Annals of Probability

Large deviations for trapped interacting Brownian particles and paths

Stefan Adams, Jean-Bernard Bru, and Wolfgang König

Full-text: Open access


We introduce two probabilistic models for N interacting Brownian motions moving in a trap in ℝd under mutually repellent forces. The two models are defined in terms of transformed path measures on finite time intervals under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The first pair interaction exhibits a particle repellency, while the second one imposes a path repellency.

We analyze both models in the limit of diverging time with fixed number N of Brownian motions. In particular, we prove large deviations principles for the normalized occupation measures. The minimizers of the rate functions are related to a certain associated operator, the Hamilton operator for a system of N interacting trapped particles. More precisely, in the particle-repellency model, the minimizer is its ground state, and in the path-repellency model, the minimizers are its ground product-states. In the case of path-repellency, we also discuss the case of a Dirac-type interaction, which is rigorously defined in terms of Brownian intersection local times. We prove a large-deviation result for a discrete variant of the model.

This study is a contribution to the search for a mathematical formulation of the quantum system of N trapped interacting bosons as a model for Bose–Einstein condensation, motivated by the success of the famous 1995 experiments. Recently, Lieb et al. described the large-N behavior of the ground state in terms of the well-known Gross–Pitaevskii formula, involving the scattering length of the pair potential. We prove that the large-N behavior of the ground product-states is also described by the Gross–Pitaevskii formula, however, with the scattering length of the pair potential replaced by its integral.

Article information

Ann. Probab., Volume 34, Number 4 (2006), 1370-1422.

First available in Project Euclid: 19 September 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations 60J65: Brownian motion [See also 58J65] 82B10: Quantum equilibrium statistical mechanics (general) 82B26: Phase transitions (general)

Large deviations interacting Brownian motions occupation measure energy functionals Gross–Pitaevskii functional


Adams, Stefan; Bru, Jean-Bernard; König, Wolfgang. Large deviations for trapped interacting Brownian particles and paths. Ann. Probab. 34 (2006), no. 4, 1370--1422. doi:10.1214/009117906000000214.

Export citation


  • Adams, S., Bru, J.-B. and König, W. (2006). Large systems of path-repellent Brownian motions in a trap at positive temperature. Electron. J. Probab. 11 460–485.
  • Adams, S. and König, W. (2006). Large symmetrised systems of Brownian bridges. Preprint arXiv:math.PR/0603702.
  • Aizenman, M. and Simon, B. (1982). Brownian motion and Harnack's inequality for Schrödinger operators. Comm. Pure Appl. Math. 35 209–271.
  • Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E. and Cornell, E. A. (1995). Observation of Bose–Einstein condensation in a dilute atomic vapor. Science 269 198–201.
  • Bolthausen, E., Deuschel, J.-D. and Schmock, U. (1993). Convergence of path measures arising from a mean field or polaron type interaction. Probab. Theory Related Fields 95 283–310.
  • Bradley, C. C., Sackett, C. A., Tollet, J. J. and Hulet, R. G. (1995). Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions. Phys. Rev. Lett. 75 1687–1690.
  • Bratteli, O. and Robinson, D. W. (1997). Operator Algebras and Quantum Statistical Mechanics II, 2nd ed. Springer, Berlin.
  • Chung, K. L. and Zhao, Z. (1995). From Brownian Motion to Schrödinger's Equation. Springer, Berlin.
  • Dalfovo, F., Giorgini, S., Pitaevskii, L. P. and Stringari, S. (1999). Theory of Bose–Einstein condensation in trapped gases. Rev. Mod. Phys. 71 463–512.
  • Davis, K. B., Mewes, M.-O., Andrews, M. R., van Druten, N. J., Durfee, D. S., Kurn, D. M. and Ketterle, W. (1995). Bose–Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75 3969–3973.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • Dickhoff, W. H. and Van Neck, D. (2005). Many-Body Theory Exposed. World Scientific, Singapore.
  • Donsker, M. D. and Varadhan, S. R. S. (1983). Asymptotics for the polaron. Comm. Pure Appl. Math. 36 505–528.
  • Fetter, A. L. and Walecka, J. D. (1971). Quantum Theory of Many Particle Systems. McGraw–Hill, New York.
  • Gantert, N., König, W. and Shi, Z. (2006). Annealed deviations for random walk in random scenery. Ann. Inst. H. Poincaré Probab. Statist. To appear.
  • Gärtner, J. (1977). On large deviations from the invariant measure. Theory Probab. Appl. 22 24–39.
  • Geman, D., Horowitz, J. and Rosen, J. (1984). A local time analysis of intersection of Brownian paths in the plane. Ann. Probab. 12 86–107.
  • Ginibre, J. (1971). Some applications of functional integration in statistical mechanics. In Statistical Mechanics and Quantum Field Theory (C. De Witt and R. Stora, eds.) 327–427. Gordon and Breach, New York.
  • Gross, E. P. (1961). Structure of a quantized vortex in boson systems. Nuovo Cimento (10) 20 454–477.
  • Gross, E. P. (1963). Hydrodynamics of a superfluid condensate. J. Math. Phys. 4 195–207.
  • Lieb, E. H. and Loss, M. (2001). Analysis, 2nd. ed. Amer. Math. Soc., Providence, RI.
  • Lieb, E. H., Seiringer, R. and Yngvason, J. (2000). Bosons in a trap: A rigorous derivation of the Gross–Pitaevskii energy functional. Phys. Rev. A 61 043602-1-13.
  • Lieb, E. H., Seiringer, R. and Yngvason, J. (2000). The ground state energy and density of interacting bosons in a trap. In Quantum Theory and Symmetries (Goslar, 1999) (H. D. Doebner, V. K. Dobrev, J. D. Hennig and W. Luecke, eds.) 101–110. World Sci., River Edge, NJ.
  • Lieb, E. H., Seiringer, R. and Yngvason, J. (2001). A rigorous derivation of the Gross–Pitaevskii energy functional for a two-dimensional Bose gas. Comm. Math. Phys. 224 17–31.
  • Lieb, E. H. and Seiringer, R. (2002). Proof of Bose–Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88 170409-1-4.
  • Lieb, E. H. and Yngvason, J. (2001). The ground state energy of a dilute two-dimensional Bose gas. J. Statist. Phys. 103 509.
  • Pitaevskii, L. P. (1961). Vortex lines in an imperfect Bose gas. Sov. Phys. JETP 13 451–454.
  • Pitaevskii, L. and Stringari, S. (2003). Bose–Einstein Condensation. Clarendon Press, Oxford.
  • Popov, V. N. (1983). Functional Integrals in Quantum Field Theory and Statistical Physics. Riedel, Dordrecht.
  • Ruelle, D. (1969). Statistical Mechanics: Rigorous Results. W. A. Benjamin, Inc., New York.