Particle Transport in Stochastic and Nonclassical Media

The last several years have seen increasing interest in particle transport in stochastic mixtures, such as Poisson tessellations, where the properties of the background material are known only in a statistical sense. The primary goal in these problems is to find the ensemble-average of the particle intensity in phase space, with current challenges centered on obtaining higher-order moments. Applications include astrophysics, nuclear medicine, reactor physics and criticality, and atmospheric sciences.

In classical transport, the probability that a particle interacts with the background medium is proportional to the path length traveled, with the proportionality constant depending on medium density and particle energy. This leads to exponential attenuation (the Beer-Lambert law). Certain real systems, however, exhibit spatial correlations or fluctuations not resolved by the model, producing nonexponential free-path distributions where classical techniques fail to preserve the true attenuation law and yield unreliable flux estimates. These correlated systems motivate the nonclassical framework developed in this project, which addresses the full range of heterogeneous media where exponential attenuation breaks down. Applications extend across reactor physics, atmospheric sciences, health physics, medical imaging, and computer graphics. Part of this research is conducted in collaboration with the Rio de Janeiro State University, funded by the Brazilian Government through a CNPq Research Grant.

Doctoral Dissertations advised on the subject:

P1 synthetic acceleration and convergence analysis for the solution of the one-speed nonclassical spectral SN equations in slab geometry
- da Silva, Alan S. (2025)
- D.Sc. in Computational Modeling, State University of Rio de Janeiro (Brazil)
- As co-advisor, with R.C. Barros as advisor
Derivation of Spherical Harmonic Approximations to the Nonclassical Particle Transport Equation
- Agbo, Sunday A. (2024)
- Ph.D. in Nuclear Engineering, The Ohio State University
Asymptotic Derivation of the Simplified PN Equations for Nonclassical Transport with Anisotropic Scattering
- Palmer, Robert K. (2020)
- Ph.D. in Nuclear Engineering, The Ohio State University

Publications

A compact introduction to a nonclassical neutral particle transport model

Leonardo R.C. Moraes, Ricardo C. Barros, Hermes Alves Filho, Richard Vasques

To appear in Integral Methods in Science and Engineering, 2026

[J29] A preliminary study of the influence of approximate boundary conditions on the accuracy of nonclassical spherical harmonic solutions

Sunday A. Agbo, Leonardo R.C. Moraes, Hermes Alves Filho, Ricardo C. Barros, Richard Vasques

Journal of Computational and Theoretical Transport, To appear (accepted), 2026

[J28] A methodology for estimating the ensemble-averaged effective multiplication factor and neutron scalar flux profile in statistically homogeneous random media using a slab-geometry, discrete ordinates and energy multigroup nonclassical neutral particle transport model

Leonardo R.C. Moraes, Ricardo C. Barros, Hermes Alves Filho, Richard Vasques

Nuclear Science and Engineering, 2025 Dec, pp. 1--24

[J27] A nonclassical model to eigenvalue neutron transport calculations

Leonardo R.C. Moraes, Ricardo C. Barros, Hermes Alves Filho, Richard Vasques

Annals of Nuclear Energy, vol. 216, 2025 Nov, p. 111250

[P43] Estimating the ensemble-average effective multiplication factor and neutron scalar flux profile in statistically homogeneous random media using a nonclassical model

Leonardo R.C. Moraes, Ricardo C. Barros, Hermes Alves Filho, Richard Vasques

Proceedings of International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering, Denver, CO, 2025 Apr

[J22] On the use of spherical harmonic approximations in nonclassical particle transport problems

Sunday A. Agbo, Leonardo R.C. Moraes, Richard Vasques

Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 324, 2024, p. 109048

[P41] Calculating the expected effective multiplication factor and neutron scalar flux profile in statistically homogeneous random media using a nonclassical model

Leonardo R.C. Moraes, Ricardo C. Barros, Richard Vasques

Proceedings of 2024 International Nuclear Atlantic Conference, Rio de Janeiro, Brazil, 2024 May

[J19] On the occurrence of linearly dependent eigenvectors in nonclassical transport calculations

Leonardo R.C. Moraes, Richard Vasques, Ricardo C. Barros

Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 295(-), 2023, p. 108407

[P40] P1 synthetic acceleration and convergence analysis for the solution of one-speed nonclassical spectral SN equations in slab geometry

Alan S. Silva, Leonardo R.C. Moraes, Ricardo C. Barros, Richard Vasques

Proceedings of International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering, Niagara Falls, Canada, 2023 Aug

[P37] Derivation of a multigroup diffusion equation for nonclassical problems

Lisa S. Enomoto, Hermes Alves Filho, Richard Vasques

Proceedings of International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering, Niagara Falls, Canada, 2023 Aug

[P36] Derivation of spherical harmonic approximations to the nonclassical particle transport equation

Sunday A. Agbo, Leonardo R.C. Moraes, Richard Vasques

Proceedings of International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering, Niagara Falls, Canada, 2023 Aug

[P35] An initial study of the convergence rate of Griffin’s pebble bed reactors algorithm

Tomas M. Paganin, Sebastian Schunert, Leonardo R.C. Moraes, Richard Vasques

Proceedings of International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering, Niagara Falls, Canada, 2023 Aug

[J18] An improved spectral approach for solving the nonclassical neutral particle transport equation

Leonardo R.C. Moraes, Japan K. Patel, Ricardo C. Barros, Richard Vasques

Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 290(-), 2022, p. 108282

[J17] On the application of the analytical discrete ordinates method to the solution of nonclassical transport problems in slab geometry

Leonardo R. C. Moraes, Liliane B. Barichello, Ricardo C. Barros, Richard Vasques

Journal of Computational Physics, vol. 455(-), 2022, p. 110982

[J16] Transport synthetic acceleration for the solution of the one-speed nonclassical spectral SN equations in slab geometry

Japan K. Patel, Leonardo R. C. Moraes, Richard Vasques, Ricardo C. Barros

Journal of Computational and Applied Mathematics, vol. 401(-), 2022 , p. 113768

[P33] A review of the deterministic solution of nonclassical transport problems

Leonardo R. C. Moraes, Ricardo C. Barros, Richard Vasques

Proceedings of International Nuclear Atlantic Conference - INAC 2021, Virtual Meeting, Brazil, 2021 Nov

[P32] The nonclassical simplified P2 and P3 equations with anisotropic scattering

Robert K. Palmer, Richard Vasques

Proceedings of The International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, Virtual Meeting (Raleigh, NC), 2021 Oct, pp. 368-378

[J14] Asymptotic derivation of the simplified PN equations for nonclassical transport with anisotropic scattering

Robert K. Palmer, Richard Vasques

Journal of Computational and Theoretical Transport, vol. 49(7), 2020 Sep, pp. 331-348

[J13] A spectral approach for solving the nonclassical transport equation

Richard Vasques, Leonardo R. C. Moraes, Ricardo Carvalho de Barros, Rachel N. Slaybaugh

Journal of Computational Physics, vol. 402(-), 2020 Feb, p. 109078

[P28] P1 synthetic acceleration for nonclassical spectral SN equations in slab geometry

Japan K. Patel, Leonardo R. C. Moraes, Richard Vasques, Ricardo C. Barros

Proceedings of 26th ICTT: International Conference on Transport Theory, Paris, France, 2019 Sep

[P27] A nonclassical Monte Carlo algorithm for transport problems in diffusive binary stochastic media

Richard Vasques, Patrick S. Brantley, Robert K. Palmer

Proceedings of 26th ICTT: International Conference on Transport Theory, Paris, France, 2019 Sep

[P26] Asymptotic derivation of the simplified PN equations for nonclassical transport with anisotropic scattering

Robert K. Palmer, Richard Vasques

Proceedings of 26th ICTT: International Conference on Transport Theory, Paris, France, 2019 Sep

[J12] Exact transport representations of the classical and nonclassical simplified PN equations

Ilker Makine, Richard Vasques, Rachel N. Slaybaugh

Journal of Computational and Theoretical Transport, vol. 47(4-6), 2018, pp. 326-349

[J11] Nonclassical particle transport in one-dimensional random periodic media

Richard Vasques, Kai Krycki, Rachel N. Slaybaugh

Nuclear Science and Engineering, vol. 185(1), 2017, pp. 78-106

[P23] Exact transport representations of the classical and nonclassical simplified PN equations

Ilker Makine, Richard Vasques, Rachel N. Slaybaugh

Proceedings of 25th ICTT: International Conference on Transport Theory, Monterey, CA, 2017 Oct

[P22] A generalized volume rendering approach for computer graphics

Magnus Wrenninge, Richard Vasques, Rachel N. Slaybaugh

Proceedings of 25th ICTT: International Conference on Transport Theory, Monterey, CA, 2017 Oct

[P21] Simplified PN equations for nonclassical transport with isotropic scattering

Richard Vasques, Rachel N. Slaybaugh

Proceedings of International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering, Jeju, South Korea, 2017 Apr

[J9] The nonclassical diffusion approximation to the nonclassical linear Boltzmann equation

Richard Vasques

Applied Mathematics Letters, vol. 53(-), 2016, pp. 63-68

[P20] Nonclassical particle transport in the 1-D diffusive limit

Richard Vasques, Rachel N. Slaybaugh, Kai Krycki

Transactions of the American Nuclear Society, vol. 114(1), 2016, pp. 361-364

[J8] The nonclassical Boltzmann equation and diffusion-based approximations to the Boltzmann equation

Martin Frank, Kai Krycki, Edward W. Larsen, Richard Vasques

Siam Journal on Applied Mathematics, vol. 75(3), 2015, pp. 1329-1345

[J7] Adjusted Levermore–Pomraning equations for diffusive random systems in slab geometry

Richard Vasques, Nitin K. Yadav

Journal of Quantitative Spectroscopy & Radiative Transfer, vol. 154(-), 2015, pp. 98-112

[P18] Boundary conditions for the 1-D non-classical transport equation

Richard Vasques, Kai Krycki

Proceedings of 24th ICTT: International Conference on Transport Theory, Taormina, Italy, 2015 Sep

[P17] On the accuracy of the non-classical transport equation in 1- D random periodic media

Richard Vasques, Kai Krycki

Proceedings of Joint International Conference on Mathematics and Computation, Supercomputing in Nuclear Applications and the Monte Carlo Method, Nashville, TN, 2015 Apr

[J6] Non-classical particle transport with angular-dependent path-length distributions. II: Application to pebble bed reactor cores

Richard Vasques, Edward W. Larsen

Annals of Nuclear Energy, vol. 70(-), 2014, pp. 301-311

[J5] Non-classical particle transport with angular-dependent path-length distributions. I: Theory

Richard Vasques, Edward W. Larsen

Annals of Nuclear Energy, vol. 70(-), 2014 Aug, pp. 292-300

[P11] Numerical schemes for a non-classical linear Boltzmann equation for transport through spatially correlated media

Kai Krycki, Richard Vasques

Proceedings of NumHyp: Numerical approximations of hyperbolic systems with source terms and applications, Aachen, Germany, 2013 Sep

[P9] Estimating anisotropic diffusion of neutrons near the boundary of a pebble bed random system

Richard Vasques

Proceedings of International Conference on Mathematics and Computational Methods Applied to Nuclear Science & Engineering, Sun Valley, ID, 2013 May

[J3] A generalized linear Boltzmann equation for non-classical particle transport

Edward W. Larsen, Richard Vasques

Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 112(4), 2011, pp. 619-631

[P7] Anisotropic diffusion in model 2-D pebble-bed reactor cores

Richard Vasques, Edward W. Larsen

Proceedings of International Conference on Advances in Mathematics, Computational Methods, and Reactor Physics, Saratoga Springs, NY, 2009 May

[P6] Particle transport in the 1-D diffusive atomic mix limit

Edward W. Larsen, Richard Vasques, Marco T. Vilhena

Proceedings of Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications, Avignon, France, 2005 Sep

[P5] State of the art of particle transport theory in stochastic media

Richard Vasques, Marco T. Vilhena, Mark Thompson, Edward W. Larsen

Proceedings of XXV CILAMCE: Iberian Latin American Congress on Computational Methods in Engineering, Recife, Brazil, 2004 Nov