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Bose-Einstein Condensation in Continuum Models

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Statement

Definitions and mathematical statement

For $N$ identical bosons in a box $\Lambda\subset{\mathbb{R}}^3$, a standard interacting Hamiltonian is \[ H_{N,\Lambda}=\sum_{i=1}^N-\Delta_{x_i}+\sum_{i<j}v(x_i-x_j) \] on symmetric wave functions. The one-particle density matrix is $\gamma(x,y)=\langle a^*(x)a(y)\rangle$. Off-diagonal long-range order means that the thermodynamic-limit kernel retains a nonzero long-distance component; in translation-invariant situations this is expressed as $\lim_{|x-y|\to\infty}\gamma(x,y)>0$. The limit sends $N,|\Lambda|\to\infty$ at fixed density before long-distance analysis. The catalogue does not specify the interaction class, ensemble, boundary conditions, or positive-temperature range. These are necessary additional model data, not grounds for claiming the assertion for every interaction.

Formulation and concept references: [S1].

Short English statement

Can low-temperature condensation be rigorously established for a specified interacting three-dimensional continuum Bose gas in the thermodynamic limit?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Phys-7-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-7

For the stated interacting continuum models, the selected sources bound the fraction occupying the constant orbital away from zero along every thermodynamic sequence. This establishes Bose condensation by the macroscopic-mode criterion, including positive temperatures in the specified dilute regime. The target additionally asks for a pointwise infinite-distance limit of the thermodynamic one-particle density kernel; that limit is not asserted by these occupation-fraction theorems. The match therefore remains partial.

Bose-Einstein Condensation in Continuum Models

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: For fixed bounded repulsive finite-range potentials, the constant-mode condensate fraction is claimed uniformly positive at sufficiently small density and temperatures up to the density squared; For sufficiently small fixed hard-sphere density, a positive volume-independent temperature is claimed to give a positive thermodynamic constant-mode condensate fraction; Every ground state of a sufficiently dilute fixed-density three-dimensional hard-sphere gas is claimed to place a uniform positive particle fraction in the constant orbital; The hard-sphere ground state is claimed to have the full scaled Bogoliubov momentum law, with depletion $\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})$; The same leading depletion coefficient is claimed for every fixed bounded nonnegative radial finite-range potential of positive scattering length, uniformly over ground-state density matrices. Together they provide the following support within the scopes described below.

Mathematical statement of the claimed result

Result 1.
Take a bounded measurable radial potential $v:\mathbb R^3\to[0,\infty)$, supported in a ball of finite radius and nonzero on a set of positive measure. On $(\mathbb R/L\mathbb Z)^3$, use the symmetric $N$-particle Hamiltonian \[ H_{N,L}=-\sum_i\Delta_i+\sum_{i<j}v_L(x_i-x_j),\qquad v_L(x)=\sum_{k\in\mathbb Z^3}v(x+Lk). \] Let $\gamma^{(1)}_{N,L,T}$ be the canonical state's one-particle matrix, of trace $N$, and $u_0=L^{-3/2}$. The claim provides $\rho_*(v),c_*(v)>0$ such that every $0<\rho<\rho_*$ has a finite $L_0(\rho,v)$, with \[ L\ge L_0,\quad\rho/2\le N/L^3\le2\rho,\quad0\le T\le\rho^2 \quad\Longrightarrow\quad \frac{\langle u_0,\gamma^{(1)}_{N,L,T}u_0\rangle}{N}\ge c_*. \] The positive fraction is uniform throughout the indicated dilute density range.
Result 2.
Fix exclusion distance $a>0$. On the three-dimensional torus of side $L$, the hard-sphere Hamiltonian is the symmetric Dirichlet kinetic operator on configurations with every interparticle torus distance greater than $a$. Let $\gamma^{(1)}_{N,L,T}$ be the canonical Gibbs one-particle matrix, of trace $N$, and $u_{0,L}=L^{-3/2}$. The source asserts that some $\rho_*(a)>0$ has the property \[ 0<\rho<\rho_*(a)\quad\Longrightarrow\quad \exists T(a,\rho)>0:\quad \liminf_{L\to\infty,\ N/L^3\to\rho} \frac{\langle u_{0,L},\gamma^{(1)}_{N,L,T}u_{0,L}\rangle}{N}>0. \] Here $a,\rho,T$ remain fixed during the thermodynamic limit.
Result 3.
For the three-dimensional torus hard-sphere Bose Hamiltonian, impose Dirichlet exclusion at pair distance $a$, with kinetic energy $-\sum_i\Delta_i$. For a normalized symmetric ground vector $\Psi$, put \[ B(\Psi)=N^{-1}\langle L^{-3/2},\gamma^{(1)}_\Psi L^{-3/2}\rangle, \qquad\operatorname{Tr}\gamma^{(1)}_\Psi=N. \] The claimed universal constants $\varepsilon_0,c_0>0$ give \[ \rho a^3<\varepsilon_0,\quad N_k,L_k\to\infty,\quad N_k/L_k^3\to\rho \quad\Longrightarrow\quad\liminf_k B(\Psi_k)\ge c_0 \] for every choice of ground vectors, allowing complex vectors and degeneracy.
Result 4.
For the three-dimensional periodic hard-sphere Bose gas of exclusion distance $a>0$, let $\mathcal G_{N,L,a}$ be all trace-one states supported on the lowest eigenspace, and $n_{\Gamma,L}(k)$ the occupation of $L^{-3/2}e^{ik\cdot x}$. Define \[ g(t)=\frac{|t|^2+1}{2\sqrt{|t|^4+2|t|^2}}-\frac12,\qquad d\nu_{\mathrm{Bog}}(t)=\frac{(8\pi)^{3/2}}{(2\pi)^3}g(t)\,dt. \] For each bounded continuous real $f$ and $\varepsilon>0$, the source gives $\rho_0(a,f,\varepsilon)>0$. If $0<\rho<\rho_0$, $\eta=\rho a^3$, and $N_j,L_j\to\infty$, $N_j/L_j^3\to\rho$, then \[ \limsup_j\sup_{\Gamma\in\mathcal G_{N_j,L_j,a}} \left|\frac{\sum_{k\in(2\pi/L_j)\mathbb Z^3,\ k\ne0} f(k/\sqrt{8\pi\rho a})n_{\Gamma,L_j}(k)}{N_j\sqrt\eta} -\int f\,d\nu_{\mathrm{Bog}}\right|\le\varepsilon. \] The measure has total mass $8/(3\sqrt\pi)$.
Result 5.
Let $v\ge0$ be bounded, measurable, radial, compactly supported, and nonzero on a positive-measure set. Its scattering length $a_v>0$ is determined by $(-\Delta+v/2)f=0$, with $f(x)=1-a_v/|x|$ outside the support. For the periodic symmetric Hamiltonian $-\sum_i\Delta_i+\sum_{i<j}v_L(x_i-x_j)$, let $\mathcal G^v_{N,L}$ contain all ground-space trace-one states and $B_\Gamma=N^{-1}\langle L^{-3/2},\gamma^{(1)}_\Gamma L^{-3/2}\rangle$. For every $\varepsilon>0$, some $\rho_0(v,\varepsilon)>0$ satisfies \[ 0<\rho<\rho_0,\ N_k,L_k\to\infty,\ N_k/L_k^3\to\rho \ \Longrightarrow\ \limsup_k\sup_{\Gamma\in\mathcal G^v_{N_k,L_k}} \left|\frac{1-B_\Gamma}{\sqrt{\rho a_v^3}}-\frac8{3\sqrt\pi}\right|\le\varepsilon. \] Volume and particle number tend to infinity while density remains fixed.

Outline of the argument

Energy and free-energy estimates control cell occupancy and isolation. A short imaginary-time slab supplies joint trajectory bounds; defective-cell handling and insertion comparisons connect distant cells and preserve a density-uniform coherence lower bound. Scaled kinetic and particle-count bounds localize the canonical state. Thermal-slab trajectory estimates permit moving paths between adjacent cells; environment exposure retains movable groups and turns their long-range coherence into the condensate lower bound.

Energy and spatial bounds produce stationary hard-core path samples. Neighbor-cell moves, retained movable groups, and routes through sparse obstacles compare phase coherence across the volume, giving a constant-mode lower bound also for complex ground states. Local energy estimates keep excitations instead of discarding them. One-body covariance and short-trajectory certificates are compared through smoothed deletion measures, while random encounter routes connect local momentum data to the thermodynamic occupation measure before the dilute limit.

Local density tails and ground-state path bounds control occupations at the relevant scattering scale. Spatial comparison transports these local estimates to the constant orbital, identifying every thermodynamic accumulation value before density tends to zero.

Formal verification

For family 267, the selected statement is a density-uniform hard-sphere ground-state condensate lower bound for pure and mixed ground states. Positive-temperature condensation and quantum-depletion asymptotics are beyond it. The listed Comparator files are HardSphere.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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Phys-7-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-7

Bose-Einstein Condensation in Continuum Models

Definitions and mathematical statement

For $N$ identical bosons in a box $\Lambda\subset{\mathbb{R}}^3$, a standard interacting Hamiltonian is \[ H_{N,\Lambda}=\sum_{i=1}^N-\Delta_{x_i}+\sum_{i<j}v(x_i-x_j) \] on symmetric wave functions. The one-particle density matrix is $\gamma(x,y)=\langle a^*(x)a(y)\rangle$. Off-diagonal long-range order means that the thermodynamic-limit kernel retains a nonzero long-distance component; in translation-invariant situations this is expressed as $\lim_{|x-y|\to\infty}\gamma(x,y)>0$. The limit sends $N,|\Lambda|\to\infty$ at fixed density before long-distance analysis. The catalogue does not specify the interaction class, ensemble, boundary conditions, or positive-temperature range. These are necessary additional model data, not grounds for claiming the assertion for every interaction.

Formulation and concept references: [S1].

Short English statement

Can low-temperature condensation be rigorously established for a specified interacting three-dimensional continuum Bose gas in the thermodynamic limit?

Research attempt

Prepared by GPT-6 Astra Ultra on 27 September 2026, with primary-source review, explicit example checks, and examination of the stated conditional reductions. This is an unresolved research attempt.
A specified model and source audit (27 September 2026).
Fix units in which the kinetic energy is $-\Delta$. Use periodic cubes $\Lambda_L=(\mathbb R/L\mathbb Z)^3$, the canonical Gibbs state at inverse temperature $\beta$, and $N=\lfloor\rho L^3\rfloor$ with $\rho=10^{-3}$. Fix the nonnegative soft-sphere interaction \[ v(x)=1_{\{|x|\le1\}}, \] periodized on the cube, with $L>2$. These data are one concrete continuum model for the catalogue question. The intended regime is fixed finite $\beta$ sufficiently large, followed by $L\to\infty$; the interaction is not rescaled with $N$.

The book [S1], available from its authors as [E1], distinguishes thermodynamic condensation from other scaling limits. Sütő [S2] claims a condensation theorem under simultaneous positivity conditions on the potential and its Fourier transform, with additional cycle analysis. That Fourier hypothesis does not hold here: in the convention $\widehat v(k)=\int e^{-ik\cdot x}v(x)\,dx$, \[ \widehat v(k)=4\pi\frac{\sin |k|-|k|\cos|k|}{|k|^3}, \qquad \widehat v(2\pi e_1)=-1/\pi. \] Its claimed theorem is therefore not invoked for this model, nor is its complete proof independently certified here. König [S3] explicitly studies a variational free-energy problem and leaves off-diagonal long-range order outside its conclusions.

The proposed route is a depletion estimate: control occupations away from zero momentum tightly enough that a positive fraction must remain in the zero mode. An explicit kinetic-density bound can be proved. The missing infrared estimate and the distinction between occupation and pointwise long-range order will remain visible.

Occupation numbers and the order of limits.
The canonical Gibbs state is translation invariant. Its one-particle density matrix is diagonal in plane waves with momenta $p\in(2\pi/L)\mathbb Z^3$. Write their occupations as $n_p\ge0$. They satisfy \[ \sum_p n_p=N,\qquad \gamma_L(0,r)=L^{-3}\sum_p n_p e^{ip\cdot r},\qquad L^{-3}\langle T\rangle=L^{-3}\sum_p |p|^2n_p. \] A sufficient condition for ordinary zero-mode condensation is $\liminf_{L\to\infty}n_0/L^3>0$. This condition refers to a macroscopic eigenvalue of the density matrix, not a large occupation at one fixed finite volume.

Define the finite momentum measures $\nu_L=L^{-3}\sum_p n_p\delta_p$. A uniform kinetic-density bound makes them tight, because their mass outside a ball of radius $R$ is at most $R^{-2}L^{-3}\langle T\rangle$. Along a weakly convergent thermodynamic subsequence, their Fourier transforms converge at every fixed separation. If its limit is \[ \nu=\rho_0\delta_0+(2\pi)^{-3}f(p)\,dp, \qquad f\in L^1(\mathbb R^3), \] the Riemann--Lebesgue lemma proves $\gamma(0,r)\to\rho_0$ as $|r|\to\infty$. The absolute continuity of the remaining momentum measure is an additional hypothesis in this implication.

Without that regularity, a spatial average still detects the atom: averaging $\gamma(0,r)$ over $[-R,R]^3$ multiplies the integrand at momentum $p$ by $\prod_j\sin(Rp_j)/(Rp_j)$. This multiplier is bounded by one and tends to zero for every $p\ne0$. Dominated convergence therefore gives the limit $\nu(\{0\})$. This averaged conclusion must not be substituted for the pointwise long-distance limit in the catalogue. Also, infinitely many small nonzero momentum modes may converge to an atom: a thermodynamic atom alone does not automatically identify one finite-volume mode.

A rigorous kinetic-energy density estimate at positive temperature.
Let $Z_N(\beta)=\operatorname{Tr}_{\mathrm{sym}}e^{-\beta H}$ and $f_N(\beta)=\log Z_N(\beta)$. The constant normalized $N$-body wavefunction has trial energy \[ E_{\mathrm{tr}}=\frac{N(N-1)}{2L^3}\widehat v(0), \qquad \widehat v(0)=4\pi/3. \] The spectral Jensen inequality gives $Z_N(\beta)\ge e^{-\beta E_{\mathrm{tr}}}$. Because $H\ge T$, the min--max principle bounds its partition function above by the free canonical partition function. This trace comparison does not require the exponential function to be operator monotone.

Convexity of $f_N$ gives \[ \langle H\rangle_\beta=-f_N'(\beta) \le\frac{2}{\beta}\left(f_N(\beta/2)-f_N(\beta)\right) \le 2E_{\mathrm{tr}}+\frac{2}{\beta}\log Z_N^{(0)}(\beta/2). \] For the ideal canonical gas, the zero mode fills whatever particles are not excited. Discarding the constraint on their total number gives \[ Z_N^{(0)}(\tau)\le \prod_{p\ne0}(1-e^{-\tau|p|^2})^{-1}. \] At fixed $\tau>0$, the logarithm of this product divided by $L^3$ converges to \[ (2\pi)^{-3}\int_{\mathbb R^3}-\log(1-e^{-\tau|p|^2})\,dp =\frac{\zeta(5/2)}{(4\pi\tau)^{3/2}}. \] Expand the logarithm into its positive series and integrate each Gaussian to obtain the equality. The logarithmic singularity at zero is integrable; shell estimates justify the corresponding Riemann-sum limit after excluding a small ball. Since the interaction is nonnegative, $\langle T\rangle\le\langle H\rangle$. Consequently \begin{equation}\label{wsref-0-62-65-63-3a-65-6e-65-72-67-79} \limsup_{L\to\infty}L^{-3}\langle T\rangle_\beta \le e_*(\beta):=\frac{4\pi}{3}\rho^2 +\frac{2\zeta(5/2)}{\beta(2\pi\beta)^{3/2}}. \end{equation} This is an actual bound for the chosen interacting canonical model. It does not by itself force a macroscopic eigenvalue: kinetic energy assigns arbitrarily small costs to arbitrarily small nonzero momenta.

A precise conditional infrared criterion.
Suppose, in addition, that for some fixed constants $A,B$ and $\kappa>0$, uniformly in sufficiently large $L$, \begin{equation}\label{wsref-0-62-65-63-3a-69-6e-66-72-61-72-65-64} n_p\le\frac{A}{|p|}+\frac{B}{\beta|p|^2} \qquad(0<|p|\le\kappa). \end{equation} Split the excited density at $\kappa$. The high-momentum contribution is at most $\kappa^{-2}L^{-3}\langle T\rangle$. Both singular functions in (bec:infrared) are integrable at zero in dimension three, and their lattice Riemann sums have limits \[ (2\pi)^{-3}\int_{|p|\le\kappa}\frac{dp}{|p|} =\frac{\kappa^2}{4\pi^2},\qquad (2\pi)^{-3}\int_{|p|\le\kappa}\frac{dp}{|p|^2} =\frac{\kappa}{2\pi^2}. \] One can control the omitted ball uniformly by grouping lattice points in shells; its contribution is $O(\varepsilon^2)$ or $O(\varepsilon)$ respectively. Thus \begin{equation}\label{wsref-0-62-65-63-3a-64-65-70-6c-65-74-69-6f-6e} \liminf_{L\to\infty}\frac{n_0}{L^3} \ge\rho-\frac{A\kappa^2}{4\pi^2} -\frac{B\kappa}{2\pi^2\beta} -\frac{e_*(\beta)}{\kappa^2}. \end{equation} Every term is measured per unit volume, with $\rho$ fixed before the thermodynamic limit.

There is a concrete nonempty budget for this sufficient criterion. For $A=1$, $\kappa=1/10$, and the chosen $\rho=10^{-3}$, the temperature-independent depletion allowance is \[ \frac{10^{-2}}{4\pi^2} +\frac{(4\pi/3)10^{-6}}{10^{-2}} <0.000673<\rho. \] Therefore a proof of (bec:infrared) with these constants and some $B$ uniform for all sufficiently large $\beta$ would imply positive zero-mode condensation at sufficiently low positive temperature. This is a conditional numerical budget, not a claimed infrared bound. The coefficients in a physically motivated approximation cannot be inserted into it as rigorous estimates without a uniform error theorem.

Why the easy substitutes do not provide that estimate.
For the ideal gas, integrating $(e^{\beta|p|^2}-1)^{-1}$ gives the critical excited density \[ \rho_c^{(0)}(\beta)=\frac{\zeta(3/2)}{(4\pi\beta)^{3/2}}. \] This benchmark explains the relevance of an integrable infrared singularity. It does not justify using ideal-gas occupations in the interacting canonical state. Operator monotonicity of occupation numbers under addition of a repulsive many-body interaction has not been proved by the partition-function comparison above.

A purely thermal estimate on all excited modes, vanishing as $\beta\to\infty$, would also discard quantum depletion. Already for two particles with this nonconstant interaction, the constant product wavefunction is not an eigenvector: applying $H$ multiplies it by $v(x_1-x_2)$. The positive finite-volume ground state therefore cannot be exactly that product. The temperature-independent term in (bec:infrared) leaves room for such excitations. This observation does not prove its coefficient $A$ or its momentum dependence.

Similarly, a bound on free energy alone is not an estimate on every $n_p$. A one-particle source $s n_p$ would make its derivative an occupation number, but extracting that derivative requires bounds uniform in the source and precise enough on both sides. No such source-dependent control follows from (bec:energy). Finite-volume periodic kernels also have no literal infinite-distance limit: first one must construct a thermodynamic correlation limit.

Verification, gap, and outcome.
The local calculations check the Fourier-transform sign, Gaussian integral constants, and the strict depletion budget. The proofs use the canonical state throughout the interacting argument. The ideal critical-density formula is a separately labelled comparison, not an ensemble-equivalence theorem for the interacting gas.

The first missing estimate is (bec:infrared), or a substitute whose integrated depletion satisfies the strict inequality in (bec:depletion). Establishing the full pointwise ODLRO target also requires a thermodynamic correlation limit and control of its noncondensed momentum part. A successful continuation could obtain source-dependent occupation estimates, prove a uniform low-momentum bound, or justify the necessary correlation regularity. None is provided by taking temperature to zero before volume to infinity.

UNRESOLVED. An explicit kinetic-density bound and a conditional condensation criterion are proved for a fixed interacting model. Neither macroscopic occupation nor pointwise ODLRO is established unconditionally. No novelty or formal proof verification is claimed.

Sources

  • S1 E. H. Lieb, R. Seiringer, J. P. Solovej, J. Yngvason, The Mathematics of the Bose Gas, Birkhäuser, 2005. https://doi.org/10.1007/3-7643-7338-9. Formulation source.
  • S2 Bose-Einstein condensation of interacting bosons: A two-step proof. https://arxiv.org/abs/2305.18959. Further reference; not a proof endorsement.
  • S3 The free energy of the interacting Bose gas: A variational description with loops and interlacements. https://www.wias-berlin.de/people/koenig/www/CondensateInterlace.pdf. Further reference; not a proof endorsement.
  • E1 Elliott H. Lieb, Robert Seiringer, Jan Philip Solovej, and Jakob Yngvason, The Mathematics of the Bose Gas and its Condensation, author version. https://arxiv.org/abs/cond-mat/0610117. Author-provided version of [S1]; consulted 27 September 2026 for the distinction between limits and models. The finite-volume energy estimate and conditional occupation bound are derived in the attempt.

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