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Quantum diffusion with drift and the Einstein relation. I

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### Abstract

We study the dynamics of a quantum particle hopping on a simple cubic lattice and driven by a constant external force. It is coupled to an array of identical, independent thermal reservoirs consisting of free, massless Bose fields, one at each site of the lattice. When the particle visits a site x of the lattice it can emit or absorb field quanta of the reservoir at x. Under the assumption that the coupling between the particle and the reservoirs and the driving force are sufficiently small, we establish the following results: The ergodic average over time of the state of the particle approaches a non-equilibrium steady state describing a non-zero mean drift of the particle. Its motion around the mean drift is diffusive, and the diffusion constant and the drift velocity are related to one another by the Einstein relation.

© 2014 AIP Publishing LLC

Received 20 June 2013
Accepted 06 December 2013
Published online 25 June 2014

Acknowledgments:
We thank A. Kupiainen and A. Pizzo for many useful discussions on related problems. We also thank D. Egli, Z. Gang, and A. Knowles for helpful comments.

The stay of J.F. at the IAS was supported by the “Fund for Math” and the “Monell foundation.” The stay of K.S. at the IAS is supported by the “Fund for Math.”. Both authors thank the IAS and Thomas Spencer for their hospitality.

W.D.R. is grateful to the DFG for financial support.

Article outline:

I. INTRODUCTION
II. DEFINITION OF THE MODEL
A. Notations and conventions
1. Banach spaces
2. Scalar products
3. Kernels
B. The particle
C. The reservoir
1. Dynamics
2. Equilibrium state
D. The interaction
E. Effective dynamics
F. Time-reversal
III. ASSUMPTIONS AND RESULTS
A. Assumptions
B. Thermodynamic limit
1. Observables of the system
2. Dynamics
C. Results
D. Correlation functions and Einstein relation
IV. STRATEGY OF PROOFS AND DISCUSSION
A. Fiber decomposition
B. Strategy of proofs of main results
1. Kinetic theory
2. Boltzmann equation
3. Perturbation around the kinetic limit
4. Einstein relation
V. KINETIC LIMIT: LINEAR BOLTZMANN EVOLUTION
A. Concepts from the theory of *C* _{0}-semigroups
B. Spectral analysis of *M* ^{0,χ}: Preliminaries
C. Refined spectral analysis of *M* ^{χ} = *M* ^{0,χ}
D. Asymptotics of the semigroup
VI. RESULTS FROM EXPANSIONS
A. Survey of expansions
B. Calculation of
C. Second order contributions
D. Analysis of
VII. ANALYSIS OF AROUND *z* = 0
A. Perturbation around the kinetic limit
VIII. PROOF OF MAIN RESULTS
A. The equilibrium regime
B. Proof of Theorems 3.2 and 3.3

/content/aip/journal/jmp/55/7/10.1063/1.4881532

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http://dx.doi.org/10.1007/BF01453800
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2014-06-25

2014-12-18

### Abstract

We study the dynamics of a quantum particle hopping on a simple cubic lattice and driven by a constant external force. It is coupled to an array of identical, independent thermal reservoirs consisting of free, massless Bose fields, one at each site of the lattice. When the particle visits a site x of the lattice it can emit or absorb field quanta of the reservoir at x. Under the assumption that the coupling between the particle and the reservoirs and the driving force are sufficiently small, we establish the following results: The ergodic average over time of the state of the particle approaches a non-equilibrium steady state describing a non-zero mean drift of the particle. Its motion around the mean drift is diffusive, and the diffusion constant and the drift velocity are related to one another by the Einstein relation.

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