A Large Deviation Principle in Many-Body Quantum Dynamics

Abstract

We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates that are consistent with central limit theorems that have been established in the last years.

Large deviation estimates for weakly interacting bosons

“…The quadratic term of (1.24) agrees with the findings from [17,Theorem 1.1]. In particular, the definition of f 0;t in (1.16) here is the same as in [17,Eq. (1.1)].…”

“…(1.1)]. 1 In contrast to [17], we prove here also a matching lower bound (ii') on the logarithmic moment generating function (resulting, together with the upper bound, in the lower bound on the rate function in Theorem 1.1(ii)). This allows to determine the rate function Λ * ψN,t up to quadratic order.…”

“…Under the same assumptions as in Theorem 1.1, (i') there exists a constant C 1 > 0 such that for all 0 ≤ λ ≤ e −e The upper bound (i') on the logarithmic moment generating function (as well as the resulting upper bound on the rate function in Theorem 1.1(i)) is an extension of the large deviation estimate obtained in [17] to more general interaction potentials. Note that the term cubic in λ in (1.24) depends on time through a double exponential, compared to a term exponential in time in [17], which is a consequence of allowing less regular interaction potentials here (entering the proof through Lemmas 3.1 and 3.2). The quadratic term of (1.24) agrees with the findings from [17,Theorem 1.1].…”

“…Note that the term cubic in λ in (1.24) depends on time through a double exponential, compared to a term exponential in time in [17], which is a consequence of allowing less regular interaction potentials here (entering the proof through Lemmas 3.1 and 3.2). The quadratic term of (1.24) agrees with the findings from [17,Theorem 1.1]. In particular, the definition of f 0;t in (1.16) here is the same as in [17,Eq.…”

A note on dependent random variables in quantum dynamics

“…Recently, for one-particle operators the probability distribution's tails were characterized through large deviation estimates [14,20], showing that…”