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Langevin dynamics in heterogeneous media and anomalous diffusion
to derive models in agreement with all the statistical features emerging from data [1, 2], but an exhaustive description is still missing.
We derive a stochastic diffusion model based on a Langevin approach, characterized by additive noise and linear friction force [3].
The complexity of the medium is parametrized by a population of the parameters, the relaxation time and diffusivity.
For proper distributions of these parameters, both Gaussian anomalous diffusion,
fractional diffusion and its generalizations can be retrieved, but characterized by a superdiffusive regime by model construction.
The inclusion of a confining potential, for example by considering the harmonic Langevin oscillator, permits to switch the process to a subdiffusive regime.
This anomalous diffusive behaviour can be reflected in the motion of the center of mass of an heterogeneous ensamble of particles [4]
and the motion of an inert tracer globally connected with such heterogeneous mesoscopic surrounding.
References
[1] M. Mura, G. Pagnini. 2008 Characterizations and simulations of a
class of stochastic processes to model anomalous diffusion. J. Phys.
A: Math. Theor. \textbf{41}, 285003.
[2] D. Molina-Garciá, T. M. Pham, P. Paradisi, C. Manzo, G. Pagnini.
2016. Fractional kinetics emerging from ergodicity breaking in random
media Phys. Rev. E \textbf{94}, 052147.
[3] S. Vitali, V. Sposini, O. Sliusarenko, P. Paradisi, G. Castellani, G.
Pagnini. 2018. J. R. Soc. Interface \textbf{15}: 20180282.
http://dx.doi.org/10.1098/rsif.2018.0282
[4] M. D’Ovidio, S. Vitali, V. Sposini, O. Sliusarenko, P. Paradisi,
G. Castellani, G. Pagnini. 2018. Centre-of-mass like superposition
of Ornstein-Uhlenbeck processes: a pathway to non-autonomous
stochastic differential equations and to fractional diffusion. Submit-
ted. https://arxiv.org/abs/1806.11351.
Fractional Calculus and the Particle in the Ring
Arbitrary order calculus is a natural generalisation of usual calculus in which the order of differentiation and integration operators in not restricted to integer numbers. In engineering fractional order derivatives are used for describing the behaviour of materials with memory (i.e. viscoelastic materials) due to the fact that these materials lay somewhere in between Hookean springs and Newtonean fluids[1,2]; there are also many dynamical systems which can be better described when arbitrary order derivatives are included [3].
Lanskin [4] formulated the first Fractional Schrödinger Equation (FSE) along with the Fractional Continuation Equation in 2002; however, we are still far from fully understanding the effect pf the FSE on physical properties such as: Tunnelling [5], Diffraction [6] and Scattering [7]. Due to the properties of fractional derivatives, many jobs have been done in which relativistic properties and effects of extrinsic magnetic fields are obtained by incorporating an arbitrary order to the kinetic energy in the Hamiltonian [8,9].
Further studies of the FSE applied on astrophysically interesting systems such as \(H_2^+\) [10] and even hydrogen atom [11] seem to be promising. We shall take the FSE for a particle in a ring (1) as a first step into this world for which the eigenvalues are (2) and the eigenfunctions (3)
\(
\left[\frac{1}{2mr^2}\right]^{\alpha-1}\left[i\hslash \partial_\theta \right]^\alpha \Psi_\alpha(\theta;r)=\lambda_\alpha\Psi_\alpha(\theta;r) \tag{1}
\)
\(
\lambda_\alpha = \left[\frac{1}{2mr^2}\right]^{1-\alpha} N^\alpha \hslash^\alpha \exp \left[ i \pi \alpha \left(n+1\right)\right]\; | \; N, \; n \in \mathbb{N}+\{0\} \tag{2}
\)
\(
\Psi_\alpha(\theta;r)=C\exp \left(-i N^\alpha\left[ \frac{\hslash}{2mr^2}\right]^{\alpha-1}\exp \left(i \pi \alpha \left(n+1\right)\right)\theta \right). \tag{3}
\)
References
[1] M. Stiassnie, 1979, Appl. Math. Modelling, 3, 300.
[2] M. Du et al., 2013, Scientific Reports, 3, 3431.
[3] V. E. Tarasov, 2013, Int. J. Mod. Phys. B, 2013, 9, 1330005.
[4]N. Laskin, Physics Review E, 2000, 66, 056108.
[5] E. Capelas et al., 2011, J. Phys. A 44, 185303.
[6] Y. Zhang et al., 2015, Scientific Reports 6, 23645.
[7] A. Liemert, 2016, Mathematics, 4, 31.
[8] J. Lorinczi and J. Malecki, 2012, J. Diff. Eq., 253, 2846.
[9] J. Blackledge and B. Babajanov, 2013, Math. Aeterna, 3, 601.
[10] A. Turbiner et al., 1999, JETP Letters, 11, 69.
[11] A. I. Arbab, 2012, J. Modern Physics, 3, 1737.
A pseudo-spectral method for a non-local Fractional Fisher-KPP equation
We develop a pseudo-spectral method to solve initial-value problems associated to PDEs involving the fractional Laplacian operator acting on the whole real line (see [1]). After a suitable representation of the operator, we perform the change of variable \(x = L\cot(s)\), \(L > 0\), to transform the real line \(\mathbb{R}\) into the interval \([0, \pi]\), where a Fourier expansion of the solution \(u(x(s))\) can be applied. We approximate the fractional Laplacian by means of the midpoint quadrature rule, improving the results with Richardson’s extrapolation, similarly as in [2]. This method deals accurately and efficiently with problems posed on \(\mathbb{R}\), and avoids truncating the domain (which requires introducing artificial
boundary conditions). In order to illustrate its applicability, we have simulated the evolution of the following non-local Fisher-KPP [3] and ZFK-Nagumo [4] models.
Keywords: Fractional Laplacian, Pseudo-spectral methods, Fourier transform, Chebyshev Polynomials.
References
[1] M. Kwaśnicki. Ten equivalent definitions of the fractional Laplace operator. Fractional Calculus and Applied Analysis, 20(1):7–51, 2017.
[2] F. de la Hoz and C. M. Cuesta. A pseudo-spectral method for a non-local KdV-Burgers equation posed on \(\mathbb{R}\). Journal of Computational Physics, 311:45–61, 2016.
[3] R. A. Fisher. The wave of advance of advantageous genes. Annals. of Eugenics, 7:355–369, 1937.
[4] Y. B. Zel’dovich and D. A. Frank-Kamenetsky. Towards the theory of uniformly propagating flames. Doklady AN SSSR, 19:693–697, 1938.
[5] J. P. Boyd. Chebyshev and Fourier Spectral Method. Springer–Verlag, XVI, 1989.
On extension problems and Hardy inequalities in the Heisenberg group.
Several new results concerning the extension problem in the Heisenberg group are also attained, including characterisations of all solutions of the extension problem satisfiying \(L^p\) integrability, and the study of the higher order extension problem.
This is a joint work with S. Thangavelu (Indian Institute of Science of Bangalore, India).
Chaotic orbits for nonlocal equations and applications to atom dislocation dynamics in crystals.
In particular, heteroclinic, homoclinic and chaotic trajectories are constructed.
This result regarding symbolic dynamics in a fractional framework is part of a study of the Peierls-Nabarro model for crystal dislocations. The associated evolution equation can be studied in the mesoscopic and macroscopic limit. Namely, the dislocation function has the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium.
These dislocation points evolve according to the external stress and an interior potential, which can be either repulsive or attractive, depending on the relative orientations of the dislocations. For opposite orientations, collisions occur, after which the system relaxes exponentially fast.
Symmetry properties for long-range phase coexistence models.
We discuss some recent results on nonlocal phase transitions
modelled by the fractional Allen-Cahn equation, also in connection with the surfaces minimising a nonlocal perimeter functional. In particular, we consider the “genuinely nonlocal regime” in which the difusion operator is of order less than 1 and present some rigidity and symmetry results.
A consistency problem in neural modelling. Coherence between input and output can be obtained using heavy tails distributions.
The coherence between the input and the output of the single units is a problem, sometimes underestimated, in network modeling.
An example in this direction is given by Integrate and Fire models used to describe the membrane potential dynamics of a neuron in a network. The focus of these models concerns the description of the inter-times between events (the InterSpike Intervals, ISIs), i.e. of the output of the neuron. This type of models describes the membrane potential evolution through a suitable stochastic process and the output of the neuron corresponds to the First Passage Time of the considered stochastic process through a boundary. However, the input mechanism determining the membrane potential dynamics often disregards its origin as function of the output of other units.
The seminal idea for these models goes back to 1964 when Gernstein and Mandelbrot proposed the Integrate and Fire model to account for the observed stable behavior of the Inter-spike Interval distribution. They suggested to model the membrane potential dynamics through a Wiener process in order to get the Inverse Gaussian distribution for the inter-times between the successive spikes of the neuron, i.e. its output. The use of the Wiener process was first motivated by its property to be the continuous limit of a random walk, later the randomized random walk was proposed to account for the continuous time characterizing the membrane potential dynamics. In this model, the arrival of inputs from the network determines jumps of fixed size on the membrane potential value. When the membrane potential attains a threshold value \(S\), the neuron releases a spike and the process restart anew. Furthermore, the inter-times between jumps are exponentially distributed.
Unfortunately, this last hypothesis is contradictory with the heavy tail distribution of the output, since the incoming inputs are output of other neurons. Later many variants of the original model appeared in the literature. Their aim was to improve the realism of the model but
unfortunately they forgot the initial clue for it, the heavy tails of the observed output distribution.
However, the ISIs models (or their variants) are generally recognized to be a good compromise between the realism and its easy use and have been proposed to model large networks. These facts motivate us to rethink the ISIs model allowing heavy tail distributions both for the ISIs of the neurons surrounding the modeled neuron and for its output.
Here, we propose to start the model formulation from the main property, i.e. the heavy tails exhibited by the ISIs. This approach allows us to propose here an Integrate and Fire model coherent with these features. The ideal framework for this rethinking involves regularly varying random variables and vectors.
We assume that each input to a unit corresponds to the output of one of \(N < \infty \) neurons of the network. The inter-times between spikes of the same neuron are independent random variables, with regularly varying distribution. Different neurons of the network are not independent, due to the network connections. The only hypothesis that we introduce to account for the dependence is very general, i.e. their ISIs determine a regularly varying vector. Based on these hypothesis we prove that the output inter-times of the considered neuron, described through an Integrate and Fire model, is a regularly varying random variable.
The next step of this modeling procedure requests a suitable rescaling of the obtained process to obtain a time fractional limit for the process describing the membrane potential evolution. We already have some result in this direction, allowing to write down the Laplace transform of the first passage time of the rescaled process through the threshold \(S\) but some mathematical step should be improved to account for the dependence between jumps times also for the limiting process.
References
[1] Bingham, N. H., Goldie, C. M. and Teugels, J. L., Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1989. ISBN 0-521-37943-1.
[2] Gal A., and Marom S. Entrainment of the Intrinsic Dynamics of Single Isola Neurons by Natural-Like Input,. The Journal of Neurosciences 33 (18) pp. 7912-7918, 2013.
[3] Gernstein G.L., Mandelbrot B. Random walk models for the activity of a single neuron., Biophys. J. 4 pp. 41-68, 1964.
[4] Holden, A.V. A Note on Convolution and Stable Distributions in the Nervous System., Biol. Cybern. 20 pp. 171-173, 1975.
[5] Jessen, A. H., and Mikosch, T. Regularly varying functions. Publ. Inst. Math. (Beograd) (N.S.), 80(94):171–192, 2006. ISSN 0350-1302.
[6] Lindner B. Superposition of many independent spike trains is generally not a Poisson process, . Physical review E 73 pp. 022901, 2006.
[7] Kyprianou, A. Fluctuations of Lévy Processes with Applications., Springer-Verlag, 2014.
[8] Persi E., Horn D., Volman V., Segev R. and Ben-Jacob E. Modeling of Synchronized Bursting Events: the importance of Inhomogeneity., Neural Computation 16 pp. 2577-2595, 2004.
[9] Tsubo Y., Isomura Y. and Fukai T. Power-Law Inter-Spike Interval Distributions infer a Conditional Maximization of Entropy in Cortical Neurons,. PLOS Computational Biology 8,4 pp. e1002461, 2012.