---
abstract: 'We introduce a gradient flow formulation of linear Boltzmann equations. Under a diffusive scaling we derive a diffusion equation by using the machinery of gradient flows.'
address:
- 'Giada Basile Dipartimento di Matematica, Università di Roma ‘La Sapienza’ P.le Aldo Moro 2, 00185 Roma, Italy'
- 'Dario Benedetto Dipartimento di Matematica, Università di Roma ‘La Sapienza’ P.le Aldo Moro 2, 00185 Roma, Italy'
- 'Lorenzo Bertini Dipartimento di Matematica, Università di Roma ‘La Sapienza’ P.le Aldo Moro 2, 00185 Roma, Italy'
author:
- Giada Basile
- Dario Benedetto
- Lorenzo Bertini
title: A gradient flow approach to linear Boltzmann equations
---

Introduction
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The Boltzmann equation describes the evolution of the one-particle distribution on position and velocity of a rarefied gas. It has become a paradigmatic equation since it encodes most of the conceptual and technical issues in the description of the statistical properties for out of equilibrium systems. In particular, from a mathematical point of view, a global existence and uniqueness result is still lacking. In the kinetic regime, some transport phenomena can be described by linear Boltzmann equations. Typical examples are the charge (or mass) transport in the Lorentz gas [@Lo], the evolution of a tagged 