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Dimension-restricted fixed-mean-density Bose condensation

Physical scope

A sequence of three-dimensional or dimension-d ideal, homogeneous, quadratic-dispersion grand states on periodic boxes whose chemical potentials are chosen to hold the mean particle density fixed. It proves the critical density, dimension restriction, and grand-ensemble zero-mode nonconcentration. It does not assert canonical-ensemble equivalence, trapping, interactions, superfluidity, a finite-volume phase transition, or empirical condensation.

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and adopt the homogeneous periodic Bose model of Finite spectral periodic Bose/Fermi gas with one internal mode ( $g = 1$ ), dimension $d \geq 1$ , $α = ℏ^{2} / (2 m) > 0$ , fixed $β > 0$ in J $^{- 1}$ and target mean number density $ρ > 0$ in m $^{- d}$ . Let $T = (k_{B} β)^{- 1}$ in K, with $k_{B} > 0$ in J K $^{- 1}$ . For each side length $L > 0$ in m, put $V : = L^{d}$ in m $^{d}$ and choose the unique $μ_{L} < 0$ solving $L^{- d} \sum_{k \in Z^{d}} (e^{β (α ∣ 2 π k / L ∣^{2} - μ_{L})} - 1)^{- 1} = ρ .$ Let $Ω_{B} : = {n : Z^{d} \to N_{0} : {k : n (k) \neq 0} is finite}$ be the Bose occupation configuration set and define the grand probability law $P_{L} (n) : = Ξ_{L}^{- 1} \exp (- β \sum_{k \in Z^{d}} (α ∣ 2 π k / L ∣^{2} - μ_{L}) n (k)), Ξ_{L} : = \sum_{n \in Ω_{B}} \exp (- β \sum_{k \in Z^{d}} (α ∣ 2 π k / L ∣^{2} - μ_{L}) n (k)),$ which is finite and positive by Explicit diagonal Fock domains and occupation trace under ${A C}_{ω}$ and its stated mode hypotheses. The zero-mode occupation random variable is $N_{0} (n) : = n (0)$ . For a $P_{L}$ -integrable real random variable $X$ write $E_{L} X : = \sum_{n} P_{L} (n) X (n)$ ; when its second moment is finite write ${Var}_{L} (X) : = E_{L} [(X - E_{L} X)^{2}]$ . Say this model has mean zero-mode condensation when $\lim_{L \to \infty} E_{L} N_{0} / V > 0$ ; this limit, when it exists, is the mean condensate density. Define the mean condensed fraction by $f_{0} : = \lim_{L \to \infty} E_{L} N_{0} / (ρ V)$ when the limit exists. Define $ζ (s) : = \sum_{ℓ = 1}^{\infty} ℓ^{- s}$ for $s > 1$ and $(x)_{+} : = \max (x, 0)$ . The critical density $ρ_{c} = (2 π)^{- d} \int_{R^{d}} (e^{β α ∣ p ∣^{2}} - 1)^{- 1} d p$ is an extended nonnegative number, with $ρ_{c} = (4 π β α)^{- d / 2} \sum_{ℓ = 1}^{\infty} ℓ^{- d / 2}$ ; this series diverges and $ρ_{c} = + \infty$ for $d \leq 2$ , while for $d > 2$ it equals $(4 π β α)^{- d / 2} ζ (d / 2)$ . For $d \leq 2$ , $E_{L} N_{0} / V \to 0$ for every finite $ρ$ . For $d > 2$ , $E_{L} N_{0} / V \to (ρ - ρ_{c})_{+}$ , so mean zero-mode condensation occurs exactly when $ρ > ρ_{c}$ . When $d = 3$ , $ρ_{c} = λ_{T}^{- 3} ζ (3 / 2)$ , $λ_{T} = (2 π ℏ^{2} β / m)^{1 / 2}$ in m, and for $ρ > ρ_{c}$ the mean condensed fraction is $f_{0} = 1 - ρ_{c} / ρ$ . At fixed $ρ$ , this holds for $T < T_{c} = [2 π ℏ^{2} / (m k_{B})] (ρ / ζ (3 / 2))^{2 / 3}$ and $f_{0} = 1 - (T / T_{c})^{3 / 2}$ .

If $d > 2$ and $ρ_{0} = ρ - ρ_{c} > 0$ , for every $x > 0$ in m $^{- d}$ the grand-state ground occupation satisfies ${Var}_{L} (N_{0} / V) \to ρ_{0}^{2}, P_{L} (N_{0} / V \geq x) \to e^{- x / ρ_{0}} (x > 0) .$ For a real random-variable family $X_{L} : Ω_{B} \to R$ with laws $P_{L}$ , convergence in probability to a constant $a$ means $P_{L} (∣ X_{L} - a ∣ > ε) \to 0$ as $L \to \infty$ for every $ε > 0$ (in the same units as $X_{L}$ ). In particular $X_{L} = N_{0} / V$ does not converge in probability to its mean limit $ρ_{0}$ in this grand ensemble.

Facts & Assumptions

Given: ${A C}_{ω}$ ; $m > 0$ in kg, $ℏ > 0$ in J s, $k_{B} > 0$ in J K $^{- 1}$ , $T > 0$ in K, $L > 0$ in m; $α$ in J m $^{2}$ , $β = (k_{B} T)^{- 1}$ in J $^{- 1}$ , continuum wave vector $p$ in m $^{- 1}$ , $V = L^{d}$ in m $^{d}$ , $ρ$ in m $^{- d}$ , and the one-mode-per-momentum grand preparation.

[A1]

Quantum statistics and model equilibrium preparation: the bosonic grand Gibbs state is an adopted preparation when its trace is finite.

[F1]

Finite spectral periodic Bose/Fermi gas, Statement: for $μ < 0$ , mode means are geometric, the finite-volume trace is finite, and scaled occupation sums converge at fixed chemical potential.

[F2]

Lattice Gaussian Riemann limits and the singular Bose endpoint, Statement: the Bose endpoint integral is the extended value $(π / (β α))^{d / 2} \sum_{ℓ \geq 1} ℓ^{- d / 2}$ in every dimension, finite exactly for $d > 2$ ; for $μ < 0$ its continuum integrals increase to this endpoint value, and for $d > 2$ the missing-origin lattice sums converge to it.

Proof

1.1givenA1F1algebra

Under the adopted grand preparation [A1] and periodic mode trace [F1], the finite-volume density as a function of $μ < 0$ is continuous: on each compact subinterval of $(- \infty, 0)$ , the Gaussian trace bound gives a uniform summable majorant for its mode series. It is strictly increasing because every summand is. As $μ \to - \infty$ it tends zero by that same Gaussian bound; as $μ ↑ 0$ , the $k = 0$ term diverges. The intermediate value theorem therefore gives exactly one $μ_{L} < 0$ for each $ρ > 0$ .

1.2givenF1F2algebra

At fixed $μ < 0$ , the continuum density is $ρ (μ) = (2 π)^{- d} \int (e^{β (α ∣ p ∣^{2} - μ)} - 1)^{- 1} d p$ by [F1]. It is continuous and strictly increasing in $μ$ by a compact-parameter Gaussian bound and pointwise strict increase. It tends zero as $μ \to - \infty$ by $f_{B} \leq e^{β μ} e^{- β α ∣ p ∣^{2}} / (1 - e^{β μ})$ , and increases to $ρ_{c}$ as $μ ↑ 0$ by monotone convergence [F2]. Thus when $d \leq 2$ , $ρ (μ)$ runs continuously from zero to infinity; when $d > 2$ it runs from zero to $ρ_{c}$ .

2.1step 1.1step 1.2F1algebra

If $ρ < ρ_{c}$ (which covers every finite $ρ$ for $d \leq 2$ ), choose the unique $μ_{*} < 0$ with $ρ (μ_{*}) = ρ$ . For any $μ_{-} < μ_{*} < μ_{+} < 0$ , fixed- $μ$ convergence from [F1] implies that for all sufficiently large $L$ , the finite-volume densities at $μ_{-}$ and $μ_{+}$ bracket $ρ$ . Strict monotonicity then traps $μ_{L}$ between them. Shrinking the bracket proves $μ_{L} \to μ_{*}$ . The ground occupation is $(e^{- β μ_{L}} - 1)^{- 1}$ , which stays bounded as $μ_{L} \to μ_{*} < 0$ ; dividing by $L^{d}$ gives zero.

2.2step 1.1step 1.2F1F2algebra

Suppose $d > 2$ and $ρ \geq ρ_{c}$ . For any $δ > 0$ , if $μ_{L} \leq - δ$ along arbitrarily large $L$ , monotonicity bounds the finite-volume density by its value at $- δ$ ; fixed-chemical-potential convergence would then give $ρ \leq ρ (- δ) < ρ_{c}$ , contradiction. Thus $μ_{L} \to 0$ . Let $ρ_{e x, L}$ be the density excluding $k = 0$ . Since $μ_{L} < 0$ , each excited occupation is at most its value at zero chemical potential, and [F2] gives ${lim sup}_{L} ρ_{e x, L} \leq ρ_{c}$ . For any fixed $μ^{'} < 0$ , eventually $μ_{L} \geq μ^{'}$ , so monotonicity and [F1] give ${lim inf}_{L} ρ_{e x, L} \geq ρ (μ^{'})$ . Let $μ^{'} ↑ 0$ and use step 1.2 to get the matching lower bound $ρ_{c}$ . Hence $ρ_{e x, L} \to ρ_{c}$ and $E N_{0} / L^{d} = ρ - ρ_{e x, L} \to ρ - ρ_{c}$ . At equality this limit is zero.

3.1step 2.2F2algebra

In $d = 3$ , evaluate $ρ_{c}$ by the series in [F2]: $(4 π β α)^{- 3 / 2} \sum_{ℓ \geq 1} ℓ^{- 3 / 2}$ . Substituting $α = ℏ^{2} / (2 m)$ gives $λ_{T}^{- 3} ζ (3 / 2)$ . Solving $ρ = ρ_{c} (T_{c})$ and using $β = (k_{B} T)^{- 1}$ gives the stated $T_{c}$ ; for $T < T_{c}$ , division of $ρ - ρ_{c} (T)$ by $ρ$ gives $1 - (T / T_{c})^{3 / 2}$ .

4.1step 2.2F1algebra∎

For $ρ_{0} > 0$ , step 2.2 gives $E_{L} N_{0} / V \to ρ_{0}$ . Write ${\overset{ˉ}{n}}_{0, L} : = E_{L} N_{0}$ . The geometric ground-mode law from [F1] has ${Var}_{L} (N_{0}) = {\overset{ˉ}{n}}_{0, L} (1 + {\overset{ˉ}{n}}_{0, L})$ , so ${Var}_{L} (N_{0} / V) \to ρ_{0}^{2}$ . Its parameter is $q_{0, L} = {\overset{ˉ}{n}}_{0, L} / (1 + {\overset{ˉ}{n}}_{0, L})$ , and the geometric tail is $P_{L} (N_{0} / V \geq x) = q_{0, L}^{⌈ x V ⌉}$ . Since $V \log q_{0, L} = - V \log (1 + 1 / {\overset{ˉ}{n}}_{0, L}) \to - 1 / ρ_{0}$ , this tail tends $e^{- x / ρ_{0}}$ . Taking $x = 2 ρ_{0}$ gives $P_{L} (∣ N_{0} / V - ρ_{0} ∣ > ρ_{0} / 2) \geq P_{L} (N_{0} / V \geq 2 ρ_{0}) \to e^{- 2} > 0$ , which rules out convergence in probability to $ρ_{0}$ . This concerns only the ideal grand ensemble; the proof supplies no canonical equivalence or superfluidity claim.

Dimension-restricted fixed-mean-density Bose condensation

Statement

Facts & Assumptions

Proof

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