Speaker : Diego del-Castillo-Negrete, Oak Ridge National Laboratory, USA

Time : 15:00 - 16.00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

YouTube : https://youtu.be/vLZC3pN8pnw?feature=shared

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: Fractional diffusion, nonlocal transport, anomalous transport, plasma physics, fusion plasmas, turbulent transport, Feynman-Kac

Abstract: There is growing experimental, numerical, and theoretical evidence of cases where local (advection-diffusion) transport models fail to describe anomalous transport (e.g., super-diffusive and sub-diffusive transport). To overcome this limitation, nonlocal models introduce nonlocal flux-gradient relations and formulate transport using integrodifferential operators in general and fractional diffusion operators in particular. In this lecture we present an overview of fractional diffusion models of nonlocal transport with emphasis on applications to plasmas and fluids. In the first part, a brief review of the statistical foundations of general nonlocal models will be presented. Following that, we will discuss experimental and numerical evidence of nonlocal transport in magnetically confined fusion plasmas and fluids, along with effective fractional diffusion models describing these phenomena. The second part of the lecture will be devoted to a recently proposed probabilistic method to solve the initial value problem and the exit-time problem of fractional diffusion equations. The method is based on the Feynman-Kac formula and reduces the computation to the evaluation of expectations (i.e., numerical quadratures), bypassing the need of sampling stochastic trajectories (as in the case of particle-based methods) or the need to deal with dense, non-sparse matrices (as in the case of continuum methods). A description of the method will be presented along with applications.

Biography: Diego del-Castillo-Negrete is a Distinguished Scientist in the Theory and Modeling Group of the Fusion Energy Division at the Oak Ridge National Laboratory in USA. He holds a Ph.D. in Physics from the University of Texas at Austin (1994). Before joining ORNL in 2000, he worked in the Theoretical Division of Los Alamos National Laboratory (1998-2000), and the Scripps Institution of Oceanography at the University of California San Diego (1994-1998). His research interests span a wide spectrum of topics in plasma physics, applied mathematics, nonlinear dynamics, computational physics, and machine learning.

Bibliography

[1] D. del-Castillo-Negrete, B.A. Carreras and V. Lynch, “Non-diffusive transport in plasma turbulence: a fractional diffusion approach”. Phys. Rev. Lett. 94, 065003 (2005).

[2] D. del-Castillo-Negrete, B.A. Carreras, and V. Lynch, “Fractional diffusion in plasma turbulence”. Phys. of Plasmas 11, 3854-3864 (2004).

[3] D. del-Castillo-Negrete, “Fractional diffusion models of nonlocal transport”. Phys. of Plasmas 13, 082308 (2006).

[4] D. del-Castillo-Negrete, “Non-diffusive, non-local transport in fluids and plasmas”. Nonlin. Processes Geophys. 17, 795-807 (2010).

[5] D. del-Castillo-Negrete and L. Chacon, “Local and nonlocal parallel transport in general magnetic fields”. Phys. Rev. Letters 106, 19, 195004 (2011).

[6] D. del-Castillo-Negrete, B.A. Carreras and V. Lynch, “Front dynamics in reaction-diffusion systems with Levy flights: A fractional diffusion approach”. Phys. Rev. Lett. 91, (1), 018302, (2003).

[7] A. Cartea and D. del-Castillo-Negrete, “Fluid limit of the continuous-time random walk with general Levy jump distribution functions”. Phys. Rev. E. 76, 041105 (2007).

[8] M. Yang, G. Zhang, D. del-Castillo-Negrete, and Y. Cao, “A probabilistic scheme for semilinear nonlocal diffusion equations with volume constrains.” SIAM Journal of Numerical Analysis 61, (6), 2718-2743 (2023)