scholarly journals Infinite product expansion of the Fokker–Planck equation with steady-state solution

2015 ◽  
Vol 471 (2179) ◽  
pp. 20150084 ◽  
Author(s):  
R. J. Martin ◽  
R. V. Craster ◽  
M. J. Kearney
Keyword(s):  
Steady State ◽  
Infinite Product ◽  
Planck Equation ◽  
Steady State Solution ◽  
State Solution ◽  
Analytical Technique ◽  
Short And Long Term ◽  
Fokker Planck Equations ◽  
Fokker Planck

We present an analytical technique for solving Fokker–Planck equations that have a steady-state solution by representing the solution as an infinite product rather than, as usual, an infinite sum. This method has many advantages: automatically ensuring positivity of the resulting approximation, and by design exactly matching both the short- and long-term behaviour. The efficacy of the technique is demonstrated via comparisons with computations of typical examples.

Non-Maxwellian kinetic equations modeling the dynamics of wealth distribution

2020 ◽  
Vol 30 (04) ◽  
pp. 685-725 ◽  
Author(s):  
Giulia Furioli ◽  
Ada Pulvirenti ◽  
Elide Terraneo ◽  
Giuseppe Toscani
Keyword(s):  
Steady State ◽  
Wealth Distribution ◽  
Kinetic Equations ◽  
Planck Equation ◽  
Logarithmic Sobolev Inequality ◽  
One Dimensional ◽  
Distribution Of Wealth ◽  
Multi Agent ◽  
Fokker Planck Equations ◽  
Fokker Planck

We introduce a class of new one-dimensional linear Fokker–Planck-type equations describing the dynamics of the distribution of wealth in a multi-agent society. The equations are obtained, via a standard limiting procedure, by introducing an economically relevant variant to the kinetic model introduced in 2005 by Cordier, Pareschi and Toscani according to previous studies by Bouchaud and Mézard. The steady state of wealth predicted by these new Fokker–Planck equations remains unchanged with respect to the steady state of the original Fokker–Planck equation. However, unlike the original equation, it is proven by a new logarithmic Sobolev inequality with weight and classical entropy methods that the solution converges exponentially fast to equilibrium.

scholarly journals The Study of a Wealth Distribution Model with a Linear Collision Kernel

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Xia Zhou ◽  
Kaili Xiang ◽  
Rongmei Sun
Keyword(s):  
Steady State ◽  
Boltzmann Equation ◽  
Substitution Rate ◽  
Wealth Distribution ◽  
Planck Equation ◽  
Steady State Solution ◽  
State Solution ◽  
Distribution Model ◽  
Collision Kernel ◽  
Continuous Trading

The wealth substitution rate, which describes the substitution relationship between agents’ investment in wealth, is introduced into the collision kernel of the Boltzmann equation to study wealth distribution. Using the continuous trading limit, the Fokker–Planck equation is derived and the steady-state solution is obtained. The results show that the inequality of wealth distribution decreases as the wealth substitution rate increases under certain assumptions. The wealth distribution has a bimodal shape if the wealth substitution rate does not equal one.

The lattice Fokker–Planck equation for models of wealth distribution

2020 ◽  
Vol 378 (2175) ◽  
pp. 20190401
Author(s):  
Shaurya Kaushal ◽  
Santosh Ansumali ◽  
Bruce Boghosian ◽  
Merek Johnson
Keyword(s):  
Steady State ◽  
Soft Matter ◽  
Wealth Distribution ◽  
Planck Equation ◽  
Agent Based ◽  
Non Local ◽  
Free Parameters ◽  
Fokker Planck Equations ◽  
Steady State Solutions ◽  
Fokker Planck

Recent work on agent-based models of wealth distribution has yielded nonlinear, non-local Fokker–Planck equations whose steady-state solutions describe empirical wealth distributions with remarkable accuracy using only a few free parameters. Because these equations are often used to solve the ‘inverse problem’ of determining the free parameters given empirical wealth data, there is much impetus to find fast and accurate methods of solving the ‘forward problem’ of finding the steady state corresponding to given parameters. In this work, we derive and calibrate a lattice Boltzmann equation for this purpose. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

BOLTZMANN–ARRHENIUS–ZHURKOV (BAZ) MODEL IN PHYSICS-OF-MATERIALS PROBLEMS

2013 ◽  
Vol 27 (13) ◽  
pp. 1330009 ◽  
Author(s):  
E. SUHIR ◽  
S.-M. KANG
Keyword(s):  
Steady State ◽  
Planck Equation ◽  
Steady State Solution ◽  
Degradation Process ◽  
Material Degradation ◽  
Fokker Planck Equation ◽  
Markovian Processes ◽  
Transient Period ◽  
The Given ◽  
Fokker Planck

Boltzmann–Arrhenius–Zhurkov (BAZ) model enables one to obtain a simple, easy-to-use and physically meaningful formula for the evaluation of the probability of failure (PoF) of a material after the given time in operation at the given temperature and under the given stress (not necessarily mechanical). It is shown that the material degradation (aging, damage accumulation, flaw propagation, etc.) can be viewed, when BAZ model is considered, as a Markovian process, and that the BAZ model can be obtained as the steady-state solution to the Fokker–Planck equation in the theory of Markovian processes. It is shown also that the BAZ model addresses the worst and a reasonably conservative situation, when the highest PoF is expected. It is suggested therefore that the transient period preceding the condition addressed by the steady-state BAZ model need not be accounted for in engineering evaluations. However, when there is an interest in understanding the physics of the transient degradation process, the obtained solution to the Fokker–Planck equation can be used for this purpose.

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