arXiv:2112.01999 · Journal of Statistical Physics

Large Deviation Estimates for Weakly Interacting Bosons

Simone Rademacher, Robert Seiringer

Published Jun 2, 2022• 8 citations

Abstract

Abstract We study the many-body dynamics of an initially factorized bosonic wave function in the mean-field regime. We prove large deviation estimates for the fluctuations around the condensate. We derive an upper bound extending a recent result to more general interactions. Furthermore, we derive a new lower bound which agrees with the upper bound in leading order.

DOI: 10.1007/s10955-022-02940-4

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arXiv:2112.01999 · Journal of Statistical Physics

Large Deviation Estimates for Weakly Interacting Bosons

Simone Rademacher, Robert Seiringer

Published Jun 2, 2022• 8 citations

arXiv HTML PDF

Abstract

DOI: 10.1007/s10955-022-02940-4

Related papers

Large Deviations for the Ground State of Weakly Interacting Bose Gases

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Abstract We consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose–Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas’ ground state correspond to dependent random variables due to the bosons’ correlation. We prove that in the limit $N \rightarrow \infty$ N → ∞ bounded one-particle operators with law given by the ground state satisfy large deviation estimates. We derive a lower and an upper bound on the rate function that match up to second order and that are characterized by quantum fluctuations around the condensate.Abstract We consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose–Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas’ ground state correspond to dependent random variables due to the bosons’ correlation. We prove that in the limit $$N \rightarrow \infty $$ N → ∞ bounded one-particle operators with law given by the ground state satisfy large deviation estimates. We derive a lower and an upper bound on the rate function that match up to second order and that are characterized by quantum fluctuations around the condensate.

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Rigorous Estimates for Correlation Functions and Existence of Phase Transitions in Some Models of Interacting Bosons

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Using the so-called method of infrared bounds and a Roepstorff's inequality we obtain a lower bound for the amount of condensate and derive an upper bound for the anomalous average [Formula: see text] of Huang–Davies (HD) model. A lower bound for the static structure factor is also obtained. We generalize the HD model and prove the existence of Bose condensation for this kind of model systems. Finally we derived upper and lower bounds for correlation functions associated to a system of Fermi-particles interacting with a field of phonons.Using the so-called method of infrared bounds and a Roepstorff's inequality we obtain a lower bound for the amount of condensate and derive an upper bound for the anomalous average [Formula: see text] of Huang–Davies (HD) model. A lower bound for the static structure factor is also obtained. We generalize the HD model and prove the existence of Bose condensation for this kind of model systems. Finally we derived upper and lower bounds for correlation functions associated to a system of Fermi-particles interacting with a field of phonons.

International Journal of Modern Physics BNov 10, 199710 citations

N \rightarrow \infty

N → ∞ bounded one-particle operators with law given by the ground state satisfy large deviation estimates. We derive a lower and an upper bound on the rate function that match up to second order and that are characterized by quantum fluctuations around the condensate.

ρ>0

from the initial BEC, the mean-field approximation error at time

t\leq ρ/v

is bounded as

ρ^{-n}

, for arbitrarily large

n\geq 1

. This is a consequence of new ballistic propagation bounds on the fluctuations around the condensate. To prove this, we develop a variant of the ASTLO (adiabatic spacetime localization observable) method for the particle non-conserving generator of the fluctuation dynamics around Hartree states.

We study the mean-field dynamics of a system of

N

interacting bosons starting from an initially condensated state. For a broad class of mean-field Hamiltonians, including models with arbitrary bounded interactions and models with unbounded interaction potentials, we prove that the probability of having

n

particles outside the condensate decays exponentially in

n

for any finite evolution time. Our results strengthen previously known bounds that provide only polynomial control on the probability of having

n

excitations.

d

d -dimensional bosons for large N. The bosons initially form a Bose–Einstein condensate and interact with each other via a pair potential of the form

(N-1)^{-1}N^{d\beta }v(N^\beta \cdot )

(N-1)-1Ndβv(Nβ·) for

\beta \in [0,\frac{1}{4d})

β∈[0,14d) . We derive a sequence of N-body functions which approximate the true many-body dynamics in

L^2({\mathbb {R}}^{dN})

L2(RdN) -norm to arbitrary precision in powers of

N^{-1}

N-1 . The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.