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A novel linear transport model with distinct scattering mechanisms for direction and speed

Martina Conte, Nadia Loy

TL;DR

The paper addresses a gap in kinetic theory by modeling linear transport with two distinct scattering mechanisms for speed and direction, whose gains depend on time-evolving marginals rather than a fixed equilibrium. It develops a kinetic equation with $\mathcal{L}=\hat{\mu}\hat{\mathcal{L}}+\tilde{\mu}\tilde{\mathcal{L}}$, analyzes kernel structures, stationary states, and non-standard pseudo-inverses in mixed Hilbert spaces, and proves entropy decay in a spatially homogeneous setting for the conditioned-speed case. Through Chapman-Enskog and multi-scale analyses, it derives macroscopic limits under equal and unequal time-scale couplings, revealing new drift-diffusion equations that reflect the interplay between speed and direction relaxation. Numerical tests illustrate how the decoupled speeds and directions yield richer, less isotropic spreading patterns than the classical linear Boltzmann model, with potential applications to cell migration in structured environments and traffic flow.

Abstract

We introduce a novel linear transport equation that models the evolution of a one-particle distribution subject to free transport and two distinct scattering mechanisms: one affecting the particle's speed and the other its direction. These scattering processes occur at different time scales and with different intensities, leading to a kinetic equation where the total scattering operator is the sum of two separate operators. Each of them depends not only on the kernel characterizing the corresponding scattering mechanism, but also explicitly on the marginal distribution of either the speed or the direction. Therefore, unlike classical settings, the gain terms in our operators are not tied to a fixed equilibrium distribution but evolve in time through the marginals. As a result, typical analytical tools from kinetic theory, such as equilibrium characterization, entropy methods, spectral analysis in Hilbert spaces, and Fredholm theory, are not applicable in a standard fashion. In this work, we rigorously analyze the properties of this new class of scattering operators, including the structure of their non-standard pseudo-inverses and their asymptotic behavior. We also derive macroscopic (hydrodynamic) limits under different regimes of scattering frequencies, revealing new effective equations and highlighting the interplay between speed and directional relaxation.

A novel linear transport model with distinct scattering mechanisms for direction and speed

TL;DR

The paper addresses a gap in kinetic theory by modeling linear transport with two distinct scattering mechanisms for speed and direction, whose gains depend on time-evolving marginals rather than a fixed equilibrium. It develops a kinetic equation with , analyzes kernel structures, stationary states, and non-standard pseudo-inverses in mixed Hilbert spaces, and proves entropy decay in a spatially homogeneous setting for the conditioned-speed case. Through Chapman-Enskog and multi-scale analyses, it derives macroscopic limits under equal and unequal time-scale couplings, revealing new drift-diffusion equations that reflect the interplay between speed and direction relaxation. Numerical tests illustrate how the decoupled speeds and directions yield richer, less isotropic spreading patterns than the classical linear Boltzmann model, with potential applications to cell migration in structured environments and traffic flow.

Abstract

We introduce a novel linear transport equation that models the evolution of a one-particle distribution subject to free transport and two distinct scattering mechanisms: one affecting the particle's speed and the other its direction. These scattering processes occur at different time scales and with different intensities, leading to a kinetic equation where the total scattering operator is the sum of two separate operators. Each of them depends not only on the kernel characterizing the corresponding scattering mechanism, but also explicitly on the marginal distribution of either the speed or the direction. Therefore, unlike classical settings, the gain terms in our operators are not tied to a fixed equilibrium distribution but evolve in time through the marginals. As a result, typical analytical tools from kinetic theory, such as equilibrium characterization, entropy methods, spectral analysis in Hilbert spaces, and Fredholm theory, are not applicable in a standard fashion. In this work, we rigorously analyze the properties of this new class of scattering operators, including the structure of their non-standard pseudo-inverses and their asymptotic behavior. We also derive macroscopic (hydrodynamic) limits under different regimes of scattering frequencies, revealing new effective equations and highlighting the interplay between speed and directional relaxation.

Paper Structure

This paper contains 21 sections, 4 theorems, 242 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Microscopic stochastic description
  3. Formal derivation from discrete-time stochastic processes
  4. The strong kinetic equation
  5. Insight on the stochastic trajectories
  6. Investigation of the operators
  7. Kernels and stationary states
  8. Hilbert spaces and pseudo-inverse operators
  9. Relaxation to Equilibrium in the Spatially Homogeneous Case
  10. Macroscopic limits
  11. Preliminaries: notation and the classical case
  12. Jump processes with same order frequencies
  13. Jump operators with different order frequencies
  14. Comparison of the macroscopic limiting equations
  15. Numerical tests
  16. ...and 6 more sections

Key Result

Proposition 3.1

Let us consider $\psi$ satisfying norm.psi,psi.L1, $q$ satisfying norm.q,q.L1, and the operator $\mathcal{L}$ defined by def:L. We have that where being where Moreover and

Figures (4)

  • Figure 1:
  • Figure 2: Test 1. Evolution of the macroscopic density $\rho$ in the three cases (K1), (K2), and (K3) at time $T=1.88$ and starting from the initial distribution defined in \ref{['rho0']}.
  • Figure 3: Test 2.1. Evolution of the macroscopic density $\rho$ in the settings (M1), (M2), (M3), and (M4). Precisely, (M1) and (M2) are shown at time $T=1.25$, (M3) at time $T=4.69$, and (M4) at time $T=0.47$. For all the simulations, we set $\mu=\tilde{\mu}=\hat{\mu}=1$.
  • Figure 4: Test 2.2. Evolution of the macroscopic density $\rho$ in the settings (M1), (M2), (M3), and (M4). Precisely, (M1) and (M2) are shown at time $T=1.875$, (M3) at time $T=5.94$, and (M4) at time $T=0.1875$. For all the simulations, we set $\mu=\tilde{\mu}=\hat{\mu}=1$.

Theorems & Definitions (10)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.1
  • Proposition 3.3
  • proof
  • Theorem 3.1
  • proof
  • Remark 4.1