Normal and anomalous diffusion in random potential landscapes

We show that some important properties of subdiffusion of unknown origin (including ones of mixed origins) can be easily assessed when finding the "fundamental moment" of the corresponding random process, i.e., the one which is additive in time. In subordinated processes, the index of the fundamental moment is inherited from the parent process and its time dependence from the leading one. In models of a particle's motion in disordered potentials, the index is governed by the structural part of the disorder while the time dependence is given by its energetic part.

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“…balance condition in equilibrium. Under quite general assumptions (see [14]) the effective diffusion coefficient in such a model is…”

Section: Arxiv:13053523v1 [Physicsdata-an] 15 May 2013mentioning

confidence: 99%

Disentangling Sources of Anomalous Diffusion

2013

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“…In some cases, the mechanism which gives origin to anomalous scaling can be different, special deterministic or random environments [5,6] or multi-particle interactions [7]. Moreover, diffusion can be strongly anomalous (E [|x(t)| q ] ∼ t qν with ν depending on q) in complex systems [8][9][10].…”

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confidence: 99%

Asymptotic properties of a bold random walk

Serva

2014

Phys. Rev. E

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In a recent paper [1] we proposed a non-Markovian random walk model with memory of the maximum distance ever reached from the starting point (home). The behavior of the walker is at variance with respect to the simple symmetric random walk (SSRW) only when she is at this maximum distance, where, having the choice to move either farther or closer, she decides with different probabilities. If the probability of a forward step is higher then the probability of a backward step, the walker is bold and her behavior turns out to be super-diffusive, otherwise she is timorous and her behavior turns out to be sub-diffusive. The scaling behavior vary continuously from sub-diffusive (timorous) to super-diffusive (bold) according to a single parameter γ ∈ R. We investigate here the asymptotic properties of the bold case in the non ballistic region γ ∈ [0, 1/2], a problem which was left partially unsolved in [1]. The exact results proved in this paper require new probabilistic tools which rely on the construction of appropriate martingales of the random walk and its hitting times.The appellative anomalous diffusion is associated to a scaling relation E [x 2 (t)] ∼ t 2ν with ν = 1/2. It may arise in random walks via diverging steps length, as in Lévy flights [2] or via long-range memory as in self avoiding random walks [3,4]. Diverging steps length and longrange memory are two different ways of violating the necessary conditions for the central limit theorem when applied to random walks.In some cases, the mechanism which gives origin to anomalous scaling can be different, special deterministic or random environments [5,6] or multi-particle interactions [7]. Moreover, diffusion can be strongly anomalous (E [|x(t)| q ] ∼ t qν with ν depending on q) in complex systems [8][9][10].There is a very large number of phenomena which exhibit anomalous diffusion as well a variety of models which have been used to describe them, (for a review of both see [11][12][13][14][15]). Nevertheless, exact solutions of non-trivial models with memory are quite scarce [16][17][18][19][20][21][22]. Motivated by this lack of exact solutions, we presented in [1] a model which is exactly treatable although genuinely non-Markovian. The model shows anomalous scaling which can be sub-diffusive, super-diffusive and also ballistic according to a single parameter γ ∈ R.It is the aim of this work to investigate here the asymptotic properties in the range γ ∈ [0, 1/2], a problem which was left partially unsolved in [1]. This range corresponds to the a non ballistic super-diffusive behavior except at the two extreme were it is ordinary SSRW (γ = 0) and ballistic (γ = 1/2).The model, as defined in [1], is one-dimensional, steps all have the same unitary length, time is discrete and the walker can only move left or right at any time step. The behavior of the random walker is modified with respect to SSRW only when she is at the maximum distance ever reached from her starting point (home). In this case, she decides with different probabilities to make a step forward ...

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“…[28,[56][57][58] However, recent work indicates that the local exponent describing transport may be dependent on the shape of the obstacles, [59] thereby leading to slowly fading transient dynamics. As such, considering diffusion on more complex potential landscapes [33,[60][61][62][63][64][65] could yield valuable insight particularly relevant to real crowded media, where the obstacles are unlikely to be spherical. Interestingly, our result is in contrast to what is seen in a periodic Lorentz gas, where softening the interactions drastically changes the nature of the transport.…”

Section: Discussionmentioning

confidence: 99%

Anomalous transport in the soft-sphere Lorentz model

Petersen

Franosch

2019

Soft Matter

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The sensitivity of anomalous transport in crowded media to the form of the inter-particle interactions is investigated through computer simulations. We extend the highly simplified Lorentz model towards realistic natural systems by modeling the interactions between the tracer and the obstacles with a smooth potential. We find that the anomalous transport at the critical point happens to be governed by the same universal exponent as for hard exclusion interactions, although the mechanism of how narrow channels are probed is rather different. The scaling behavior of simulations close to the critical point confirm this exponent. Our result indicates that the simple Lorentz model may be applicable to describing the fundamental properties of long-range transport in real crowded environments.

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Normal and anomalous diffusion in random potential landscapes

Cited by 18 publications

References 13 publications

Disentangling Sources of Anomalous Diffusion

Disentangling Sources of Anomalous Diffusion

Asymptotic properties of a bold random walk

Anomalous transport in the soft-sphere Lorentz model

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