Non-Gaussian behavior of reflected fractional Brownian motion

Wada, Alexander H. O.; Warhover, Alex; Vojta, Thomas

doi:10.1088/1742-5468/ab02f1

Cited by 13 publications

(19 citation statements)

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“…This depletion of particles close to the reflecting wall for the FLE with anti-persistent noise is analogous to the behavior of FBM with anti-persistent noise [22,23]. However, for FBM, the probability density actually vanishes right at the wall whereas Fig.…”

Section: A Anti-persistent Noisesupporting

confidence: 54%

“…To summarize, motivated by recent observations of a non-Gaussian probability density for reflected FBM [22,23] as well as corresponding deviations from a uniform stationary probability density for FBM on a finite interval [24], we have studied FLE motion confined by reflecting potential barriers. The main part of our work focuses on the case when the random and damping forces fulfill the fluctuation-dissipation theorem.…”

Section: Discussionmentioning

confidence: 99%

“…Recently, large-scale computer simulations of FBM confined by reflecting walls have demonstrated that the interplay between the long-time correlations and the confinement strongly affects the probability density function P (x, t) of the diffusing particle. Specifically, if the motion is restricted to a semi-infinite interval by a reflecting wall at the origin, the probability density becomes highly non-Gaussian and develops a power-law singularity, P ∼ x κ at the wall [22,23]. For persistent noise, particles accumulate at the wall, κ < 0.…”

Section: Introductionmentioning

confidence: 99%

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Probability density of the fractional Langevin equation with reflecting walls

2019

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We investigate anomalous diffusion processes governed by the fractional Langevin equation and confined to a finite or semi-infinite interval by reflecting potential barriers. As the random and damping forces in the fractional Langevin equation fulfill the appropriate fluctuation-dissipation relation, the probability density on a finite interval converges for long times towards the expected uniform distribution prescribed by thermal equilibrium. In contrast, on a semi-infinite interval with a reflecting wall at the origin, the probability density shows pronounced deviations from the Gaussian behavior observed for normal diffusion. If the correlations of the random force are persistent (positive), particles accumulate at the reflecting wall while anti-persistent (negative) correlations lead to a depletion of particles near the wall. We compare and contrast these results with the strong accumulation and depletion effects recently observed for non-thermal fractional Brownian motion with reflecting walls, and we discuss broader implications. arXiv:1907.08188v1 [cond-mat.stat-mech]

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Section: A Anti-persistent Noisesupporting

confidence: 54%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Probability density of the fractional Langevin equation with reflecting walls

2019

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“…A particularly important problem for biological sciences is the behavior of FBM in bounded domains (e.g., in two-or three-dimensional shapes). Reflected BM is well understood (Ito and McKean, 1965), but it is not until very recently that the first description of the properties of reflected FBM (rFBM) have become available, in onedimensional domains (Wada and Vojta, 2018;Guggenberger et al, 2019;Wada et al, 2019). The present study is the first application of this theoretical framework to serotonergic fiber distributions, on the whole-brain scale.…”

Section: Introductionmentioning

confidence: 92%

Serotonergic Axons as Fractional Brownian Motion Paths: Insights into the Self-organization of Regional Densities

Janušonis

Detering

Metzler

et al. 2019

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All vertebrate brains contain a dense matrix of thin fibers that release serotonin (5hydroxytryptamine), a neurotransmitter that modulates a wide range of neural, glial, and vascular processes. Perturbations in the density of this matrix have been associated with a number of mental disorders, including autism and depression, but its self-organization and plasticity remain poorly understood. We introduce a model based on reflected Fractional Brownian Motion (FBM), a rigorously defined stochastic process, and show that it recapitulates some key features of regional serotonergic fiber densities. Specifically, we use supercomputing simulations to model fibers as FBM-paths in two-dimensional brain-like domains and demonstrate that the resultant steady state distributions approximate the fiber distributions in physical brain sections immunostained for the serotonin transporter (a marker for serotonergic axons in the adult brain). We suggest that this framework can support predictive descriptions and manipulations of the serotonergic matrix and that it can be further extended to incorporate the detailed physical properties of the fibers and their environment.

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“…Por fim, apresentaremos a teoria e as simulações do movimento Browniano fracionário com uma parede refletora em = 0 [83,84], e contrastaremos os efeitos da parede refletora com os resultados da difusão normal e do movimento Browniano fracionário não confinado.…”

Section: Ponto Multicríticounclassified

Transições de fase em modelos populacionais com desordem espacial e temporal

Wada¹,

Oliveira²

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São Paulo 2019ii iii Agradecimentos Agradeço à Fapesp (processos 2015/02100-8 e 2017/08631-0) e ao CNPq pelo apoio financeiro.Agradeço ao meu orientador Dr. Mário José de Oliveira, pois sem sua orientação nada do que segue seria possível. Sob sua orientação não só aprendi muita Física, mas também pude melhorar muitas outras habilidades necessárias para um pesquisador.Agradeço também ao Dr. Thomas Vojta por me receber em seu grupo de pesquisa e me orientar durante o estágio no exterior. A supervisão do Dr. Vojta também foi essencial para este trabalho e a experiência de trabalhar no seu grupo de pesquisa contribuiu muito para a minha formação.Por fim agradeço a minha família e meus amigos pelo apoio durante esta jornada.Apesar de todos serem de grande importância, gostaria de agradecer especialmente a minha amiga Carolina Feher da Silva pela ajuda com linguagens de programação. iv v Resumo Nesta tese estudamos os efeitos da desordem espacial e temporal na transição de fase entre a sobrevivência e extinção de populações biológicas. Na primeira parte estudamos um modelo epidemiológico com quatro estados. Apesar deste modelo não conter desordem, concluímos que seu comportamento crítico é o mesmo do processo de contato com desordem (espacial) quenched. Na segunda parte estudamos o movimento Browniano fracionário refletido, onde vimos que a combinação dos efeitos do ruído com correlações de longo alcance e a parede refletora cria uma singularidade em lei de potência na densidade de probabilidade da posição do caminhante. Por fim, estudamos a equação logística com desordem temporal através do mapeamento no movimento Browniano fracionário refletido. Neste último estudo vimos como as correlações de longo alcance mudam o comportamento crítico deste sistema. Palavras-chave: Modelos para populações biológicas, desordem espacial, desordem temporal, fase de Griffiths, percolação direcionada, percolação dinâmica vi vii AbstractWe have studied the efects of spatial and temporal disorder at the phase transition between survival and extinction of biological populations. In the first part we studied a four states biological population model. Despite having no disorder, we have seen that its critical behavior is the same of the contact process with (spatial) quenched disorder.In the second part, we studied the reflected fractional Brownian motion, where the interplay between the correlated noise and the reflecting wall results in a power-law singularity in the probability density of the position of the walker. Finally, we deduced the critical properties of the logistic equation with temporal disorder by mapping it onto the reflected fractional Brownian motion. This mapping allow us to understand how long-range correlations change the critical behavior of this system.

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Non-Gaussian behavior of reflected fractional Brownian motion

Cited by 13 publications

References 45 publications

Probability density of the fractional Langevin equation with reflecting walls

Probability density of the fractional Langevin equation with reflecting walls

Serotonergic Axons as Fractional Brownian Motion Paths: Insights into the Self-organization of Regional Densities

Transições de fase em modelos populacionais com desordem espacial e temporal

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