Derivation of the Onsager symmetry in many textbooks and papers is as follows: First, assume that the correlation function of two state variables,##a_i## and ##a_j## satifsies for sufficiently small time interval ##t## that
$$\langle a_i(t) a_j(0) \rangle = \langle a_i(-t) a_j(0) \rangle =...