Approximate Hamiltonian Statistics in One-dimensional Driven Dissipative Many-Particle Systems

TL;DR

This paper derives the steady-state velocity and distance distributions of driven particles on a ring, showing that asymmetric interactions can approximate equilibrium distributions of Hamiltonian systems, confirmed by simulations.

Contribution

It demonstrates that steady-state distributions in driven dissipative systems can be approximated by Hamiltonian systems, even with asymmetric interactions.

Findings

Asymmetric interactions yield distributions similar to symmetric Hamiltonian systems.

Analytical results are confirmed by computer simulations.

The approach helps understand queueing system departure times.

Abstract

This contribution presents a derivation of the steady-state distribution of velocities and distances of driven particles on a onedimensional periodic ring. We will compare two different situations: (i) symmetrical interaction forces fulfilling Newton's law of "actio = reactio" and (ii) asymmetric, forwardly directed interactions as, for example in vehicular traffic. Surprisingly, the steady-state velocity and distance distributions for asymmetric interactions and driving terms agree with the equilibrium distributions of classical many-particle systems with symmetrical interactions, if the system is large enough. This analytical result is confirmed by computer simulations and establishes the possibility of approximating the steady state statistics in driven many-particle systems by Hamiltonian systems. Our finding is also useful to understand the various departure time distributions of queueing systems as a possible effect of interactions among the elements in the respective queue [D. Helbing et al., Physica A 363, 62 (2006)].