials, including positive-temperature condensation and zero-temperature depletion. Their central convention is the ordinary thermodynamic limit at fixed physical parameters, followed, where needed, by a separate dilute limit.

2. Background and essential definitions

Place $N$ identical bosons on a periodic cube of side $L$, mathematically the torus $(\mathbb R/L\mathbb Z)^3$. Their wave function is symmetric under particle exchange. Density is $\rho=N/L^3$, and the thermodynamic limit sends $N,L\to\infty$ while this ratio tends to a fixed positive number.

For a bounded repulsive potential $v$, the Hamiltonian is

$$H_{N,L}=-\sum_{i=1}^N\Delta_i +\sum_{i<j}v_L(x_i-x_j),$$

where $v_L$ is the periodic version of $v$. Units are chosen so that $\hbar^2/(2m_{\mathrm{particle}})=1$, and the thermal papers also set Boltzmann’s constant to one.

For hard spheres, configurations with two centers at distance at most $a$ are forbidden. The Hamiltonian is the kinetic operator on the allowed configurations, with Dirichlet boundary conditions at contact. The parameter $a$ is the exclusion distance between centers, not a separately doubled geometric radius.

The scattering length summarizes low-energy two-body repulsion. For a bounded radial potential it is determined by the zero-energy scattering solution

$$\left(-\Delta+\frac12v\right)f=0, \qquad f(x)=1-\frac{a}{|x|}$$

outside the potential’s range. For the hard-core model, the scattering length equals the exclusion distance. The dimensionless gas parameter is $\eta=\rho a^3$; small $\eta$ means dilute.

For an $N$-particle state $\Gamma$, its one-particle density matrix is $\gamma^{(1)}=N\operatorname{Tr}_{2,\ldots,N}\Gamma$, with trace $N$. Let $u_0=L^{-3/2}$ be the normalized constant orbital. Its occupation fraction is

$$B=\frac{\langle u_0,\gamma^{(1)}u_0\rangle}{N}.$$

A positive lower limit of $B$ establishes condensation in that orbital. The fraction $1-B$ is its depletion. At positive temperature the exact canonical state is $e^{-H/T}/\operatorname{Tr}(e^{-H/T})$; at zero temperature the papers use ground states or density matrices supported on the ground eigenspace.

3. Development history and previous partial results

Bose’s counting and Einstein’s extension to material particles led to the ideal-gas condensation prediction. Penrose and Onsager formulated the interacting version through a macroscopic eigenvalue of the one-particle density matrix. Bogoliubov’s quasiparticle theory then predicted excitations and depletion around a macroscopically occupied mode. Lee, Huang and Yang calculated the leading hard-sphere energy correction and depletion coefficient.

Rigorous control of the energy came through Dyson’s bounds and Lieb–Yngvason’s leading dilute asymptotics. Later Lee–Huang–Yang estimates reached the next energy term. Fournais and Solovej’s general lower bound includes hard cores; the manuscripts credit Basti, Brooks, Cenatiempo, Olgiati and Schlein with the matching sharp hard-sphere upper bound. These are crucial energy inputs, but the occupation of a single orbital is a separate observable.

The reason is visible in a large box. The first nonzero one-particle kinetic energy is of order $L^{-2}$. Moving particles into extremely long-wavelength modes can cost little energy as $L$ grows. An accurate energy formula alone therefore need not distinguish the global constant orbital from nearby modes.

Lieb and Seiringer established complete condensation in the trapped Gross–Pitaevskii regime. Further work established thermal condensation and Bogoliubov depletion in other coupled scaling regimes, where scattering length, density or box size vary together. Those results have their own substantial content. They do not automatically permit unrestricted growth of volume at a fixed positive gas parameter.

Fixed-parameter condensation also has precedents for lattice gases and continuum interactions under different assumptions, including positivity of both the potential and its Fourier transform. A literal hard core falls outside some such hypotheses. The present family’s claimed advance is exact canonical or ground-state occupation in the stated fixed-interaction thermodynamic setting, followed by sharp depletion information.

4. What the new manuscripts claim

Condensation for hard spheres

M5 claims absolute constants $\eta_0,c_0>0$ such that, whenever fixed $a,\rho>0$ satisfy $\rho a^3<\eta_0$, every thermodynamic sequence of hard-sphere ground states obeys

$$\liminf B\geq c_0.$$

This includes arbitrary complex ground vectors and mixtures supported on the ground eigenspace. The positive fraction is uniform throughout the sufficiently dilute interval. The volume threshold may depend on $a$ and $\rho$. M5 itself establishes a positive fraction, without claiming the optimal depletion law.

M1 adds positive temperature. For every fixed $a>0$, each sufficiently small fixed density admits a fixed temperature $T(a,\rho)>0$ at which the exact canonical Gibbs state has a positive lower limiting constant-orbital fraction. Exclusion distance, density and temperature all stay fixed while volume grows. The theorem does not locate the transition temperature or state a sharp thermal depletion formula.

A uniform bound for bounded repulsion

M4 assumes $v$ is bounded, measurable, nonnegative, radial, finite-range and not zero almost everywhere. It claims constants $\rho_*(v),c_*(v)>0$ such that, for each $0<\rho<\rho_*$, all sufficiently large boxes with

$$\rho/2\leq N/L^3\leq2\rho$$

satisfy $B\geq c_*$ for every $0\leq T\leq\rho^2$, in its stated units. The interaction is not weakened with particle number. The fraction bound is independent of density within the admitted interval, although the required box size is not. Neither radial monotonicity nor positivity of the Fourier transform is assumed.

The leading quantum depletion

At zero temperature, M2 for hard spheres and M3 for fixed bounded potentials claim

$$1-B=\frac{8}{3\sqrt\pi}\sqrt{\rho a^3} +o\bigl(\sqrt{\rho a^3}\bigr).$$

The thermodynamic limit is taken first at each fixed density, then $\rho\downarrow0$. More precisely, all thermodynamic accumulation values have this same dilute asymptotic, uniformly over the stated ground-state density matrices. The claim does not require that occupation have a unique thermodynamic limit at every fixed positive density.

The bounded-potential theorem keeps the potential itself fixed throughout both limits. Its hypotheses are the same type of bounded, nonnegative, radial finite-range repulsion, with positive scattering length. M3 reports uniqueness of the finite-volume ground state in that class but formulates occupation using density matrices.

Where the depleted hard-sphere particles go

M2 claims the full scaled nonzero-momentum occupation measure for hard spheres. For periodic momenta $k\in(2\pi/L)\mathbb Z^3$, let $n(k)$ be the occupation of $L^{-3/2}e^{ik\cdot x}$. Define

$$\nu_{N,L}=\frac1{N\sqrt\eta} \sum_{k\ne0}n(k)\,\delta_{k/\sqrt{8\pi\rho a}}.$$

In the same thermodynamic-then-dilute sense, this converges against every bounded continuous test function to the finite measure

$$\nu_{\mathrm{Bog}}(dt)=\frac{(8\pi)^{3/2}}{(2\pi)^3}g(t)\,dt,$$

$$g(t)=\frac{|t|^2+1}{2\sqrt{|t|^4+2|t|^2}}-\frac12.$$

The value at $t=0$ is irrelevant to the integral. The density is integrable both near zero and at infinity, and the total measure is $8/(3\sqrt\pi)$. Testing with the constant function one gives the depletion coefficient. This momentum-profile theorem is asserted for hard spheres; M3’s stated headline is the total depletion for bounded potentials.

5. Why the advance matters

Provisional importance: 5/5. Condensation in a fixed-density interacting continuum gas is a central many-body question. The claims address coherence across arbitrarily large samples, not just an energy expansion or a regime in which interactions simultaneously vanish with the box size.

The family then reaches the next level of precision: a universal leading depletion coefficient depending on the interaction through scattering length. For hard spheres it identifies the distribution of those depleted particles on the natural momentum scale, rather than just their total number.

The positive-temperature result is also substantial, but its role is an existence theorem at a sufficiently low fixed temperature. It should not be inflated into a determination of the critical temperature or a complete phase diagram. The importance comes from the stated coherence and occupation claims themselves.

6. The main idea

The core obstacle is long-distance coherence. Local energy estimates can show that particles prefer a nearly constant mode inside a moderate box. They do not automatically make the phases or amplitudes in far-apart boxes agree.

The papers use positive imaginary-time path representations. Roughly, a quantum state or thermal kernel is represented using Brownian trajectories with interaction weights; hard spheres enforce avoidance of forbidden contacts. One compares configurations in which a distinguished particle has been moved between distant cells while the observed remaining particles—the bath—are matched.

Multiplying a fixed loss at every step of a long route would destroy the desired bound as the sample grows. Instead, the construction averages over many routes before taking a second moment. Exact conditional normalizations cancel away from encounters between two sampled routes. The remaining cost depends on how often the routes meet, rather than on their total lengths. Probabilistic paths with controlled intersection tails, plus detours around rare unusable cells, provide the geometric input.

M5 first handles the modulus of a ground wave function and then controls phase transfer to cover arbitrary complex ground states. M1 must additionally handle a mixed thermal state. It factors the Gibbs kernel through its square root, retaining the same bath in the two factors whose overlap gives constant-orbital occupation. Joint spatial and trajectory estimates ensure that enough movable particles remain available under the interacting thermal law.

The depletion papers separate two contributions. Let $P_\ell$ project a one-particle function onto functions constant on cubes of side $\ell$, and let $P_0$ project onto the global constant. Since $P_0\leq P_\ell$, for a normalized bosonic ground vector,

$$1-B=\langle\Psi,(1-P_\ell)_1\Psi\rangle +\|(P_\ell-P_0)_1\Psi\|^2.$$

The subscript means action on one particle. The first term measures variation inside cubes; the second measures differences between their averages.

Local energy bounds retaining positive quasiparticle terms control the first contribution at the Bogoliubov scale. Sharper comparisons of particle-deletion measures control the second below the depletion scale, using fourth moments locally and second moments over long distances. M2 then tests one-body observables to recover the whole momentum distribution. M3 adapts the method to bounded interactions while preserving their exact Boltzmann weights.

The companion roles are explicit. M5 uses M4’s geometric route construction; M1 develops the additional thermal hard-core estimates; M2 uses M5’s spatial and path inputs; M3 uses bounded-potential density tails from M4 and route/comparison methods from M2. These are related arguments with model-specific work, not five interchangeable formulations of one theorem.

7. Scope, limitations, and open questions

All five papers concern three-dimensional homogeneous periodic systems and the specified repulsive interactions. They do not directly cover attractive gases, arbitrary long-range forces, disordered systems or every trapping geometry. The hard-core and bounded-potential hypotheses must remain separate.

Condensation, superfluid response, energy asymptotics and a critical-temperature formula are different properties. The family does not claim all of them from the same occupation bound. The temperature produced for hard spheres is positive and volume-independent, but not claimed optimal. The exact depletion law is a zero-temperature statement.

The sharp occupation asymptotics retain their order of limits and accumulation-value interpretation. No unique fixed-density thermodynamic occupation limit is assumed or supplied by that wording. Rates of convergence, finite-temperature depletion coefficients and transition-temperature shifts remain additional questions beyond the displayed claims.

8. Sources and further reading

The primary links in section 3 lead to Lieb–Seiringer’s trapped condensation theorem and Fournais–Solovej’s energy analysis including hard cores. The manuscripts’ bibliographies give the original Bogoliubov and Lee–Huang–Yang predictions and the later sharp upper bounds. M4 and M5 introduce the fixed-parameter condensation mechanisms, M1 extends the hard-core setting to positive temperature, and M2–M3 give the sharper zero-temperature occupation statements.

Source manuscripts

• M1. OpenAI, Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas (October 5, 2026).
• M2. OpenAI, Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas (October 5, 2026).
• M3. OpenAI, Quantum Depletion for Fixed Bounded Repulsive Potentials (October 5, 2026).
• M4. OpenAI, A density-uniform condensate bound for dilute Bose gases (September 27, 2026).
• M5. OpenAI, Ground-state condensation in the dilute hard-sphere gas (September 24, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.