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The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics

Gas in external fields: the weird case of the logarithmic trap

L. Ferrari

Published Dec 2, 2023• 1 citation
Publisher

Abstract

The effects of an attractive logarithmic potential u(r)=u0ln⁡(r/r0) (u0>0 being the trap strength and r0 an arbitrary length scale) on a gas of N non interacting particles (Bosons or Fermions), in a box of volume VD, are studied in D = 2, 3 dimensions. The unconventional behaviour of the gas challenges the current notions of thermodynamic limit and size independence. When VD and N diverge, with finite density N/VD<∞ and finite u0, the gas collapses in the ground state, independently from the bosonic/fermionic nature of the particles, at any temperature. If, instead, N/VD→0, there exists a critical temperature Tc, such that the gas remains in the ground state at any T<Tc, and ‘evaporates’ above, in a non-equilibrium state of borderless diffusion. For the gas to exhibit a conventional Bose–Einstein condensation (BEC) or a finite Fermi level, the strength u0 must vanish with VD→∞, according to a complicated exponential relationship, as a consequence of the exponentially increasing density of states, specific of the logarithmic trap.

DOI
10.1080/14786435.2023.2285368

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The effects of an attractive logarithmic potential u(r)=u0ln⁡(r/r0) (u0>0 being the trap strength and r0 an arbitrary length scale) on a gas of N non interacting particles (Bosons or Fermions), in a box of volume VD, are studied in D = 2, 3 dimensions. The unconventional behaviour of the gas challenges the current notions of thermodynamic limit and size independence. When VD and N diverge, with finite density N/VD<∞ and finite u0, the gas collapses in the ground state, independently from the bosonic/fermionic nature of the particles, at any temperature. If, instead, N/VD→0, there exists a critical temperature Tc, such that the gas remains in the ground state at any T<Tc, and ‘evaporates’ above, in a non-equilibrium state of borderless diffusion. For the gas to exhibit a conventional Bose–Einstein condensation (BEC) or a finite Fermi level, the strength u0 must vanish with VD→∞, according to a complicated exponential relationship, as a consequence of the exponentially increasing density of states, specific of the logarithmic trap.
The effects of an attractive logarithmic potential u0ln⁡(r/r0)u_0\ln(r/r_0)u0​ln(r/r0​) on a gas of NNN non interacting particles (Bosons or Fermions), in a box of volume VDV_DVD​, are studied in D=2,3D=2,3D=2,3 dimensions. The unconventional behavior of the gas challenges the current notions of thermodynamic limit and size independence. When VDV_DVD​ and NNN diverge, with finite density N/VD<∞N/V_D<\inftyN/VD​<∞ and finite trap strength u0>0u_0>0u0​>0, the gas collapses in the ground state, independently from the bosonic/fermionic nature of the particles, at \emph{any} temperature. If, instead, N/VD→0N/V_D\rightarrow0N/VD​→0, there exists a critical temperature TcT_cTc​, such that the gas remains in the ground state at any $T
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2L^2L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01\textit{Full-text embargoed until: 2026-01-01}Full-text embargoed until: 2026-01-01
We analytically calculate the statistics of Bose-Einstein condensate (BEC) fluctuations in an interacting gas trapped in a three-dimensional cubic or rectangular box with the Dirichlet, fused or periodic boundary conditions within the mean-field Bogoliubov and Thomas-Fermi approximations. We study a mesoscopic system of a finite number of trapped particles and its thermodynamic limit. We find that the BEC fluctuations, first, are anomalously large and non-Gaussian and, second, depend on the trap's form and boundary conditions. Remarkably, these effects persist with increasing interparticle interaction and even in the thermodynamic limit-only the mean BEC occupation, not BEC fluctuations, becomes independent on the trap's form and boundary conditions.
For an ideal Bose gas whose density of states grows like εᵃ⁻¹, every equilibrium quantity at the condensation point is a Mellin moment Γ(a)ζ(a) or Γ(a+1)ζ(a+1). The exponent is a = d/2 + d/δ for a gas in d dimensions in a potential r^δ: a = 3/2 in a box, a = 3 in a harmonic trap. We make the arithmetic content of this classical rule explicit. The density of states chooses the zeta argument, and the only pole of ζ, at s = 1, decides three things. Condensation occurs if and only if a > 1, because Liₐ(1) = ζ(a) is finite. The heat capacity jumps at T_c if and only if a > 2, by a²ζ(a)/ζ(a−1), because ζ(a−1) is finite; the value below T_c is a(a+1)ζ(a+1)/ζ(a). In the borderline case a = 1, the one-dimensional harmonic trap, the pole turns into a logarithm: the excited-state sum is the Lambert series Σ d(n)e^(−n/t) of the divisor function with Mellin transform Γ(s)ζ(s)², and its double pole gives t(log t + γ) + 1/4 − 1/(144t) + …, with 1/4 = ζ(0)² and 1/144 = ζ(−1)². The finite-size shift −ζ(2)/(2ζ(3)^(2/3)) · N^(−1/3) of a three-dimensional trap is the next residue of the same kind of transform. Fermions never condense, because η(s) has no pole. The Debye law belongs to the same scheme with an even argument: 12π⁴/5 = 216 ζ(4), and its T³ form is exact at low temperature with the directional average of Σᵢ cᵢ⁻³ over the three sound branches. For copper at 0 K the elastic constants give Θ_D = 344.4 K against the calorimetric 343 K, an independent test of the prefactor at the level of one per cent. The value ζ(3/2) of the box has no known closed form. We also record that the first condensates of 1995 were made in harmonic traps and so test ζ(3), not ζ(3/2). Accompanying code repository contents (SHA256SUMS): a14986048687f94245c87d98670266179a8671cbeeae4af368a675388f778ede README.mdd685a7bcfe008d42d08d4f1546809ddfba3a5bcd4244a92b905b8f7e0ca76c6f p4_checks.py67eca726ed7311cedaf292189e4c55a6cc1c8ca54763ed2e91359617050a4660 p4_checks.log1be23f68b50ea329f2e74db39924ccc60902b3c2db149b921af8d2995fd304ca run_all_checks.py63de8f343011fd3c5e921f67371a96630a5b4ccfaf4194877fd10d963276da61 run_all_checks.log
We study condensation of trapped bosons in the limit when the number of particles tends to infinity. For the noninteracting gas we prove that there is no phase transition in any dimension, but in any dimension at any temperature the system is 100% condensated into the one-particle ground state. In the case of an interacting gas we show that for a family of suitably scaled pair interactions, the Gross-Pitaevskii scaling included, a less-than-100% condensation into a one-particle eigenstate, which may depend on the interaction strength, persists at all temperatures.
Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number NNN of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation temperature Tc{T}_{c}Tc​ in terms of a power series in ${x}_{0}={\ensuremath{\varepsilon}}_{0}/{k}_{B}{T}_{c}$, where ${\ensuremath{\varepsilon}}_{0}$ denotes the zero-point energy of the trapping potential. This expression, which applies in Cartesian, cylindrical, and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of x0{x}_{0}x0​ up to the second order. We show that, for a harmonic trap, the well-known first-order shift of the critical temperature $\ensuremath{\Delta}{T}_{c}/{T}_{c}\ensuremath{\propto}{N}^{\ensuremath{-}1/3}$ is inaccurate when $N\ensuremath{\leqslant}{10}^{5}$, the next order (proportional to ${N}^{\ensuremath{-}1/2}$) being significant. We also show that finite-size effects on the condensation temperature cancel out in a cubic trapping potential, e.g., $V(r)\ensuremath{\propto}{r}^{3}$. Finally, we show that in a generic power-law potential of higher order, e.g., $V(r)\ensuremath{\propto}{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>3$, the shift of the critical temperature becomes positive. This effect provides a large increase of Tc{T}_{c}Tc​ for relatively small atom numbers. For instance, an increase of about $+$40% is expected with 104{10}^{4}104 atoms in a $V(r)\ensuremath{\propto}{r}^{12}$ trapping potential.