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

Martina Conte; Nadia Loy

doi:10.3934/krm.2026011

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2026, Volume 22: 197-243. Doi: 10.3934/krm.2026011

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

Martina Conte^, and
Nadia Loy

Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Italy

^*Corresponding author: Martina Conte
^*Corresponding author: Martina Conte

Received: October 2025

Published online: March 31, 2026

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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.

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Mathematics Subject Classification: Primary: 35Q20, 35Q70, 35Q84.

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Figure 1. Graphical representation of the stochastic trajectories

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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 (134)

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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 $

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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 $

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Table 1. Summary of the main symbols used throughout the manuscript

Symbol	Meaning
$ \tilde{v} $	Speed of motion.
$ \hat{v} $	Direction of motion.
$ v := \tilde{v} \hat{v} $	Velocity vector with magnitude $ \tilde{v} $ and direction $ \hat{v} $.
$ f(t,x, \tilde{v}, \hat{v}) $	Joint probability density function of $ ( \tilde{v}, \hat{v}) $ at $ (t,x)\in\mathbb{R}_+\times\mathbb{R}^d $.
$ \tilde{f}(t,x, \tilde{v}) $	Marginal distribution with respect to the speed $ \tilde{v} $.
$ \hat{f}(t,x, \hat{v}) $	Marginal distribution with respect to the direction $ \hat{v} $.
$ \tilde{\mu} $	Frequency associated with the speed–jump process.
$ \hat{\mu} $	Frequency associated with the direction–jump process.
$ \psi( \tilde{v}\| \hat{v}) $	Conditional transition probability density of the speed–jump process given direction $ \hat{v} $.
$ q( \hat{v}) $	Transition probability density of the direction–jump process.
$ \psi_q( \tilde{v}) $	Marginal equilibrium distribution of the speed, defined in (51).
$ \psi_q^c( \tilde{v}\| \hat{v}) $	Conditional stationary distribution of the speed given direction $ \hat{v} $, defined in (50).
$ T( \tilde{v}, \hat{v}) $	Transition probability density defining the equilibrium of $ \mathcal{L} $ in (31).

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