Master equation

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

Noise in Gene Regulatory Networks

Abstract: Life processes in single cells and at the molecular level are inherently stochastic. Quantifying the noise is, however, far from trivial, as a major contribution comes from intrinsic fluctuations, arising from the randomness in the times between discrete jumps. It is shown in this paper how a noise-filtering setup with an operator theoretic interpretation can be relevant for analyzing the intrinsic stochasticity in jump processes described by master equations. Such interpretation naturally exists in linear noise approximations, but it also provides an exact description of the jump process when the transition rates are linear. As an important example, it is shown in this paper how, by addressing the proximity of the underlying dynamics in an appropriate topology, a sequence of coupled birth-death processes, which can be relevant in gene expression, tends to a pure delay; this implies important limitations in noise suppression capabilities. Despite the exactness, in a linear regime, of the analysis of noise in conjunction with the network dynamics, we emphasize in this paper the importance of also analyzing dynamic behavior when transition rates are highly nonlinear; otherwise, steady-state solutions can be misinterpreted. The examples are taken from systems with macroscopic models leading to bistability. It is discussed that bistability in the deterministic mass action kinetics and bimodality in the steady-state solution of the master equation neither always imply one another nor do they necessarily lead to efficient switching behaviours: the underlying dynamics need to be taken into account. Finally, we explore some of these issues in relation to a model of the lac operation.

Non-additive dissipation in open quantum networks out of equilibrium

Abstract: We theoretically study a simple non-equilibrium quantum network whose dynamics can be expressed and exactly solved in terms of a time-local master equation. Specifically, we consider a pair of coupled fermionic modes, each one locally exchanging energy and particles with an independent, macroscopic thermal reservoir. We show that the generator of the asymptotic master equation is not additive, i.e. it cannot be expressed as a sum of contributions describing the action of each reservoir alone. Instead, we identify an additional interference term that generates coherences in the energy eigenbasis, associated with the current of conserved particles flowing in the steady state. Notably, non-additivity arises even for wide-band reservoirs coupled arbitrarily weakly to the system. Our results shed light on the non-trivial interplay between multiple thermal noise sources in modular open quantum systems.

The Expansion of the Master Equation

Abstract: VII. Semiconductor , . . . . . . . . 260 VIII. Higher-Order Corrections . . . . . . 263 IX. The Malthus-Verhulst Problem . . . . . . . 265 X. The Transition Region . . . . . . . , . . 268 X1. The Langcvin Approach . . . . . . 270 XII. The Generation and Recombination Currents . . . . . 213 XIII. Example of a Multivariate Master Equation . . . . . . 215 XIV. Solution of the Multivariate Fokker-Planck Equation . . . . 279 XV. Fokker-Planck Equation with Constant Coefficients . . . . 282 XVI. Chemical Reactions . . . . . , . 284 . . . . 286 XVII. Two-step Chemical Reactions . . . . . XVIII. Fluctuations about a Limit Cycle . . . . . . 289 XIX. The Random Walk Picture . . . . . 293 XX. PhaseTransitions . . . . . . . . 297 XXI. Critical Fluctuations . . . . . . . 302

Microscopic Quantum Hydrodynamics of Systems of Fermions: Part I

Abstract: The fundamental equations of the microscopic quantum hydrodynamics of fermions in an external electromagnetic field (i.e., the particle balance equation, the momentum balance equation, the energy balance equation, and the magnetic moment balance equation) are derived using the Schrodinger equation. The form of the spin–spin interaction Hamiltonian is specified. To close the system of the balance equations for a multiparticle fermion system, the effective one-particle Schrodinger equation must be introduced.

Classical theory for shot noise in resonant tunneling.

Abstract: We show that shot noise for electrons can be suppressed in resonant tunneling through a double barrier, using a classical description based on the rate equation for ``sequential'' tunneling. The suppression is greatest when the escape rates through the two barriers are equal, in agreement with experiment and with the quantum-mechanical ``coherent'' model of resonant tunneling. A master equation is needed to calculate the noise, but cannot be uniquely derived from the rate equation; choices differ in the way that they describe transport between the emitter and the resonant state. Our choice for the rates, which are consistent with the exclusion principle, gives a suppression of the shot noise. We briefly discuss the results of choosing rates that are consistent with classical or Bose statistics instead of Fermi statistics. Finally, we apply our results to the two-state regime of the classical Coulomb blockade.