Critical Dynamics of Dilute Helium Mixture —Equation of Motion Approach with Mode Coupling—

Abstract

A simplified method of mode coupling equation is applied to the critical dynamics near the lambda line of dilute solution of ³He in ⁴He. The order parameter, the ³He concentration, and the entropy are the relevant modes as discussed by Siggia and Nelson. Using the concept of the self-consistent self-energy, developed by Ferrell and Bhattacharjee for ⁴He, we obtain the characteristic frequencies of these modes, the fluctuation spectrum, and the damping constant. The self-consistent self-energies satisfy the dynamic scaling hypothesis and the exponent z turns out to be d⁄2, d being the dimension of the system. The ³He concentration dependence of the amplitudes of the self-energies is found to be very peculiar. The fluctuation spectrum is shown to depend on the amplitude ratios very much. It never takes the double humped form.