A closed probability density function evolution equation for stochastic diffusion

1994 ◽  
Vol 6 (9) ◽  
pp. 3195-3196
Author(s):  
Javier Ros ◽  
Luis Valiño
Keyword(s):  
Probability Density Function ◽  
Evolution Equation ◽  
Probability Density ◽  
Density Function ◽  
Stochastic Diffusion

scholarly journals A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems

2011 ◽  
Vol 468 (2139) ◽  
pp. 759-783 ◽  
Author(s):  
D. Venturi ◽  
T. P. Sapsis ◽  
H. Cho ◽  
G. E. Karniadakis
Keyword(s):  
Probability Density Function ◽  
Evolution Equation ◽  
Probability Density ◽  
Density Function ◽  
Random Noise ◽  
Functional Integral ◽  
Excitation Density ◽  
Numerical Verification ◽  
Markovian Systems ◽  
Excitation Probability

By using functional integral methods, we determine a computable evolution equation for the joint response-excitation probability density function of a stochastic dynamical system driven by coloured noise. This equation can be represented in terms of a superimposition of differential constraints, i.e. partial differential equations involving unusual limit partial derivatives, the first one of which was originally proposed by Sapsis & Athanassoulis. A connection with the classical response approach is established in the general case of random noise with arbitrary correlation time, yielding a fully consistent new theory for non-Markovian systems. We also address the question of computability of the joint response-excitation probability density function as a solution to a boundary value problem involving only one differential constraint. By means of a simple analytical example, it is shown that, in general, such a problem is undetermined, in the sense that it admits an infinite number of solutions. This issue can be overcome by completing the system with additional relations yielding a closure problem, which is similar to the one arising in the standard response theory. Numerical verification of the equations for the joint response-excitation density is obtained for a tumour cell growth model under immune response.

scholarly journals Some remarks on modelling the PDF of the concentration of a dispersing scalar in turbulence

2002 ◽  
Vol 13 (1) ◽  
pp. 95-108 ◽  
Author(s):  
P. C. CHATWIN
Keyword(s):  
Probability Density Function ◽  
Evolution Equation ◽  
Turbulent Flow ◽  
Probability Density ◽  
Density Function ◽  
Molecular Diffusion ◽  
Concentration Field ◽  
Similarity Solutions ◽  
Small Scale ◽  
Probability Density Function Pdf

The paper deals with the probability density function (PDF) of the concentration of a scalar within a turbulent flow. Following some comments about the overall structure of the PDF, and its approach to a limit at large times, attention focusses on the so-called small scale mixing term in the evolution equation for the PDF. This represents the effect of molecular diffusion in reducing concentration uctuations, eventually to zero. Arguments are presented which suggest that this quantity could, in certain circumstances, depend inversely upon the PDF, and a particular example of this leads to a new closure hypothesis. Consequences of this, especially similarity solutions, are explored for the case when the concentration field is statistically homogeneous.

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