A comparison of master equation and Fokker-Planck equation in the thermodynamic limit

1986 ◽  
Vol 64 (2) ◽  
pp. 247-252 ◽  
Author(s):  
Hu Gang
Keyword(s):  
Master Equation ◽  
Thermodynamic Limit ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck

FOKKER–PLANCK-BASED CONTROL OF A TWO-LEVEL OPEN QUANTUM SYSTEM

2013 ◽  
Vol 23 (11) ◽  
pp. 2039-2064 ◽  
Author(s):  
M. ANNUNZIATO ◽  
A. BORZI
Keyword(s):  
Master Equation ◽  
Quantum System ◽  
Open Quantum System ◽  
Planck Equation ◽  
Open Loop ◽  
Optimality System ◽  
Environment Interaction ◽  
Open Quantum ◽  
Fokker Planck Equation ◽  
Fokker Planck

The control of a two-level open quantum system subject to dissipation due to environment interaction is considered. The evolution of this system is governed by a Lindblad master equation which is augmented by a stochastic term to model the effect of time-continuous measurements. In order to control this stochastic master equation model, a Fokker–Planck control framework is investigated. Within this strategy, the control objectives are defined based on the probability density functions of the two-level stochastic process and the controls are computed as minimizers of these objectives subject to the constraints represented by the Fokker–Planck equation. This minimization problem is characterized by an optimality system including the Fokker–Planck equation and its adjoint. This optimality system is approximated by a second-order accurate, stable, conservative and positive-preserving discretization scheme. The implementation of the resulting open-loop controls is realized with a receding-horizon algorithm over a sequence of time windows. Results of numerical experiments demonstrate the effectiveness of the proposed approach.

scholarly journals Lévy flights and Lévy-Schrödinger semigroups

Open Physics ◽  
2010 ◽  
Vol 8 (5) ◽  
Author(s):  
Piotr Garbaczewski
Keyword(s):  
Master Equation ◽  
Langevin Equation ◽  
Probability Function ◽  
Planck Equation ◽  
Levy Process ◽  
Stationary Probability ◽  
Lévy Flights ◽  
Fokker Planck Equation ◽  
Levy Flights ◽  
Fokker Planck

AbstractWe analyze two different confining mechanisms for Lévy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Lévy-Schrödinger semigroups which induce so-called topological Lévy processes (Lévy flights with locally modified jump rates in the master equation). Given a stationary probability function (pdf) associated with the Langevin-based fractional Fokker-Planck equation, we demonstrate that generically there exists a topological Lévy process with the same invariant pdf and in reverse.

scholarly journals Nonlinear Kinetics on a Lattice Featuring Anomalous Diffusion

2018 ◽  
Author(s):  
Giorgio Kaniadakis ◽  
Dionissios T. Hristopulos
Keyword(s):  
Master Equation ◽  
Continuum Limit ◽  
Numerical Solutions ◽  
Planck Equation ◽  
Master Equations ◽  
Discretization Scheme ◽  
Partial Differential ◽  
Fokker Planck Equation ◽  
The Continuum ◽  
Fokker Planck

Master equations define the dynamics that govern the time evolution of various physical processes on lattices. In the continuum limit, master equations lead to Fokker-Planck partial differential equations that represent the dynamics of physical systems in continuous spaces. Over the last few decades, nonlinear Fokker-Planck equations have become very popular in condensed matter physics and in statistical physics. Numerical solutions of these equations require the use of discretization schemes. However, the discrete evolution equation obtained by the discretization of a Fokker-Planck partial differential equation depends on the specific discretization scheme. In general, the discretized form is different from the master equation that has generated the respective Fokker-Planck equation in the continuum limit. Therefore, the knowledge of the master equation associated with a given Fokker-Planck equation is extremely important for the correct numerical integration of the latter, since it provides a unique, physically motivated discretization scheme. This paper shows that the Kinetic Interaction Principle (KIP) that governs the particle kinetics of many body systems, introduced in [G. Kaniadakis, Physica A 296, 405 (2001)], univocally defines a very simple master equation that in the continuum limit yields the nonlinear Fokker-Planck equation in its most general form.

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