Energy dissipation in the deterministic and nondeterministic Nagel–Schreckenberg models

In this paper, we numerically study energy dissipation per unit time per car Ed in the Nagel–Schreckenberg (NS) model. Numerical results show that there is a critical density over which energy dissipation occurs, but below which no energy loss happens in the deterministic NS model. Energy dissipation depends on both the car density and initial configuration in the deterministic case. The energy dissipation rate, Ed, calculated starting from a completely jammed state whose value is minimum, decreases monotonously with the increase of car density above the critical density. In the nondeterministic case, however, there is no critical density and energy dissipation happens in the whole density region, and initial configuration has no effects on energy dissipation. The dissipated energy has two different contributions: one coming from the interactions and another from the braking noise in the stochastic case. The relative contributions of the two dissipation mechanisms are presented. In the free-flow state, Ed is proportional to p(1−p) where p is the stochastic braking probability. In the case of vmax=1,Ed is proportional to the mean density of “go and stop” car per time step ρgs, which is equal to n0(1−n0) where n0 is the fraction of stopped car. Theoretical analyses give an excellent agreement with numerical results.