The correlation time of mesoscopic chemical clocks

Abstract

A formula is proved for the correlation time of nonequilibrium chemical clocks in the presence of molecular noise. The correlation time is defined as the inverse of the damping rate of the autocorrelation functions of the chemical concentrations. Using the Hamilton–Jacobi method for stochastic systems as well as a Legendre transform from the Onsager–Machlup action to a reduced action depending only on the Hamilton–Jacobi pseudoenergy, the correlation time is given in the weak-noise limit in terms of the extensivity parameter, the period of oscillations, as well as the derivative of the period with respect to the pseudoenergy. Using this result, an estimation is obtained for the minimum number of molecules required for the oscillations of the chemical concentrations to remain correlated in time. This estimation puts a fundamental lower limit on the size of chemical clocks. For typical oscillators, the minimum number of molecules is estimated between ten and one hundred, which essentially corresponds to nanometric systems.