Fluctuation-dissipation theorem for chemical reactions near a critical point

In a fluid mixture near a critical point, there are long-range fluctuations in the component concentrations that exceed the range of the intermolecular forces. If the components are linked by a chemical reaction, then the fluctuations in the concentrations of the reactants and products have their origin in the fluctuation in the extent of reaction xi. The fluctuation in xi about the position of chemical equilibrium can be expressed by the statistical variance, var(xi(e)), where the subscript "e" denotes equilibrium. We show that var(xi(e)) is inversely proportional to (partial derivativeDeltaG/partial derivativexi)(e), where DeltaG is the Gibbs energy difference separating products from reactants. Because the relaxation time, tau, that governs the rate of approach of the reaction to equilibrium is also inversely proportional to (partial derivativeDeltaG/partial derivativexi)(e), tau is proportional to var(xi(e)). This latter relation constitutes a fluctuation-dissipation theorem. Under circumstances near a critical point where var(xi(e)) --> infinity, the theorem predicts that the specific relaxation rate 1/tau, should go to zero and that the rate of approach of the reaction to chemical equilibrium should slow.