On Gauss-Markov arbitrary kinetic level stochastic dynamics of plasmas I

Dense fluids and liquids molecules are in constant interaction; hence, they do not fit into the Boltzmann’s picture of a clearcut separation between free-streaming and collisional interactions. Since the interactions are soft and do not involve large scattering angles, an effective way of describing dense fluids is to formulate stochastic models of particle motion, as pioneered by Einstein’s theory of Brownian motion and later extended by Paul Langevin. Besides its practical value for the study of the kinetic theory of dense fluids, Brownian motion bears a central place in the historical development of kinetic theory. Among others, it provided conclusive evidence in favor of the atomistic theory of matter. This chapter introduces the basic notions of stochastic dynamics and its connection with other important kinetic equations, primarily the Fokker–Planck equation, which bear a complementary role to the Boltzmann equation in the kinetic theory of dense fluids.

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Stochastic Dynamics in Computational Biology

10.1007/978-3-030-62387-6 ◽

2020 ◽

Author(s):

Stefanie Winkelmann ◽

Christof Schütte

Keyword(s):

Computational Biology ◽

Stochastic Dynamics

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Neural network aided approximation and parameter inference of non-Markovian models of gene expression

Nature Communications ◽

10.1038/s41467-021-22919-1 ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

Qingchao Jiang ◽

Xiaoming Fu ◽

Shifu Yan ◽

Runlai Li ◽

Wenli Du ◽

...

Keyword(s):

Neural Network ◽

Stochastic Dynamics ◽

Stochastic Simulations ◽

Parameter Inference ◽

Biochemical Processes ◽

Markovian Models ◽

The Neural Network ◽

Large Numbers ◽

Small Set ◽

Noisy Measurements

AbstractNon-Markovian models of stochastic biochemical kinetics often incorporate explicit time delays to effectively model large numbers of intermediate biochemical processes. Analysis and simulation of these models, as well as the inference of their parameters from data, are fraught with difficulties because the dynamics depends on the system’s history. Here we use an artificial neural network to approximate the time-dependent distributions of non-Markovian models by the solutions of much simpler time-inhomogeneous Markovian models; the approximation does not increase the dimensionality of the model and simultaneously leads to inference of the kinetic parameters. The training of the neural network uses a relatively small set of noisy measurements generated by experimental data or stochastic simulations of the non-Markovian model. We show using a variety of models, where the delays stem from transcriptional processes and feedback control, that the Markovian models learnt by the neural network accurately reflect the stochastic dynamics across parameter space.

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Generalized fractional Poisson process and related stochastic dynamics

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0034 ◽

2020 ◽

Vol 23 (3) ◽

pp. 656-693 ◽

Cited By ~ 2

Author(s):

Thomas M. Michelitsch ◽

Alejandro P. Riascos

Keyword(s):

Poisson Process ◽

Continuous Time ◽

Counting Process ◽

Stochastic Dynamics ◽

Waiting Times ◽

Fractional Diffusion ◽

Diffusion Limit ◽

Fractional Operators ◽

Fractional Poisson Process ◽

Erlang Process

AbstractWe survey the ‘generalized fractional Poisson process’ (GFPP). The GFPP is a renewal process generalizing Laskin’s fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the ‘well-scaled’ diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.

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