# Quantum Mechanical Harmonic Chain Attached to Heat Baths

## Abstract

We investigate a finite linear chain of *N* equal particles connected by equal harmonic springs whose left and right ends are in contact with independent stochastic heat baths at temperatures *T* _{1} and *T* _{ N }, respectively. The heat baths itself can be modelled as a system of coupled oscillators that introduce both fluctuations and dissipation in the chain [1]. In its classical version this model has been studied in [2]. Our starting point are the corresponding quantum Langevin equations [1] for the operators *x* _{ n }(*t*), *P* _{ n }(*t*) of the displacement of the n-th particle out of its equilibrium position and its conjugate momentum, respectively. Exploiting the linearity of the system we derive the equations of motion for the equal time correlation functions of the operators *x* _{ n }, *p* _{ n }, i.e. < *x* _{ n }(*t*)*x* _{ m }(*t*) >, < *x* _{ n }(*t*)*p* _{ m }(*t*) >, etc. For the stationary state we have explicitely determined the covariance matrix in the limit *N* → ∞, from which all statistical properties of the chain can be inferred. If both temperatures are equal, *T* _{1} = *T* _{ n }, these correlation functions are determined for vanishing damping constant by the standard weak coupling expression < o_{n}o_{m} > = *Z* ^{-1} *tro* _{ n } *o* _{ m }exp( — *H/k* _{ B } *T*), where o_{n} is an operator of the particle at site n, *H* is the Hamiltonian of the chain and *Z* denotes the partition function. Classically these expressions are exact for all damping constants and yield e.g. equipartitioning of the kinetic energy. In the quantum case finite coupling corrections to the weak coupling expressions are still small well inside the chain but grow towards the end of the chain, where < *p* _{ n } ^{2} > for *n* = 1,*N* diverges logarthmically with an upper cutoff on the frequencies of the heat baths. If the heat baths are at different temperatures, *T* _{1} ≠ *T* _{ N }, the chain is in a stationary nonequilibrium state, in which the quantum mechanical heat flux is decreased compared to the classical case. We shall discuss the equilibrium and nonequilibrium properties of the harmonic chain in greater detail elsewhere [3].

## Keywords

Heat Flux Kinetic Energy Correlation Function Covariance Matrix Stationary State## References

- [1]G. W. Ford, M. Kac, and P. Mazur, J.Math.Phys.6,504 (1965)MathSciNetADSMATHCrossRefGoogle Scholar
- [2]Z. Rieder, J L. Lebowitz, and E.H. Lieb, J.Math.Phys.8,1073 (1967)ADSCrossRefGoogle Scholar
- [3]U. Zürcher, P. Talkner, to be publishedGoogle Scholar