arXiv:2208.08931

Condensation of interacting bosons

András Sütö

Published Aug 18, 2022• 0 citations

Abstract

In this third paper of a series that started with arXiv:2106.10032 [math-ph] and continued with arXiv:2108.02659 [math-ph] we show that in $d\geq 3$ dimensions at low temperatures or high densities bosons interacting via pair potentials that are both positive and positive type form permutation cycles whose length diverges proportionally with the number of particles. Based on the second-cited paper, this implies Bose-Einstein condensation.

DOI: 10.48550/arxiv.2208.08931

Related papers

Simultaneous occurrence of off-diagonal long-range order and infinite permutation cycles in systems of interacting atoms

András Sütö

arXiv:2108.02659Aug 5, 20210 citations

Bose-Einstein condensation of interacting bosons: A two-step proof

András Sütö

arXiv:2305.18959May 30, 20231 citation

Infinite cycles of interacting bosons

András Sütö

arXiv:2409.19695Physica ScriptaOct 11, 20240 citations

Percolation transition in the Bose gas

András Sütö

Journal of Physics A Mathematical and GeneralSep 21, 199359 citations

A Theoretical Investigation of Low-Dimensional Bose-Einstein Condensations

Ching‐Fuh Lin

Chinese Journal of PhysicsApr 1, 20060 citations

Equilibrium and Non-Equilibrium Properties of Dilute Bose Gases

Oldenburg, Jakob Felix

Dec 9, 20250 citations

arXiv:2208.08931

Condensation of interacting bosons

András Sütö

Published Aug 18, 2022• 0 citations

arXiv HTML PDF

Abstract

In this third paper of a series that started with arXiv:2106.10032 [math-ph] and continued with arXiv:2108.02659 [math-ph] we show that in

d\geq 3

dimensions at low temperatures or high densities bosons interacting via pair potentials that are both positive and positive type form permutation cycles whose length diverges proportionally with the number of particles. Based on the second-cited paper, this implies Bose-Einstein condensation.

DOI: 10.48550/arxiv.2208.08931

Related papers

Simultaneous occurrence of off-diagonal long-range order and infinite permutation cycles in systems of interacting atoms

András Sütö

Based on the paper "Fourier formula for quantum partition functions", arXiv:2106.10032 [math-ph], we show that in an infinite system of identical bosons interacting via a positive-type pair potential there is off-diagonal long-range order if and only if a nonzero fraction of the particles form infinite permutation cycles. In particular, there is Bose-Einstein condensation if and only if the diverging cycle lengths increase at least as fast with $N$ , the number of particles, as $N^{2/d}$ in $d\geq 3$ dimensions. This extends a similar result known for the ideal Bose gas.Based on the paper "Fourier formula for quantum partition functions", arXiv:2106.10032 [math-ph], we show that in an infinite system of identical bosons interacting via a positive-type pair potential there is off-diagonal long-range order if and only if a nonzero fraction of the particles form infinite permutation cycles. In particular, there is Bose-Einstein condensation if and only if the diverging cycle lengths increase at least as fast with $N$, the number of particles, as $N^{2/d}$ in $d\geq 3$ dimensions. This extends a similar result known for the ideal Bose gas.

arXiv:2108.02659Aug 5, 20210 citations

Bose-Einstein condensation of interacting bosons: A two-step proof

András Sütö

We prove two equilibrium properties of a system of interacting atoms in three or higher dimensional continuous space. (i) If the particles interact via pair potentials of a nonnegative Fourier transform, their self-organization into infinite permutation cycles is simultaneous with off-diagonal long-range order. If the cycle lengths tend to infinity not slower than the square of the linear extension of the system, there is also Bose-Einstein condensation. (ii) If the pair potential is also nonnegative, cycles composed of a nonzero fraction of the total number of particles do appear if the density exceeds a temperature-dependent threshold value. The two together constitute the proof that in such a system Bose-Einstein condensation takes place at high enough densities.We prove two equilibrium properties of a system of interacting atoms in three or higher dimensional continuous space. (i) If the particles interact via pair potentials of a nonnegative Fourier transform, their self-organization into infinite permutation cycles is simultaneous with off-diagonal long-range order. If the cycle lengths tend to infinity not slower than the square of the linear extension of the system, there is also Bose-Einstein condensation. (ii) If the pair potential is also nonnegative, cycles composed of a nonzero fraction of the total number of particles do appear if the density exceeds a temperature-dependent threshold value. The two together constitute the proof that in such a system Bose-Einstein condensation takes place at high enough densities.

arXiv:2305.18959May 30, 20231 citation

Infinite cycles of interacting bosons

András Sütö

Abstract In the first-quantized description of bosonic systems permutation cycles formed by the particles play a fundamental role. In the ideal Bose gas Bose-Enstein condensation (BEC) is signaled by the appearance of infinite cycles. When the particles interact, the two phenomena may not be simultaneous, the existence of infinite cycles is necessary but not sufficient for BEC. We demonstrate that their appearance is always accompanied by a singularity in the thermodynamic quantities which in three and four dimensions can be as strong as a one-sided divergence of the isothermal compressibility. Arguments are presented that long-range interactions can give rise to unexpected results, such as the absence of infinite cycles in three dimensions for long-range repulsion or their presence in one and two dimensions if the pair potential has a long attractive tail.Abstract In the first-quantized description of bosonic systems permutation cycles formed by the particles play a fundamental role. In the ideal Bose gas Bose-Enstein condensation (BEC) is signaled by the appearance of infinite cycles. When the particles interact, the two phenomena may not be simultaneous, the existence of infinite cycles is necessary but not sufficient for BEC. We demonstrate that their appearance is always accompanied by a singularity in the thermodynamic quantities which in three and four dimensions can be as strong as a one-sided divergence of the isothermal compressibility. Arguments are presented that long-range interactions can give rise to unexpected results, such as the absence of infinite cycles in three dimensions for long-range repulsion or their presence in one and two dimensions if the pair potential has a long attractive tail.

arXiv:2409.19695Physica ScriptaOct 11, 20240 citations

Percolation transition in the Bose gas

András Sütö

The canonical partition function of a Bose gas gives rise to a probability distribution over the permutations of N particles. The author studies the probability and mean value of the cycle lengths in the cyclic permutations, their relation to physical quantities like pair correlations, and their thermodynamic limit. He shows that in the ground state of most interacting boson gases the mean cycle length diverges in the bulk limit and the particles form macroscopic cycles. In the free Bose gas Bose-Einstein condensation is accompanied by a percolation transition: the appearance of infinite cycle with nonvanishing probability.The canonical partition function of a Bose gas gives rise to a probability distribution over the permutations of N particles. The author studies the probability and mean value of the cycle lengths in the cyclic permutations, their relation to physical quantities like pair correlations, and their thermodynamic limit. He shows that in the ground state of most interacting boson gases the mean cycle length diverges in the bulk limit and the particles form macroscopic cycles. In the free Bose gas Bose-Einstein condensation is accompanied by a percolation transition: the appearance of infinite cycle with nonvanishing probability.

Journal of Physics A Mathematical and GeneralSep 21, 199359 citations

A Theoretical Investigation of Low-Dimensional Bose-Einstein Condensations

Ching‐Fuh Lin

Low-dimensional Bose-Einstein condensations (BEC) for g(e) (the density of states) = Cde(superscript d-1) with d ≤ 1, corresponding to a dimensionality of not larger than 2, are theoretically analyzed. When the energy quantization of the first excited state is taken into account, the boson number in the excited states is finite even for d ≤ 1 with the density of states remaining proportional toe(superscript d-1). The condensation temperature increases with the particle number faster for d ≤ 1 than for d ＞ 1. The condensation depends highly on the quantized energye1 of the first excited state. Whene1 just deviates slightly from zero, the condensation temperature quickly increases withe1. Theoretical estimation shows that a low-dimensional BEC can occur at a temperature above 1 K for excitons in semiconductor quantum wells even without the harmonic-oscillator potential.Low-dimensional Bose-Einstein condensations (BEC) for g(e) (the density of states) = Cde(superscript d-1) with d ≤ 1, corresponding to a dimensionality of not larger than 2, are theoretically analyzed. When the energy quantization of the first excited state is taken into account, the boson number in the excited states is finite even for d ≤ 1 with the density of states remaining proportional toe(superscript d-1). The condensation temperature increases with the particle number faster for d ≤ 1 than for d ＞ 1. The condensation depends highly on the quantized energye1 of the first excited state. Whene1 just deviates slightly from zero, the condensation temperature quickly increases withe1. Theoretical estimation shows that a low-dimensional BEC can occur at a temperature above 1 K for excitons in semiconductor quantum wells even without the harmonic-oscillator potential.

Chinese Journal of PhysicsApr 1, 20060 citations

Equilibrium and Non-Equilibrium Properties of Dilute Bose Gases

Oldenburg, Jakob Felix

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 $L^2$ -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. $\textit{Full-text embargoed until: 2026-01-01}$ 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 $L^2$-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. $\textit{Full-text embargoed until: 2026-01-01}$

Dec 9, 20250 citations

In this third paper of a series that started with arXiv:2106.10032 [math-ph] and continued with arXiv:2108.02659 [math-ph] we show that in

d\geq 3

N

, the number of particles, as

N^{2/d}

d\geq 3

dimensions. This extends a similar result known for the ideal Bose gas.

L^2

-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.

\textit{Full-text embargoed until: 2026-01-01}