Master equation

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

Nonequilibrium phase transitions in chemical reactions

Abstract: The far from equilibrium steady states of a simple nonlinear chemical system are analyzed. A standard macroscopic analysis shows that the nonlinearity introduces an instability which causes a transition analogous to a thermodynamic second-order phase transition. Fluctuations are introduced into this model through a stochastic master equation approach. The solution of this master equation in the steady state reveals that the system goes into a more ordered state above the transition point. An analogy is drawn with the nonequilibrium phase transition occurring in the laser at threshold.

Stability for linearized gravity on the Kerr spacetime

Abstract: In this paper we prove integrated energy and pointwise decay estimates for solutions of the vacuum linearized Einstein equation on the domain of outer communication of the Kerr black hole spacetime. The estimates are valid for the full subextreme range of Kerr black holes, provided integrated energy estimates for the Teukolsky Master Equation holds. For slowly rotating Kerr backgrounds, such estimates are known to hold, due to the work of one of the authors. The results in this paper thus provide the first stability results for linearized gravity on the Kerr background, in the slowly rotating case, and reduce the linearized stability problem for the full subextreme range to proving integrated energy estimates for the Teukolsky equation. This constitutes an essential step towards a proof of the black hole stability conjecture, i.e. the statement that the Kerr family is dynamically stable, one of the central open problems in general relativity.

Determination of the Spectral Gap for Kac's Master Equation and Related Stochastic Evolutions

Abstract: We present a method for bounding, and in some cases computing, the spectral gap for systems of many particles evolving under the influence of a random collision mechanism. In particular, the method yields the exact spectral gap in a model due to Mark Kac of energy conserving collisions with one dimensional velocities. It is also sufficiently robust to provide qualitatively sharp bounds also in the case of more physically realistic momentum and energy conserving collisions in three dimensions, as well as a range of related models.

Master Equation Analysis of Thermal Activation Reactions: Energy-Transfer Constraints on Falloff Behavior in the Decomposition of Reactive Intermediates with Low Thresholds

Abstract: This paper deals with the high-temperature decomposition of reactive intermediates with low reaction thresholds. If these intermediates are created in situ, for example, through radical chain processes, their initial molecular distribution functions may be characteristic of the bath temperature and, under certain circumstances, peak at energies above the reaction threshold. Such an ordering of reaction thresholds and distribution functions has some similarities to that found during chemical activation. This leads to consequences that are essenially the inverse (larger rate constants than those deduced from steady-state distributions) of the situation for stable compounds under shock-heated conditions and hence reduces falloff effects. To study this behavior, rate constants for the unimolecular decomposition of allyl, ethyl, n-propyl, and n-hexyl radicals have been determined on the basis of the solution of the time-dependent master equation with specific rate constants from RRKM calculations. The time req...

Staying positive: going beyond Lindblad with perturbative master equations

Abstract: The perturbative master equation (Bloch-Redfield) is extensively used to study dissipative quantum mechanics - particularly for qubits - despite the 25 year old criticism that it violates positivity (generating negative probabilities). We take an arbitrary system coupled to an environment containing many degrees-of-freedom, and cast its perturbative master equation (derived from a perturbative treatment of Nakajima-Zwanzig or Schoeller-Schon equations) in the form of a Lindblad master equation. We find that the equation's parameters are time-dependent. This time-dependence is rarely accounted for, and invalidates Lindblad's dynamical semigroup analysis. We analyze one such Bloch-Redfield master equation (for a two-level system coupled to an environment with a short but non-vanishing memory time), which apparently violates positivity. We show analytically that, once the time-dependence of the parameters is accounted for, positivity is preserved.