Master equation

About: Master equation is a research topic. Over the lifetime, 10541 publications have been published within this topic receiving 276095 citations.

Emergent Hierarchical Structures in Complex-System Dynamics.

Abstract: A method is introduced for studying thermal relaxation in multiminima energy landscapes. All the configurations connected to a given energy minimum by paths never exceeding a chosen energy lid are found, each equipped with a set of pointers to its neighbours. This information defines a phase space pocket around the minimum, in which the master equation for the relaxation process is directly solved. As an example we analyse some instances of the Travelling-Salesman Problem. We find that i) the number of configurations accessible from a given suboptimal tour grows exponentially with the energy lid, ii) the density of states within the pocket also shows exponential growth, iii) the low-temperature dynamical behaviour is characterized by a sequence of local equilibrations in increasingly larger regions of phase space and finally iv) the propagator decays algebraically with a temperature-dependent exponent. These observations are related to both theoretical models and experimental findings on relaxation in complex systems.

Theory of relaxation in viscous liquids and glasses

Abstract: A theory of relaxation (the time‐dependent change of macroscopic properties) of nonpolymeric viscous liquids and glasses is presented. The fluid is described by a set of quasiequilibrium structures, and a master equation gives the transitions among these structures. Any structural change is presumed to require a cooperative rearrangement involving many atoms, and this rearrangement entails a fluctuation to a high‐energy transition state. The structure of the fluid varies from point to point, and the rate of this transformation depends crucially on the local structure. The resulting kinetic equation describes very well the main features of observed relaxation—namely, the broad distribution of relaxation times and the nonlinearity (in ΔT) of relaxation following a temperature jump ΔT, where the apparent activation energy for relaxation depends on time. The kinetic equation is solved exactly, and the resulting solution is exhibited for a particular set of the parameters. The resulting relaxation function for energy relaxation goes as e −t 1 / 4 , except at the very shortest times. (The precise exponent— 1/4 in this case—will depend on the parameters used in the particular calculation.) A detailed comparison with other theories is made, and suggestions of how the theory can be used to develop a phenomenological model of relaxation of glass are given.

Energy master equation: A low-temperature approximation to Bässler's random-walk model.

Abstract: The first part of this paper deals with the justification of B\"assler's phenomenological random-walk model for viscous liquids [Phys. Rev. Lett. 58, 767 (1987)], which considers the random walk of a ``particle'' representing the liquid state on a d-dimensional infinite cubic lattice with site energies chosen randomly according to a Gaussian. The random-walk model is here derived from Newton's laws by making a number of simplifying assumptions. In the second part of the paper an approximate low-temperature description of energy fluctuations in the random-walk model---the energy master equation (EME)---is arrived at. The EME is one dimensional and involves only energy; it is derived by arguing that percolation dominates the relaxational properties of the random-walk model at low temperatures. The approximate EME description of the random-walk model is expected to be valid at low temperatures at long times in high dimensions. However, computer simulations show that the EME works well already in two dimensions and at only moderately low temperatures. The EME has no randomness and no fitting parameters. The EME is completely specified from the density of states and the attempt frequency of the random-walk model. The EME allows a calculation of the energy probability distribution at realistic laboratory time scales for an arbitrarily varying temperature as function of time. The EME is probably the only realistic equation available today with this property that is also explicitly consistent with statistical mechanics. The final part of the paper gives a comprehensive discussion, comparing the EME to related work and listing the EME's qualitatively correct predictions, its new predictions, and some ``wrong'' predictions, most of which go against the common picture of viscous liquids and the glass transition without violating experiments.

Hierarchical quantum master equation approach to electronic-vibrational coupling in nonequilibrium transport through nanosystems

Abstract: Quantum transport in nanosystems is often characterized by strong coupling between electronic and vibrational degrees of freedom. Examples include single-molecule junctions, nanoelectromechanical systems, and suspended carbon nanotubes. Electronic-vibrational coupling manifests itself in vibronic structures in the transport characteristics and results in a multitude of nonequilibrium phenomena, such as current-induced local heating and cooling, multistability, switching, hysteresis, and decoherence. The theoretical study of quantum transport, in particular in the strong-coupling regime, requires nonperturbative approaches that can be systematically converged, i.e., numerically exact methods. In this work, the authors outline how the hierarchical quantum master equation approach, which generalizes perturbative master equation methods by including higher-order contributions as well as non-Markovian memory, can be used in this context. The results show that vibrational nonequilibrium effects are important in a broad spectrum of scenarios, which range from the nonadiabatic to the adiabatic regime and include both resonant and off-resonant transport.