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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.
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.