Power-law Fokker–Planck equation of unimolecular reaction based on the approximation to master equation

2016 ◽  
Vol 463 ◽  
pp. 445-451 ◽  
Author(s):  
Yanjun Zhou ◽  
Cangtao Yin
Keyword(s):  
Master Equation ◽  
Power Law ◽  
Planck Equation ◽  
Unimolecular Reaction ◽  
Fokker Planck Equation ◽  
Fokker Planck

scholarly journals Nonlinear Fokker–Planck Equation Approach to Systems of Interacting Particles: Thermostatistical Features Related to the Range of the Interactions

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 163 ◽  
Author(s):  
Angel Plastino ◽  
Roseli Wedemann
Keyword(s):  
Power Law ◽  
Evolution Equations ◽  
Planck Equation ◽  
Nonlinear Evolution Equations ◽  
Present Contribution ◽  
Gaussian Form ◽  
Newtonian Gravitation ◽  
Fokker Planck Equation ◽  
Many Body Systems ◽  
Fokker Planck

Nonlinear Fokker–Planck equations (NLFPEs) constitute useful effective descriptions of some interacting many-body systems. Important instances of these nonlinear evolution equations are closely related to the thermostatistics based on the S q power-law entropic functionals. Most applications of the connection between the NLFPE and the S q entropies have focused on systems interacting through short-range forces. In the present contribution we re-visit the NLFPE approach to interacting systems in order to clarify the role played by the range of the interactions, and to explore the possibility of developing similar treatments for systems with long-range interactions, such as those corresponding to Newtonian gravitation. In particular, we consider a system of particles interacting via forces following the inverse square law and performing overdamped motion, that is described by a density obeying an integro-differential evolution equation that admits exact time-dependent solutions of the q-Gaussian form. These q-Gaussian solutions, which constitute a signature of S q -thermostatistics, evolve in a similar but not identical way to the solutions of an appropriate nonlinear, power-law Fokker–Planck equation.

scholarly journals GENERALIZED MODEL OF MIGRATION-DRIVEN AGGREGATE GROWTH — ASYMPTOTIC DISTRIBUTIONS, POWER LAWS AND APPARENT FRACTALITY

2012 ◽  
Vol 26 (01) ◽  
pp. 1250010 ◽  
Author(s):  
YURI G. GORDIENKO
Keyword(s):  
Exact Solution ◽  
Power Law ◽  
Rate Equation ◽  
Planck Equation ◽  
Asymptotic Distributions ◽  
Cumulative Distribution ◽  
Aggregation Kinetics ◽  
Scaling Analysis ◽  
Fokker Planck Equation ◽  
Fokker Planck

The rate equation for exchange-driven aggregation of monomers between clusters of size n by power-law exchange rate (~ nα), where detaching and attaching processes were considered separately, is reduced to Fokker–Planck equation. Its exact solution was found for unbiased aggregation and agreed with asymptotic conclusions of other models. Asymptotic transitions were found from exact solution to Weibull/normal/exponential distribution, and then to power law distribution. Intermediate asymptotic size distributions were found to be functions of exponent α and vary from normal (α = 0) through Weibull (0 < α < 1) to exponential (α =1) ones, that gives the new system for linking these basic statistical distributions. Simulations were performed for the unbiased aggregation model on the basis of the initial rate equation without simplifications used for reduction to Fokker–Planck equation. The exact solution was confirmed, shape and scale parameters of Weibull distribution (for 0 < α < 1) were determined by analysis of cumulative distribution functions and mean cluster sizes, which are of great interest, because they can be measured in experiments and allow to identify details of aggregation kinetics (like α). In practical sense, scaling analysis of evolving series of aggregating cluster distributions can give much more reliable estimations of their parameters than analysis of solitary distributions. It is assumed that some apparent power and fractal laws observed experimentally may be manifestations of such simple migration-driven aggregation kinetics even.

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.

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