Chaos and its impact on the foundations of statistical mechanics

In this work we present a brief derivation of the periodic orbit expansion for simple dynamical systems, and then we apply it to the study of a classical statistical mechanical model, the Lorentz gas, both at equilibrium and in a nonequilibrium steady state. The results are compared with those obtained through standard molecular dynamics simulations, and they are found to be in good agreement. The form of the average using the periodic orbit expansion suggests the definition of a new dynamical partition function, which we test numerically. An analytic formula is obtained for the Lyapunov numbers of periodic orbits for the nonequilibrium Lorentz gas. Using this formula and other numerical techniques we study the nonequilibrium Lorentz gas as a dynamical system and obtain an estimate of the upper bound on the external field for which the system remains ergodic.