For a nonchemical-equilibrium state of an isolated system A that has r constituents with initial amounts na = {n1a, n2a, …, nra}, and that is subject to τ chemical reaction mechanisms, temperature, pressure, and chemical potentials cannot be defined. As time evolves, the values of the amounts of constitutents vary according to the stoichiometric relations ni(t) = nia + Σj=1τ νi(j) εj(t), where νi(j) is the stoichiometric coefficient of the ith constituent in the j-reaction mechanism and εj(t) the reaction coordinate of the jth reaction at time t. For such a state, we approximate the values of all the properties at time t with the corresponding properties of the stable equilibrium state of a surrogate system B consisting of the same constituents as A with amounts equal to ni(t) for i = 1, 2, …, r, but experiencing no chemical reactions. Under this approximation, the rate of entropy generation is given by the expression S˙irr = ε˙ · Y, where ε˙ is the row vector of the τ rates of change of the reaction coordinates, ε˙ = { ε˙1, …, ε˙τ }, Y the column vector of the τ ratios aj/Toff for j = 1, 2, …, τ, aj = −Σi=1r νi(j) μi,off, that is, the jth affinity of the stable equilibrium state of the surrogate system B, and μi,off, and Toff are the chemical potential of the ith constituent and the temperature of the stable equilibrium state of the surrogate system. Under the same approximation, by further assuming that ε˙ can be represented as a function of Y only that is, ε˙(Y), with ε˙(0) = 0 for chemical equilibrium, we show that ε˙ = L·Y + (higher order terms in Y), where L is a τ × τ matrix that must be non-negative definite and symmetric, that is, such that the matrix elements Lij satisfy the Onsager reciprocal relations, Lij = Lji. It is noteworthy that, for the first time, the Onsager relations are proven without reference to microscopic reversibility. In our view, if a process is irreversible, microscopic reversibility does not exist.
Finite-dimensional maps describing the dynamics of a logistic equation with delay
