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Fluctuations in a kinetic transport model for quantum friction

Roland Bauerschmidt, Wojciech de Roeck, Jürg Fröhlich

Abstract

We consider a linear Boltzmann equation that arises in a model for quantum friction. It describes a particle that is slowed down by the emission of bosons. We study the stochastic process generated by this Boltzmann equation and we show convergence of its spatial trajectory to a multiple of Brownian motion with exponential scaling. The asymptotic position of the particle is finite in mean, even though its absolute value is typically infinite. This is contrasted to an approximation that neglects the influence of fluctuations, where the mean asymptotic position is infinite.

Fluctuations in a kinetic transport model for quantum friction

Abstract

We consider a linear Boltzmann equation that arises in a model for quantum friction. It describes a particle that is slowed down by the emission of bosons. We study the stochastic process generated by this Boltzmann equation and we show convergence of its spatial trajectory to a multiple of Brownian motion with exponential scaling. The asymptotic position of the particle is finite in mean, even though its absolute value is typically infinite. This is contrasted to an approximation that neglects the influence of fluctuations, where the mean asymptotic position is infinite.

Paper Structure

This paper contains 13 sections, 7 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Jump process
  4. Mean field approximation vs. fluctuations
  5. Derivation from first principles
  6. Result
  7. Proof of Theorem \ref{['thm:convergence']}
  8. Stochastic decomposition
  9. Proof of Proposition \ref{['prop:lln']}
  10. Proof of Proposition \ref{['prop:clt']}
  11. Proof of Theorem \ref{['thm:position']}
  12. Martingale functional central limit theorem
  13. Proof of Proposition \ref{['prop:markovboltzmann']}

Key Result

Theorem 2.1

There are explicit positive constants $\theta<1$ and $\sigma$ such that, for an arbitrary choice of initial conditions $(X_0,K_0) = (x,k) \in \mathbb{R}^3 \times \mathbb{R}^3$, in distribution, in the topology of uniform convergence on bounded intervals of the variable $s$.

Figures (1)

  • Figure 1: The surfaces $k'\in ak+(1-a)|k|S^{2}$ for $a < {\frac{1}{2}}$ (left) and $a> {\frac{1}{2}}$ (right).

Theorems & Definitions (14)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof : Proof of Theorem \ref{['thm:convergence']}
  • proof : Proof of Proposition \ref{['prop:lln']}
  • Lemma 3.4
  • proof
  • proof : Proof of Proposition \ref{['prop:clt']}
  • ...and 4 more