Topic
Master equation
About: Master equation is a research topic. Over the lifetime, 10541 publications have been published within this topic receiving 276095 citations.
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TL;DR: In this paper, an algebraic structure related to discrete zero curvature equations is established, which is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations.
Abstract: An algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries. The key of the theory is to generate nonisospectral flows (λt=λl, l⩾0) from the discrete spectral problem associated with a given system of discrete evolution equations. Three examples are given.
171 citations
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TL;DR: This work establishes a range of closed form results for linear growth processes, including the scaling behaviours of the maximum growth rate and of the reaction end-point and shows that a self-consistent approach applied to the master equation of filamentous growth allows the determination of the evolution of the shape of the length distribution.
Abstract: Nucleated polymerisation processes are involved in many growth phenomena in nature, including the formation of cytoskeletal filaments and the assembly of sickle hemoglobin and amyloid fibrils. Closed form rate equations have, however, been challenging to derive for these growth phenomena in cases where secondary nucleation processes are active, a difficulty exemplified by the highly non-linear nature of the equation systems that describe monomer dependent secondary nucleation pathways. We explore here the use of fixed point analysis to provide self-consistent solutions to such growth problems. We present iterative solutions and discuss their convergence behaviour. We establish a range of closed form results for linear growth processes, including the scaling behaviours of the maximum growth rate and of the reaction end-point. We further show that a self-consistent approach applied to the master equation of filamentous growth allows the determination of the evolution of the shape of the length distribution including the mean, the standard deviation, and the mode. Our results highlight the power of fixed-point approaches in finding closed form self-consistent solutions to growth problems characterised by the highly non-linear master equations.
171 citations
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TL;DR: In this paper, the authors presented an approach to the effective simulation of the dynamics of open quantum many-body systems based on machine-learning techniques and derived a variational Monte-Carlo algorithm for their time evolution and stationary states.
Abstract: In experimentally realistic situations, quantum systems are never perfectly isolated and the coupling to their environment needs to be taken into account. Often, the effect of the environment can be well approximated by a Markovian master equation. However, solving this master equation for quantum many-body systems becomes exceedingly hard due to the high dimension of the Hilbert space. Here we present an approach to the effective simulation of the dynamics of open quantum many-body systems based on machine-learning techniques. We represent the mixed many-body quantum states with neural networks in the form of restricted Boltzmann machines and derive a variational Monte Carlo algorithm for their time evolution and stationary states. We document the accuracy of the approach with numerical examples for a dissipative spin lattice system.
171 citations
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TL;DR: In this article, the Schrodinger, Heisenberg, and Weyl-Wigner-Moyal representations of the Lindblad equaton are given explicitly and the solutions of the differential equations for the variances are put in a new synthetic form, suggested by a direct computation of the variance from the time dependent Weyl operators.
169 citations
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TL;DR: In this paper, an approximate Fokker-Planck model for nonlinear macroscopic systems is presented, which is superior to the conventional method based on the truncated Kramers-Moyal expansion.
Abstract: Relaxation and fluctuations of nonlinear macroscopic systems, which are frequently described by means of Fokker-Planck or Langevin equations, are studied on the basis of a master equation. The problem of an approximate Fokker-Planck modeling of the dynamics is investigated. A new Fokker-Planck modeling is presented which is superior to the conventional method based on the truncated Kramers-Moyal expansion. The new approach is shown to give the correct transition rates between deterministically stable states, while the conventional method overestimates these rates. An application to the Schl\"ogl models for first- and second-order nonequilibrium phase transitions is given.
168 citations