Quasiinvariants of the motion and existence of the $\varepsilon$-limit in the nonequilibrium statistical operator method

In the framework of the axiomatic approach to the thermodynamic limit developed by Ruelle [6] and Haag et al. [7], an investigation is made of the existence of a nonequilibrium stationary state generated by a retarded solution of the Liouville equation, i.e., of the limit as $\varepsilon\to+0$ of states generated by quasiinvariants of the motion obtained by causal smoothing of the coarse-grained statistical operator [2, 3]. It is shown that the $\varepsilon$-limit exists if the coarse-grained state and the operators of time evolution of the variables at positive times in the thermodynamic limit satisfy a definite condition, which is intimately related to the condition of correlation weakening. The proof is based on the use of the $n$-quasiinvariants of the motion [3] and the Yosida–Kakutani ergodic theorem.