Digital Object Identifier (DOI) 10.1007/s00220-002-0765-5 Commun. Math. Phys. 233, 545–569 (2003) Communications in Mathematical Physics On the Motion of a Charged Particle Interacting with an Infinitely Extended System * Paolo Butt` a, Emanuele Caglioti, Carlo Marchioro Dipartimento di Matematica, Universit` a di Roma ‘La Sapienza’, P.leAldo Moro 2, 00185 Roma, Italy. E-mail: butta@mat.uniroma1.it; caglioti@mat.uniroma1.it; marchior@mat.uniroma1.it Received: 7 March 2002 / Accepted: 23 September 2002 Published online: 8 January 2003 – © Springer-Verlag 2003 Abstract: We study the time evolution of a charged particle moving in a medium under the action of a constant electric field E. In the framework of fully Hamiltonian mod- els, we discuss conditions on the particle/medium interaction which are necessary for the particle to reach a finite limit velocity. We first consider the case when the charged particle is confined in an unbounded tube of R 3 . The electric field E is directed along the symmetry axis of the tube and the particle also interacts with an infinitely many particle system. The background system initial conditions are chosen in a set which is typical for any reasonable thermodynamic (equilibrium or non-equilibrium) state. We prove that, for large E and bounded interactions between the charged particle and the background, the velocity v(t) of the charged particle does not reach a finite limit velocity, but it increases to infinite as: |v(t) - Et |≤ C 0 (1 + t), where C 0 is a constant independent of E. As a corollary we obtain that, if the initial conditions of the background system are distributed according to any Gibbs state, then the average velocity of the charged particle diverges as time goes to infinite. This result is obtained for E large enough in comparison with the mean energy of the Gibbs state. We next study the one-dimensional case, in which the estimates can be improved. We finally discuss, at an heuristic level, the existence of a finite limit velocity for unbounded interactions, and give some suggestions about the case of small electric fields. 1. Introduction In the present paper we study the time evolution of a charged particle moving in a me- dium under the action of a constant electric field E. If the particle moves in the vacuum the motion is trivial and the velocity of the particle grows linearly in time. Our goal is to discuss some conditions on the interaction between the charged particle and the medium under which the former may reach a bounded asymptotic velocity. In kinetic theory it is ∗ Work partially supported by the GNFM-INDAM and the Italian Ministry of the University.