Hostname: page-component-cb9f654ff-kl2l2 Total loading time: 0 Render date: 2025-08-20T00:22:13.185Z Has data issue: false hasContentIssue false
  • English
  • Français

Bose–Einstein condensation for particles with repulsive short-range pair interactions in a Poisson random external potential in $\mathbb{R}^{d}$

Part of: Applications to specific physical systems Equilibrium statistical mechanics Stochastic processes

Published online by Cambridge University Press:  28 October 2022

Joachim Kerner  and
Maximilian Pechmann
Joachim Kerner*
Affiliation:
FernUniversität in Hagen
Maximilian Pechmann*
Affiliation:
University of Tennessee
*
*Postal address: Fakultät für Mathematik und Informatik, FernUniversität in Hagen, 58084 Hagen, Germany. Email: joachim.kerner@fernuni-hagen.de
**Postal address: Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA. Email: mpechmann@utk.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We study Bose gases in $d \ge 2$ dimensions with short-range repulsive pair interactions at positive temperature, in the canonical ensemble and in the thermodynamic limit. We assume the presence of hard Poissonian obstacles and focus on the non-percolation regime. For sufficiently strong interparticle interactions, we show that almost surely there cannot be Bose–Einstein condensation into a sufficiently localized, normalized one-particle state. The results apply to the canonical eigenstates of the underlying one-particle Hamiltonian.

Keywords

Interacting Bose gasstrong repulsive pair interactionshard Poissonian obstaclesnon-percolation regime

MSC classification

Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Secondary: 60G55: Point processes 81V70: Many-body theory; quantum Hall effect

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 2 , June 2023 , pp. 382 - 393
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Adhikari, A., Brennecke, C. and Schlein, B. (2021). Bose–Einstein condensation beyond the Gross–Pitaevskii regime. Ann. Inst. H. Poincaré Prob. Statist. 22, 1163–1233.CrossRefGoogle ScholarPubMed
Aizenman, M. and Barsky, D. J. (1987). Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108, 489–526.CrossRefGoogle Scholar
Aonghusa, P. M. and Pulé, J. V. (1987). Hard cores destroy Bose–Einstein condensation. Lett. Math. Phys. 14, 117–121.CrossRefGoogle Scholar
Boccato, C., Brennecke, C., Cenatiempo, S. and Schlein, B. (2018). Complete Bose–Einstein condensation in the Gross–Pitaevskii regime. Commun. Math. Phys. 359, 975–1026.CrossRefGoogle Scholar
Boccato, C., Brennecke, C., Cenatiempo, S. and Schlein, B. (2020). Optimal rate for Bose–Einstein condensation in the Gross–Pitaevskii regime. Commun. Math. Phys. 376, 1311–1395.CrossRefGoogle Scholar
Bolte, J. and Kerner, J. (2016). Instability of Bose–Einstein condensation into the one-particle ground state on quantum graphs under repulsive perturbations. J. Math. Phys. 57, 043301.CrossRefGoogle Scholar
Bose, S. N. (1924). Plancks Gesetz und Lichtquantenhypothese. Z. Phys. 26, 178–181.CrossRefGoogle Scholar
de Smedt, P. (1986). The effect of repulsive interactions on Bose–Einstein condensation. J. Stat. Phys. 45, 201–213.CrossRefGoogle Scholar
Dimonte, D., Falconi, M. and Olgiati, A. (2020). On some rigorous aspects of fragmented condensation. Nonlinearity 34, 1–32.CrossRefGoogle Scholar
Einstein, A. (1924). Quantentheorie des einatomigen idealen Gases. In Sitzungsberichte der Preussischen Akademie Wissenschaften, pp. 261–267.Google Scholar
Einstein, A. (1925). Quantentheorie des einatomigen idealen Gases, Zweite Abhandlung. Sitzungsberichte der Preussischen Akademie Wissenschaften, pp. 3–14.Google Scholar
Fournais, S. (2021). Length scales for BEC in the dilute Bose gas. In Partial Differential Equations, Spectral Theory, and Mathematical Physics, eds P. Exner, H. Holden, R. L. Frank, T. Weidl and F. Gesztesy. EMS Press, Berlin, pp. 115–133.CrossRefGoogle Scholar
Gouéré, J. B. (2008). Subcritical regimes in the Poisson Boolean model of continuum percolation. Ann. Prob. 36, 1209–1220.CrossRefGoogle Scholar
Hammersley, J. M. (1957). Percolation processes: Lower bounds for the critical probability. Ann. Math. Stat. 28, 790–795.CrossRefGoogle Scholar
Kac, M. and Luttinger, J. M. (1973). Bose–Einstein condensation in the presence of impurities. J. Math. Phys. 14, 1626–1628.CrossRefGoogle Scholar
Kac, M. and Luttinger, J. M. (1974). Bose–Einstein condensation in the presence of impurities. II. J. Math. Phys. 15, 183–186.CrossRefGoogle Scholar
Kerner, J. and Pechmann, M. (2021). On the effect of repulsive pair interactions on Bose–Einstein condensation in the Luttinger–Sy model. Proc. Amer. Math. Soc. 149, 3499–3513.CrossRefGoogle Scholar
Kerner, J., Pechmann, M. and Spitzer, W. (2019). Bose–Einstein condensation in the Luttinger–Sy model with contact interaction. Ann. Inst. H. Poincaré Prob. Statist. 20, 2101–2134.CrossRefGoogle Scholar
Kesten, H. (2002). Some highlights of percolation. Preprint, arXiv:math/0212398.Google Scholar
Kingman, J. (1992). Poisson Processes, Vol. 3. Clarendon Press, Oxford.Google Scholar
Klein, A., Germinet, F. and Hislop, P. D. (2007). Localization for Schrödinger operators with Poisson random potential. J. Eur. Math. Soc. 9, 577–607.CrossRefGoogle Scholar
Last, G. and Penrose, M. (2018). Lectures on the Poisson Process, Vol. 7. Cambridge University Press.Google Scholar
Lauwers, J., Verbeure, A. and Zagrebnov, V. A. (2003). Proof of Bose–Einstein condensation for interacting gases with a one-particle gap. J. Phys. A 36, 169–174.CrossRefGoogle Scholar
Lenoble, O. and Zagrebnov, V. A. (2007). Bose–Einstein condensation in the Luttinger–Sy model. Markov Process. Relat. Fields 13, 441–468.Google Scholar
Lieb, E. H. and Seiringer, R. (2002). Proof of Bose–Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88, 170409.CrossRefGoogle ScholarPubMed
Lieb, E. H. and Seiringer, R. (2010). The Stability of Matter in Quantum Mechanics. Cambridge University Press.Google Scholar
Martin, P. A. and Rothen, F. (2004). Many-Body Problems and Quantum Field Theory. Springer, Berlin.CrossRefGoogle Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation, Vol. 119. Cambridge University Press.CrossRefGoogle Scholar
Menshikov, M. V. (1986). Coincidence of critical points in percolation problems. Soviet Math. Dokl. 33, 856–859.Google Scholar
Michelangeli, A. (2007). Reduced density matrices and Bose–Einstein condensation. Preprint, SISSA 39/2007/MP.Google Scholar
Ruelle, D. (1999). Statistical Mechanics: Rigorous Results. Imperial College Press and World Scientific Publishing.CrossRefGoogle Scholar
Sznitman, A.-S. (1998). Brownian Motion, Obstacles and Random Media. Springer, Berlin.CrossRefGoogle Scholar