Thermodynamic Limit of Time-Dependent Correlation Functions for One-Dimensional Systems

We investigate the time evolution of the correlation functions of a nonequilibrium system when the size of the system becomes very large. At the initial time t = 0, the system is represented by an equilibrium grand canonical ensemble with a Hamiltonian consisting of a kinetic energy part, a pairwise interaction potential energy between the particles, and an external potential. At time t = 0 the external field is turned off and the system is permitted to evolve under its internal Hamiltonian alone. Using the ``time‐evolution theorem'' for a 1‐dimensional system with bounded finite‐range pair forces, we prove the existence of infinite‐volume time‐dependent correlation functions for such systems, limρΛ(t;q1,p1;⋯;qn,pn), as Λ→∞, where Λ is the size of the finite system. We also show that these infinite‐volume correlation functions satisfy the infinite BBGKY hierarchy in the sense of distributions