scholarly journals Steady state solution of the Fokker-Planck equation combined with unidirectional quasilinear diffusion under detailed balance conditions

1985 ◽  
Vol 109 (7) ◽  
pp. 325-330
Author(s):  
Kyriakos Hizanidis
Keyword(s):  
Steady State ◽  
Detailed Balance ◽  
Planck Equation ◽  
Steady State Solution ◽  
State Solution ◽  
Fokker Planck Equation ◽  
Quasilinear Diffusion ◽  
Fokker Planck

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.

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.

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.

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