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

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)].

💡 Research Summary

The paper investigates the steady‑state statistical properties of a one‑dimensional driven‑dissipative many‑particle system placed on a periodic ring. Each particle i is described by its position xi and velocity vi and obeys a Langevin‑type equation that includes a constant driving force F0, a linear friction term γvi, stochastic white‑noise ξi(t) with strength D, and interaction forces fij with neighboring particles. Two interaction scenarios are examined. In the first (symmetric) case the forces derive from a pair potential U(s) (s = xi−xj) and satisfy Newton’s third law, i.e., fij = −∂U/∂s = −fji. In the second (asymmetric) case the interaction is forward‑only, mimicking vehicular traffic: a particle feels a repulsive force only from the particle directly ahead, fij = −∂U/∂(xi−x_{i+1}), while fji = 0. This asymmetry breaks the usual Hamiltonian structure and introduces a non‑conservative, dissipative dynamics.

Starting from the Langevin equations, the authors derive the corresponding Fokker‑Planck equation for the joint probability density P({xi},{vi},t). By assuming a large number of particles (N≫1) and that distance fluctuations around the mean spacing (\bar{s}=L/N) are small, they show that the stationary solution factorizes into independent single‑particle velocity and gap distributions: \