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arXiv:0705.4479 · Physical Review Letters

Nonequilibrium Quantum Impurities: From Entropy Production to Information Theory

Pankaj Mehta, Natan Andrei

Published Feb 26, 2008• 22 citations
arXivHTMLPDF

Abstract

Nonequilibrium steady-state currents, unlike their equilibrium counterparts, continuously dissipate energy into their physical surroundings leading to entropy production and time-reversal symmetry breaking. This Letter discusses these issues in the context of quantum impurity models. We use simple thermodynamic arguments to define the rate of entropy production sigma and show that sigma has a simple information-theoretic interpretation in terms of nonequilibrium distribution functions. This allows us to show that the entropy production is strictly positive for any nonequilibrium steady state. We conclude by applying these ideas to the resonance level model and the Kondo model.

DOI
10.1103/physrevlett.100.086804

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Nonequilibrium steady-state currents, unlike their equilibrium counterparts, continuously dissipate energy into their physical surroundings leading to entropy production and time-reversal symmetry breaking. This Letter discusses these issues in the context of quantum impurity models. We use simple thermodynamic arguments to define the rate of entropy production sigma and show that sigma has a simple information-theoretic interpretation in terms of nonequilibrium distribution functions. This allows us to show that the entropy production is strictly positive for any nonequilibrium steady state. We conclude by applying these ideas to the resonance level model and the Kondo model.
The microscopic features of bosonic quantum transport in a nonequilibrium steady state, which breaks time reversal invariance spontaneously, are investigated. The analysis is based on the probability distributions, generated by the correlation functions of the particle current and the entropy production operator. The general approach is applied to an exactly solvable model with a point-like interaction driving the system away from equilibrium. The quantum fluctuations of the particle current and the entropy production are explicitly evaluated in the zero frequency limit. It is shown that all moments of the entropy production distribution are non-negative, which provides a microscopic version of the second law of thermodynamics. On this basis a concept of efficiency, taking into account all quantum fluctuations, is proposed and analyzed. The role of the quantum statistics in this context is also discussed.
In this thesis we develop an approach to nonequilibrium quantum-statistical mechanics which is based on the consideration of the thermodynamic entropy as being a quantum observable with an associated hermitian operator and with its own equation of motion, from which the rate of entropy production can be studied. The relationship of this quantum observable---the expectation value of which is proved to obey the laws of thermodynamics and is thus called the thermodynamic entropy---with heat dissipation in quantum many-body systems is investigated in detail. After showing how the classical theory of nonequilibrium thermodynamics is obtained from our entropy-production formalism in the limit of very weakly coupled subsystems of a larger isolated quantum system, the solution of the equation of motion for the thermodynamic entropy operator is formally obtained and applied to the case of electrons interacting with phonons and being driven by an external electric field, arriving at an explicit expression for the Joule heat without any a priori consideration of the rate of change of the energy of the system. Finally, the formalism is applied to solve a puzzle introduced with the most basic model for atom-light interactions: the Jaynes-Cummings model. Without using the correct thermodynamic arguments implied by our entropy-production theory, this model leads to the conclusion that a stream of two-level atoms sent to a cavity filled with monochromatic photons---in the so called one-atom maser configuration---thermalize, under off-resonant conditions, to a temperature different from that of the photons, casting doubts on the validity of the principle of conservation of energy.
We consider the generic model of a finite-size quantum electron system connected to two (temperature and particle) reservoirs. The quantum open system is driven out of equilibrium by the presence of both potential temperature and chemical differences between the two reservoirs. The nonequilibrium (NE) thermodynamical properties of such a quantum open system are studied for the steady state regime. In such a regime, the corresponding NE density matrix is built on the so-called generalised Gibbs ensembles. From different expressions of the NE density matrix, we can identify the terms related to the entropy production in the system. We show, for a simple model, that the entropy production rate is always a positive quantity. Alternative expressions for the entropy production are also obtained from the Gibbs–von Neumann conventional formula and discussed in detail. Our results corroborate and expand earlier works found in the literature.
The thermodynamic entropy production for the scattering processes of noninteracting bosons and fermions in mesoscopic systems is shown to be related to the difference between the Connes–Narnhofer–Thirring entropy per unit time, characterizing temporal disorder in the motion of quantum particles, and the associated time-reversed coentropy per unit time. Under nonequilibrium conditions, the positivity of thermodynamic entropy production can thus be interpreted as a time-reversal symmetry breaking in the temporal disorder of the quantum transport process. Moreover, the full counting statistics of both fermionic and bosonic quantum transport is formulated in relation with the energy and particle currents producing thermodynamic entropy in nonequilibrium steady states.
We discuss fundamental features of the local expression for the entropy production rate of the nonequilibrium quantum dynamics of open systems and its relations to memory effects and the spectrum of the generator of the dynamics. Defining the entropy production rate as negative rate of change of the relative entropy with respect to an instantaneous fixed point, it is shown that positivity of the entropy production rate for all possible initial states implies that the real parts of the eigenvalues of the time-local generator for the quantum master equation are always negative. It is further demonstrated that Markovian dynamics, identified as P-divisibility of the quantum dynamical map, implies positivity of entropy production rate, thus providing a kind of generalized second law in the nonequilibrium regime. We also prove by means of the counterexample of a phase covariant quantum master equation that the converse of this statement is not true, i.e., there are non-Markovian dynamics for which the entropy production rate is always positive. Thus, we conclude that the emergence of negative entropy production rates is a sufficient but not necessary condition for non-Markovianity of the quantum dynamics. Finally, we also consider a recently introduced map-based notion of entropy production and show the equivalence between its positivity and Markovianity for general finite-dimensional systems.