Positive-temperature Bose–Einstein condensation and exact quantum depletion

2 · Going further

The main problem concerns the exact canonical Gibbs state of a three-dimensional hard-sphere Bose gas. For each fixed exclusion distance $a$ and every sufficiently small fixed density $\rho$, the main result establishes a temperature $T>0$, independent of volume, at which the thermodynamic-limit condensate fraction is positive. The occupied condensate orbital is constant in space. The fixed-density condition is essential: dilution is not being used to change the density while taking this large-system limit.

At zero temperature, Manuscript 2 shows in the manuscript the full scaled Bogoliubov momentum distribution of depleted particles for hard spheres, uniformly over pure and mixed ground states. The thermodynamic limit is taken first at each fixed density, followed by the dilute limit. All thermodynamic accumulation values have the same dilute asymptotic, and the limiting scaled nonzero-momentum occupation measure has total mass $8/(3\sqrt{\pi })$. Consequently, the depletion fraction is $\frac{8}{3\sqrt{\pi }}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})$.

Manuscript 3 shows in the manuscript the same leading depletion law for fixed bounded, nonnegative, radial, finite-range interactions with positive scattering length $a$. Its assertion is uniform over ground-state density matrices and does not require a unique thermodynamic limit of the occupation. Manuscript 4 supplies a complementary density-uniform lower condensate bound for fixed bounded measurable, nonnegative, radial, finite-range interactions that are not zero almost everywhere, for sufficiently small densities and temperatures $0\le T\le {\rho }^2$. Manuscript 5 shows in the manuscript ground-state condensation for hard spheres, including arbitrary complex ground states, with a positive fraction independent of the sufficiently small gas parameter.

The abstracts organize the family around positive lower condensate bounds and precise dilute asymptotics for particles outside the constant orbital. They do not describe the argument's technical mechanism, so no proof strategy can be reconstructed from the supplied text. The main result has a Lean formalization; the supplied material does not identify its detailed scope or establish formal verification of the companion results.

Useful notions

• Quantum states and occupation numbers
• Equilibrium states and ground states
• Density and interaction range
• Thermodynamic and dilute limits
• Leading asymptotic formulas

What was known before / what is new

Limits and assumptions

All results concern three dimensions and the stated dilute regimes. The hard-sphere positive-temperature assertion gives some strictly positive temperature for each fixed exclusion distance and sufficiently small fixed density; the abstract does not give a full temperature range or a critical temperature. The bounded-potential results require their respective boundedness, nonnegativity, radial symmetry, finite-range, and nontriviality or positive-scattering-length assumptions. The depletion formulas are leading dilute asymptotics, not exact formulas at arbitrary density, and the order of limits is thermodynamic first, dilute second. The main result has a Lean formalization, but its detailed scope is not supplied, and formalization of the companions is not specified. The source lists no explicit remaining open problems; in particular, the accumulation-value formulations should not be read as assertions of a unique thermodynamic occupation limit.

Mini glossary

Condensate fraction

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The fraction of particles occupying the condensate orbital, which is spatially constant in these results.

Quantum depletion

—

The fraction of particles outside the condensate orbital at zero temperature.

Canonical Gibbs state

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The equilibrium quantum state at a specified temperature and fixed particle number.

Thermodynamic limit

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The large-system limit in which particle number and volume grow while density remains fixed.

Dilute limit

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The limit of decreasing density with the interaction fixed, taken after the thermodynamic limit in the depletion results.

Thermodynamic accumulation value

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A value approached along a sequence of growing systems, without requiring every such sequence to have the same limit.

Keywords : Bose–Einstein condensation · Quantum depletion · Dilute Bose gases

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family