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.