Langevin equation in complex media and anomalous diffusion

Vitali, Silvia; Sposini, Vittoria; Sliusarenko, Oleksii; Paradisi, Paolo; Castellani, Gastone; Pagnini, Gianni

doi:10.1098/rsif.2018.0282

Cited by 43 publications

(56 citation statements)

References 65 publications

(130 reference statements)

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“…We also note here that there exist other classes of anomalous diffusion models such as semi-Markovian continuous time random walks with scale-free waiting time statistic [65], Markovian continuous time random walks with time scale populations [66], scaled Brownian motion [67], heterogeneous diffusion processes [68], generalised grey Brownian motion [56,69], or a recent approach using heterogeneous Brownian particle ensembles [70]. The use of either model depends on the physical situation.…”

Section: Discussionmentioning

confidence: 99%

“…Indeed, in the limit 0 l  the noise autocorrelation function (70) approaches the one of fractional Gaussian noise [36,57], as can be derived by using the small argument expansion (67) of the Bessel function. In this limit λ→0 for any finite τ the autocorrelation function (70)…”

Section: Direct Tempering Of Mandelbrot's Fractional Brownian Motionmentioning

confidence: 99%

“…in which we have to find the Laplace transformation of the autocorrelation function (70). We assume that σ 2 =1 for simplicity from now on.…”

Section: Fractional Langevin Equation With Directly Tempered Fractionmentioning

confidence: 99%

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Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers

et al. 2018

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The emerging diffusive dynamics in many complex systems show a characteristic crossover behaviour from anomalous to normal diffusion which is otherwise fitted by two independent power-laws. A prominent example for a subdiffusive-diffusive crossover are viscoelastic systems such as lipid bilayer membranes, while superdiffusive-diffusive crossovers occur in systems of actively moving biological cells. We here consider the general dynamics of a stochastic particle driven by so-called tempered fractional Gaussian noise, that is noise with Gaussian amplitude and power-law correlations, which are cut off at some mesoscopic time scale. Concretely we consider such noise with built-in exponential or power-law tempering, driving an overdamped Langevin equation (fractional Brownian motion) and fractional Langevin equation motion. We derive explicit expressions for the mean squared displacement and correlation functions, including different shapes of the crossover behaviour depending on the concrete tempering, and discuss the physical meaning of the tempering. In the case of power-law tempering we also find a crossover behaviour from faster to slower superdiffusion and slower to faster subdiffusion. As a direct application of our model we demonstrate that the obtained dynamics quantitatively describes the subdiffusion-diffusion and subdiffusion-subdiffusion crossover in lipid bilayer systems. We also show that a model of tempered fractional Brownian motion recently proposed by Sabzikar and Meerschaert leads to physically very different behaviour with a seemingly paradoxical ballistic long time scaling. 11 A more consistent approach using the smoothening procedure of fractional Brownian motion over infinitesimally small time intervals à la Mandelbrot and van Ness [36] shows that the weak divergence of the autocorrelation function (6) at τ=0 does not lead to a change of the MSD. 12 The power-law correlations in the autocorrelation function (6) contrast the sharp δ-correlation of relation (8) [38,39]. We note that in this combination of the Langevin equation (2) and the autocorrelation function (6) the fluctuation dissipation theorem is not satisfied, and the noise ξ(t) can be considered as an external noise [40], see also the discussion of the generalised Langevin equation below.

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Section: Discussionmentioning

confidence: 99%

Section: Direct Tempering Of Mandelbrot's Fractional Brownian Motionmentioning

confidence: 99%

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Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers

et al. 2018

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“…The heterogeneous ensemble of Brownian particle (HEBP) approach [1,2] is based on the idea that a population of scales in the system in which particles are diffusing may generate the anomalous diffusing behaviour observed in many physical and biological systems [3][4][5][6]. Long time and space correlation, characteristics of many anomalous diffusion processes [7][8][9], are often described through the introduction of memory kernels and integral operators [10,11], as the fractional derivatives are [12], leading in general to non-Markovianity and/or non-locality of the processes.…”

Section: Introductionmentioning

confidence: 99%

“…In the Langevin description of HEPB [1], one of the scales contributing to the anomalous behaviour is the time scale τ. In particular, the presence of a population of time scales, described by a carefully chosen distribution, generates a process with the same one-time one-point probability density function (PDF) of the fractional Brownian motion (fBm), i.e.…”

Section: Introductionmentioning

confidence: 99%

The Role of the Central Limit Theorem in the Heterogeneous Ensemble of Brownian Particles Approach

Vitali¹,

Budimir²,

Runfola³

et al. 2019

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The central limit theorem (CLT) and its generalization to stable distributions have been widely described in literature. However, many variations of the theorem have been defined and often their applicability in practical situations is not straightforward. In particular, the applicability of the CLT is essential for a derivation of heterogeneous ensemble of Brownian particles (HEBP). Here, we analyze the role of the CLT within the HEBP approach in more detail and derive the conditions under which the existing theorems are valid.

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Understanding biochemical processes in the presence of sub-diffusive behavior of biomolecules in solution and living cells

2019

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In order to maintain cellular function, biomolecules like protein, DNA, and RNAs have to diffuse to the target spaces within the cell. Changes in the cytosolic microenvironment or in the nucleus during the fulfillment of these cellular processes affect their mobility, folding, and stability thereby impacting the transient or stable interactions with their adjacent neighbors in the organized and dynamic cellular interior. Using classical Brownian motion to elucidate the diffusion behavior of these biomolecules is hard considering their complex nature. The understanding of biomolecular diffusion inside cells still remains elusive due to the lack of a proper model that can be extrapolated to these cases. In this review, we have comprehensively addressed the progresses in this field, laying emphasis on the different aspects of anomalous diffusion in the different biochemical reactions in cell interior. These experiment-based models help to explain the diffusion behavior of biomolecules in the cytosolic and nuclear microenvironment. Moreover, since understanding of biochemical reactions within living cellular system is our main focus, we coupled the experimental observations with the concept of sub-diffusion from in vitro to in vivo condition. We believe that the pairing between the understanding of complex behavior and structure-function paradigm of biological molecules would take us forward by one step in order to solve the puzzle around diseases caused by cellular dysfunction.

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Langevin equation in complex media and anomalous diffusion

Cited by 43 publications

References 65 publications

Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers

Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers

The Role of the Central Limit Theorem in the Heterogeneous Ensemble of Brownian Particles Approach

Understanding biochemical processes in the presence of sub-diffusive behavior of biomolecules in solution and living cells

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