scholarly journals Log-Normal Superstatistics for Brownian Particles in a Heterogeneous Environment

Physics ◽  
2020 ◽  
Vol 2 (4) ◽  
pp. 571-586
Author(s):  
Maike Antonio Faustino dos Santos ◽  
Luiz Menon Junior
Keyword(s):  
Brownian Motion ◽  
Diffusion Processes ◽  
Single Particle Tracking ◽  
Heterogeneous Environment ◽  
Density Distribution Function ◽  
Biological Matter ◽  
Brownian Particles ◽  
Points Of View ◽  
Non Gaussian ◽  
Log Normal

Superstatistical approaches have played a crucial role in the investigations of mixtures of Gaussian processes. Such approaches look to describe non-Gaussian diffusion emergence in single-particle tracking experiments realized in soft and biological matter. Currently, relevant progress in superstatistics of Gaussian diffusion processes has been investigated by applying χ2-gamma and χ2-gamma inverse superstatistics to systems of particles in a heterogeneous environment whose diffusivities are randomly distributed; such situations imply Brownian yet non-Gaussian diffusion. In this paper, we present how the log-normal superstatistics of diffusivities modify the density distribution function for two types of mixture of Brownian processes. Firstly, we investigate the time evolution of the ensemble of Brownian particles with random diffusivity through the analytical and simulated points of view. Furthermore, we analyzed approximations of the overall probability distribution for log-normal superstatistics of Brownian motion. Secondly, we propose two models for a mixture of scaled Brownian motion and to analyze the log-normal superstatistics associated with them, which admits an anomalous diffusion process. The results found in this work contribute to advances of non-Gaussian diffusion processes and superstatistical theory.

SMOLYANOV–WEIZSÄCKER SURFACE MEASURES GENERATED BY DIFFUSIONS ON THE SET OF TRAJECTORIES IN RIEMANNIAN MANIFOLDS

2008 ◽  
Vol 11 (01) ◽  
pp. 21-31 ◽  
Author(s):  
ILYA V. TELYATNIKOV
Keyword(s):  
Brownian Motion ◽  
Euclidean Space ◽  
Diffusion Process ◽  
Riemannian Manifolds ◽  
Marginal Distribution ◽  
Diffusion Processes ◽  
Ambient Space ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Limit Surface

We consider surface measures on the set of trajectories in a smooth compact Riemannian submanifold of Euclidean space generated by diffusion processes in the ambient space. A construction of surface measures on the path space of a smooth compact Riemannian submanifold of Euclidean space was introduced by Smolyanov and Weizsäcker for the case of the standard Brownian motion. The result presented in this paper extends the result of Smolyanov and Weizsäcker to the case when we consider measures generated by diffusion processes in the ambient space with nonidentical correlation operators. For every partition of the time interval, we consider the marginal distribution of the diffusion process in the ambient space under the condition that it visits the manifold at all times of the partition, when the mesh of the partition tends to zero. We prove the existence of some limit surface measures and the equivalence of the above measures to the distribution of some diffusion process on the manifold.

scholarly journals Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

2021 ◽  
Author(s):  
Tianyu Ma ◽  
Vladimir S. Matveev ◽  
Ilya Pavlyukevich
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Random Walks ◽  
Diffusion Processes ◽  
Riemannian Metric ◽  
Finsler Manifold ◽  
Finsler Manifolds ◽  
Bounded Geometry ◽  
And Brownian Motion

AbstractWe show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

scholarly journals Exponential functionals of Brownian motion, II: Some related diffusion processes

2005 ◽  
Vol 2 (0) ◽  
pp. 348-384 ◽  
Author(s):  
Hiroyuki Matsumoto ◽  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Diffusion Processes

Quantification of Model-Form Uncertainty in Drift-Diffusion Simulation Using Fractional Derivatives

2015 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Diffusion Processes ◽  
Planck Equation ◽  
Optimization Procedure ◽  
Physical World ◽  
Drift Diffusion ◽  
Long Range Correlations ◽  
Modeling Process ◽  
Model Form ◽  
Generic Description ◽  
Non Gaussian

In modeling and simulation, model-form uncertainty arises from the lack of knowledge and simplification during modeling process and numerical treatment for ease of computation. Traditional uncertainty quantification approaches are based on assumptions of stochasticity in real, reciprocal, or functional spaces to make them computationally tractable. This makes the prediction of important quantities of interest such as rare events difficult. In this paper, a new approach to capture model-form uncertainty is proposed. It is based on fractional calculus, and its flexibility allows us to model a family of non-Gaussian processes, which provides a more generic description of the physical world. A generalized fractional Fokker-Planck equation (fFPE) is proposed to describe the drift-diffusion processes under long-range correlations and memory effects. A new model calibration approach based on the maximum accumulative mutual information is also proposed to reduce model-form uncertainty, where an optimization procedure is taken.

The Level-Crossing Problem: First-Passage, Escape and Extremes

2014 ◽  
Vol 13 (04) ◽  
pp. 1430001 ◽  
Author(s):  
Jaume Masoliver
Keyword(s):  
Brownian Motion ◽  
Diffusion Processes ◽  
Extreme Values ◽  
Level Crossing ◽  
First Passage ◽  
One Dimensional ◽  
Absolute Value ◽  
Dimensional Diffusion

We review the level-crossing problem which includes the first-passage and escape problems as well as the theory of extreme values (the maximum, the minimum, the maximum absolute value and the range or span). We set the definitions and general results and apply them to one-dimensional diffusion processes with explicit results for the Brownian motion and the Ornstein–Uhlenbeck (OU) process.

scholarly journals On the macroscopic limit of Brownian particles with local interaction

2020 ◽  
Vol 20 (06) ◽  
pp. 2040007
Author(s):  
Franco Flandoli ◽  
Marta Leocata ◽  
Cristiano Ricci
Keyword(s):  
Explicit Form ◽  
Diffusion Processes ◽  
Particle System ◽  
Local Interaction ◽  
Nonlinear Pde ◽  
Rigorous Results ◽  
Precise Form ◽  
Interacting Particle ◽  
Brownian Particles ◽  
Main Ideas

An interacting particle system made of diffusion processes with local interaction is considered and the macroscopic limit to a nonlinear PDE is investigated. Few rigorous results exists on this problem and in particular the explicit form of the nonlinearity is not known. This paper reviews this subject, some of the main ideas to get the limit nonlinear PDE and provides both heuristic and numerical informations on the precise form of the nonlinearity which are new with respect to the literature and coherent with the few known informations.

scholarly journals Subtle changes in crosslinking drive diverse anomalous transport characteristics in actin-microtubule networks

2020 ◽  
Author(s):  
Sylas Anderson ◽  
Jonathan Garamella ◽  
Ryan McGorty ◽  
Rae Robertson-Anderson
Keyword(s):  
Intracellular Transport ◽  
Single Particle Tracking ◽  
Anomalous Transport ◽  
Brownian Diffusion ◽  
Complex Environments ◽  
Materials Engineering ◽  
Microtubule Networks ◽  
Non Gaussian ◽  
Fluid Rheology ◽  
The Impact

Abstract Anomalous diffusion in crowded and complex environments is widely studied due to its importance in intracellular transport, fluid rheology and materials engineering. Specifically, diffusion through the cytoskeleton, a network comprised of semiflexible actin filaments and rigid microtubules that interact both sterically and via crosslinking, plays a principal role in viral infection, vesicle transport and targeted drug delivery. Here, we elucidate the impact of crosslinking on particle diffusion in composites of actin and microtubules with actin-actin, microtubule-microtubule and actin-microtubule crosslinking. We analyze a suite of complementary transport metrics by coupling single-particle tracking and differential dynamic microscopy. Using these orthogonal techniques, we find that particles display non-Gaussian and non-ergodic subdiffusion that is markedly enhanced by cytoskeletal crosslinking of any type, which we attribute to suppressed microtubule mobility. However, the extent to which transport deviates from normal Brownian diffusion depends strongly on the crosslinking motif – with actin-microtubule crosslinking inducing the most pronounced anomalous characteristics – due to increased actin fluctuation heterogeneity. Our results reveal that subtle changes to actin-microtubule interactions can have dramatic impacts on diffusion in the cytoskeleton, and suggest that less mobile and more locally heterogeneous networks lead to more strongly anomalous transport.

Diffusion Processes The Wiener Process and Brownian Motion

2021 ◽  
pp. 1-14
Author(s):  
Gardiner Crispin
Keyword(s):  
Brownian Motion ◽  
Wiener Process ◽  
Diffusion Processes ◽  
And Brownian Motion

Transition functions, paths and path integrals

1991 ◽  
Vol 434 (1890) ◽  
pp. 41-63 ◽  
Keyword(s):  
Brownian Motion ◽  
Diffusion Processes ◽  
Nonlinear Dynamical Systems ◽  
Diffusion Theory ◽  
Main Theme ◽  
Nonlinear Dynamical ◽  
Transition Functions ◽  
Sample Paths ◽  
Itô Calculus ◽  
Expository Paper

The main theme of this expository paper is the relation between analysis and probability in the context of diffusion theory. Section 1 discusses in rather heuristic fashion the very satisfying solution to the problem of describing diffusion processes which Kolmogorov achieved via PDE theory (the theory of partial differential equations) and his criterion for path continuity. Section 2 describes how Itô calculus totally transformed the subject by allowing us to construct the sample paths of a diffusion process X by solving an SDE (stochastic differential equation) driven by brownian motion. (Of course, SDEs have great intrinsic importance too as noisy perturbations of nonlinear dynamical systems.) Though §2 begins heuristically, the mathematics is then tightened up. This paper is, after all, a tribute to the man whose greatest contribution to science is his setting probability theory on a rigorous foundation. Once Kolmogorov’s precise language is available, §2 then takes a quick sight-seeing trip through some of the great developments by Doob, Itô and their successors. (In an age in which so many do simulations of Itô equations, I have explained precisely in the briefest possible fashion what the exact theory is. It is easy and usable.) You will just have time for a snapshot of how brownian motion on the orthonormal frame bundle is linked to index theorems, and of what the Malliavin calculus is about. You will, however, be advised on guide-books on these and other areas (including physicists’ favourites: large deviations, measure-valued diffusions, etc.), so that you can later explore at your leisure. Confession : the paper consists very largely of selected tracks (remixed!) from the album (Rogers & Williams 1987 Diffusions, Markov processes and martingales ; Chichester: Wiley). Tributes to Kolmogorov’s work in probability and statistics have appeared in (every book ever written on probability and in) Ann. Probability 17 (1989), 815-964, Ann. Statist. 18 (1990), 987-1031, Bull. Lond. math. Soc. . 22 (1990), 31-100, Teor. Veroyatnost Primenen 34 (1989), no. 1, Usp. mat. Nauk 43 (1988), no. 6. The official biography by Shiryaev will be a wonderful volume. For me, this paper is a further expression of my thanks to Kolmogorov and (as he would have wished) to Lévy, Doob and Itô too.

Brownian Motion and Diffusion Processes

1996 ◽  
pp. 233-253
Author(s):  
Denis Bosq ◽  
Hung T. Nguyen
Keyword(s):  
Brownian Motion ◽  
Diffusion Processes ◽  
And Diffusion
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