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Can We Still Get Bose-Einstein Condensates If the Noise Is Not Thermal?

Martin Bier

Published Jan 3, 2024• 1 citation
Publisher

Abstract

Bosons are particles that can occupy the same quantum state. Identical classical particles are considered distinguishable, but with bosons this is no longer the case. With bosons 1 and 2 in states α and β, the compound state function, ΨS=(1/2)Ψα(1)Ψβ(2)+Ψβ(1)Ψα(2), does not change when the particles are interchanged, i.e., there is one state where classically there would be two. This leads to boson statistics, which is different from Boltzmann statistics in that it favors putting more particles in the same state. For a sufficiently low temperature, it is possible that all bosons aggregate in the ground state where they effectively form a single megaparticle. We cover boson statistics and show how and why Bose-Einstein condensation can occur. We also show how growing networks like the WWW, the economy, or citation networks in scientific literature can follow boson statistics. Analyses of Bose-Einstein condensation have generally assumed a bath with a temperature, i.e., a thermal equilibrium where random collisions lead to a Gaussian-noise-term. However, many setups in physics involve conversion or transport of energy, i.e., nonequilibrium. Nonequilibrium noise is commonly characterized by the frequent occurrence of large kicks, i.e., outliers. We model this by implementing α-stable noise. No temperature exists in that case. We find that Bose-Einstein condensation is still theoretically possible, but probably much harder to engineer. Social and economic traffic is also often best modeled as nonequilibrium and our results may be significant in these contexts.

DOI
10.20944/preprints202401.0282.v1

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Bosons are particles that can occupy the same quantum state. Identical classical particles are considered distinguishable, but with bosons this is no longer the case. With bosons 1 and 2 in states α and β, the compound state function, ΨS=(1/2)Ψα(1)Ψβ(2)+Ψβ(1)Ψα(2), does not change when the particles are interchanged, i.e., there is one state where classically there would be two. This leads to boson statistics, which is different from Boltzmann statistics in that it favors putting more particles in the same state. For a sufficiently low temperature, it is possible that all bosons aggregate in the ground state where they effectively form a single megaparticle. We cover boson statistics and show how and why Bose-Einstein condensation can occur. We also show how growing networks like the WWW, the economy, or citation networks in scientific literature can follow boson statistics. Analyses of Bose-Einstein condensation have generally assumed a bath with a temperature, i.e., a thermal equilibrium where random collisions lead to a Gaussian-noise-term. However, many setups in physics involve conversion or transport of energy, i.e., nonequilibrium. Nonequilibrium noise is commonly characterized by the frequent occurrence of large kicks, i.e., outliers. We model this by implementing α-stable noise. No temperature exists in that case. We find that Bose-Einstein condensation is still theoretically possible, but probably much harder to engineer. Social and economic traffic is also often best modeled as nonequilibrium and our results may be significant in these contexts.
Abstract Bose–Einstein condensation occurs when bosons aggregate to effectively form a single megaparticle. Analyses of Bose–Einstein condensation have generally assumed a bath with a temperature, i.e., a thermal equilibrium where random collisions lead to a Gaussian-noise-term. However, many setups in physics involve conversion or transport of energy, i.e., nonequilibrium. Nonequilibrium noise is commonly characterized by the frequent occurrence of large kicks and, as such, can be effectively modeled by the implementation of α\alpha α α -stable noise, also called Lévy noise. No temperature exists in that case. We analyze the simple case of bosons in a double-well potential subjected to α\alpha α α -stable noise. A formula for the distribution over the two wells is derived. It is found that Bose–Einstein condensation can still occur, but is probably much harder to engineer. Our results could be significant for understanding the obviously nonequilibrium quark-gluon plasmas that form after high-energy collisions of heavy nuclei.
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
This chapter will focus on the application of kinetic and hydrodynamic approaches to Bose–Einstein condensates. A Bose–Einstein condensate is a state of matter, formed of identical bosons (usually atoms behaving like bosons), so that at a sufficiently low temperature, many of the particles making up the system occupy the lowest energy state (ground state). This is only possible for bosons (particles with an integer spin) unlike fermions (particles with a half-integer spin) due to the Pauli exclusion principle.
Bose–Einstein condensation can be understood as a macroscopic occupation of a single-particle state. But an alternative definition, which requires only a macroscopic occupation of an arbitrarily small energy band of single-particle states, is also known, referred to as generalized Bose–Einstein condensation, and considered to be thermodynamically more stable. Depending on the number of macroscopically occupied single-particle states in the condensate, three types are then distin-guished. If the number of macroscopically occupied single-particle states is finite but at least one, then the condensation is said to be of type-I. A type-II condensate occurs if infinitely many single-particle states are macroscopically occupied. If a generalized Bose–Einstein condensation without any macroscopically occupied single-particle states is present, then one speaks of a type-III con-densation. In noninteracting Bose gases, it is generally easier to prove the occurrence of generalized Bose–Einstein condensation than to determine its type. While a main step for the former task is to show that the critical density is finite, quite detailed knowledge of the energy gaps of the cor-responding single-particle Hamiltonian at the lower edge of the spectrum is needed for the latter one. In this dissertation, we study the occurrence of a Bose–Einstein condensate in Bose gases that are placed in metrically transitive random potentials. We mainly consider the Poisson random potential. The thermodynamic limit and Dirichlet boundary conditions are used throughout this work. We explore noninteracting Bose gases in arbitrary dimension at positive temperatures and a one-dimensional interacting Bose gas at zero temperature. We provide fairly general sufficient conditions for the random potential that imply the almost sure occurrence of generalized Bose–Einstein condensation in noninteracting Bose gases. These conditions are fulfilled by a Poisson random potential, depending on its single-impurity potential. They are also met by the one-dimensional Luttinger–Sy model, which is a central system in the research area of Bose gases in random potentials, and a generalization thereof. Moreover, we state sufficient energy gap conditions such that one can conclude a type-I condensation in Poisson random potentials. We consider different probabilistic notions here, such as convergence in the rth mean, r P N, in probability, and with probability almost one. In particular, we prove that in the Luttinger–Sy model with infinitely strong random potential a type-I condensation occurs in the rth mean, r P N, if the particle density is greater than the critical one. For the Luttinger–Sy model with a finitely strong random potential, we show the same statement, but with probability almost one. Finally, we discuss the Luttinger–Sy model with a repulsive contact interaction between the particles at zero temperature. We provide sufficient requirements such that a generalized Bose–Einstein condensate almost surely occurs. We also show that the type of the condensate is altered by the interparticle interaction. In particular, we demonstrate a transition from a type-I to a type-III condensate when the strength of the contact interaction is increased. Furthermore, we prove that the interparticle interaction reduces the local particle density.
Bose–Einstein condensation (BEC) is discussed at the level of an advanced course of statistical thermodynamics, clarifying some formal and physical aspects that are usually not covered by the standard pedagogical literature. The non-conventional approach adopted starts by showing that the continuum limit, in certain cases, cancels out the crucial role of the bosonic ground level. If so, a correct treatment of the problem, including the ground level population N 0 by construction, leads to BEC in a straightforward way. For a density of states of the form G ( )∝ γ , the chemical potential µ is explicitly calculated as a function of the temperature T and of the number N of bosons, for various significant values of the positive exponent γ. In the thermodynamic limit, in which the boson number N diverges and BEC is a sharp process, the chemical potential µ is a singular function of T at the critical temperature T B , determined by an appropriate critical exponent. The condensate population N 0 is studied analytically and numerically as a function of the temperature, for various values of N and for different γ. This provides an accurate description of the way BEC approaches the character of a sharp phase transition. Some aspects of the real experiments on BEC, involving a finite number of bosons, are also illustrated.