Exact stationary solution of the master equation for systems far from thermal equilibrium in detailed balance

If the master equation (Chapman-Kolmogorov equation) has a unique solution and fulfills the principle of detailed balance it may be solved explicitly by mere summations or, in the continuous case, by quadratures.     

Distribution function for classical and quantum systems far from thermal equilibrium

We consider a system composed of many subsystems which are coupled to individual reservoirs at different temperatures. We show how the solution of a many-dimensional Fokker-Planck equation may be reduced to a Fokker-Planck equation of dimensionn, wheren is the number of relevant constants of motion. We treat also a Fokker-Planck equation with continuously many variables and the time-dependent one. The usefulness of the present procedure to determine explicitly distribution functions is exhibited by several examples. If all temperatures are equal the Boltzman distribution function is obtained as a special case. Using the method of quantum-classical correspondence, the distribution function for quantum systems may be found.     

The Onsager relations for quantum systems weakly coupled to KMS reservoirs

We show that the Onsager relations for energy and matter flows through small quantum systems are generically direct consequences of the KMS (Kubo-Martin-Schwinger) property of the reservoir (Gibbs) equilibrium states in the (van Hove) weak coupling limit.     

Correlations in classical and quantum systems far from thermal equilibrium

We consider classical systems described by a Fokker-Planck equation or a generalized Fokker-Planck equation and quantum systems described by a density matrix equation or by a generalized Fokker-Planck equation using the principle of quantum classical correspondence. We split the corresponding operators of the equation of motion into a part which refers to the proper system and another one which describes the coupling of the proper system to the external world (reservoirs). We demonstrate that by use of conservation laws, referring to the proper systems, exact relations hold for certain moments, valid for all temperatures and coupling constants of the reservoirs. Using the concepts of a previous paper we describe then a perturbation theoretical approach which allows in a simple manner to determine a number of important correlation functions (moments of the total system). The time dependent case is briefly discussed. The applicability and usefulness of the present procedure is demonstrated by the example of the single-mode laser yielding e.g. expressions for the atom-field correlation.     

Distribution function for systems far from thermal equilibrium

We present the general stationary solution of the Fokker-Planck equation for a system of interacting subsystems weakly coupled to reservoirs at different temperatures, T_j. For T_j = T, it reduces to the Boltzman distribution function.     

Phase transitions far from thermal equilibrium in spin systems

The nonlinear effects in the spin resonance due to the negative differential r.f. magnetic susceptibility are considered. The r.f. magnetization is shown to exhibite a kind of first order phase transition. A set of scaling relations, valid near the appropriate critical point, is obtained.     

On the assumption of initial factorization in the master equation for weakly coupled systems I: General framework

We analyze the dynamics of a quantum mechanical system in interaction with a reservoir when the initial state is not factorized. In the weak-coupling (van Hove) limit, the dynamics can be properly described in terms of a master equation, but a consistent application of Nakajima-Zwanzig’s projection method requires that the reference (not necessarily equilibrium) state of the reservoir be endowed with the mixing property.     

On the assumption of initial factorization in the master equation for weakly coupled systems II: Solvable models

We analyze some solvable models of a quantum mechanical system in interaction with a reservoir when the initial state is not factorized. We apply Nakajima-Zwanzig’s projection method by choosing a reference state of the reservoir endowed with the mixing property. In van Hove’s limit, the dynamics is described in terms of a master equation. We observe that Markovianity becomes a valid approximation for timescales that depend both on the form factors of the interaction and on the observables of the reservoir that can be measured.     

Higher order effects in the master equation for coupled systems

The usual derivations of the master equation for coupled systems such as the laser make an assumption of weak coupling both for the coupling of the components to their heat baths and for the internal coupling between the components. It is this second condition that we wish to relax. In the usual derivation of the irreversible part of the master equation the approximation is made that the density operator for the coupled system factorizes into a product of the density operators for the two components. However when strong coupling is present, such as in high intensity lasers this approximation is no longer valid. To illustrate how the irreversible part of the master equation may be derived without making the factorization ansatz we consider the case of two coupled boson fields. Our derivation leads to additional terms in the usual master equation arising from correlations between the heat baths introduced by the coupling. This modified master equation yields the correct stationary solution for the density operator of the coupled system, whereas the usual master equation leads to a stationary solution for the density operator correct for the free components only.     

Reduced dynamics and the master equation of open quantum systems

An exact reduced density operator of a quantum system interacting with a bosonic thermal reservoir is derived by means of the simple algebraic method. The necessary and sufficient condition is found that the time-convolutionless master equation becomes exact up to the second order with respect to the system-reservoir interaction. The result is examined by means of the boson-detector model. The reduced dynamics of a quantum system interacting with a classical reservoir is also discussed.     

A master equation description of coupled systems. II

Dynamical correlations between two coupled systems are described by means of an infinite hierarchy of equations. The condition for a consistent truncation of the hierarchy is given. It allows to derive a closed equation describing the dynamics of one system. The structure of the equations for the stationary and non-stationary case is discussed and a comparison with other approaches is made.     

Return to thermal equilibrium by the solution of a quantum Langevin equation

A quantum-mechanical treatment of the evolution of an anharmonic oscillator coupled to a heat bath is given. It is shown that for a certain class of anharmonic potentials the heat bath drives the oscillator to an equilibrium state, close to the quantum Gibbs state associated to the potential. Thus a partial proof is provided for a conjecture of R. Benguria and M. Kac.This paper contains part of the author's Ph.D. work, done at the Institute for Theoretical Physics of Groningen State University, Groningen, the Netherlands.     

Traffic jams and ordering far from thermal equilibrium

The recently suggested correspondence between domain dynamics of traffic models and the asymmetric chipping model is reviewed. It is observed that in many cases traffic domains perform the two characteristic dynamical processes of the chipping model, namely chipping and diffusion. This correspondence indicates that jamming in traffic models in which all dynamical rates are non-deterministic takes place as a broad crossover phenomenon, rather than a sharp transition. Two traffic models are studied in detail and analyzed within this picture.     

Criticality in quark-gluon systems far beyond thermal and chemical equilibrium

Experimental evidence and theoretical arguments for the existence of self-organized criticality in systems of gluons and quarks are presented. It is observed that the existing data for high-transverse-momentum jet production exhibit striking regularities; and it is shown that such regularities can be used not only to probe the possible compositeness of quarks but also to obtain direct evidence for, or against, the existence of critical temperature and/or critical chemical potential in quark-gluon systems when hadrons are squeezed together.     

Number-resolved master equation approach to quantum measurement and quantum transport

In addition to the well-known Landauer–Büttiker scattering theory and the nonequilibrium Green’s function technique for mesoscopic transports, an alternative (and very useful) scheme is quantum master equation approach. In this article, we review the particle-number (n)-resolved master equation (n-ME) approach and its systematic applications in quantum measurement and quantum transport problems. The n-ME contains rich dynamical information, allowing efficient study of topics such as shot noise and full counting statistics analysis. Moreover, we also review a newly developed master equation approach (and its n-resolved version) under self-consistent Born approximation. The application potential of this new approach is critically examined via its ability to recover the exact results for noninteracting systems under arbitrary voltage and in presence of strong quantum interference, and the challenging non-equilibrium Kondo effect.     

Some properties of stochastic equations for coupled chemical reactions far from equilibrium

Solutions of master equations for coupled chemical reactions far from equilibrium with one varying molecule species are studied and used for getting information about nonlinear Fokker-Planck equations and slow time-dependent processes such as extinction to an absorbing state and transition between several steady states. The Fokker-Planck equation solution is compared to that of the master equation in a relative sense and it is shown that they agree quite well in some important situations but that in general the cases can deviate considerably, when, e.g., accounting for the mutual importance of two probability maxima.