Correlation-Function Approach to the Transport Coefficients near the Critical Point. I

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Abstract
A general theory is presented for the transport coefficients exhibiting anomalous peaks near the critical point, employing correlation-function expressions for transport coefficients. Recognizing that the anomaly should arise from the anomalous increase in the fluctuations of certain macroscopic variables, we attempt an expansion of the flux entering the correlation-function expression in powers of the macroscopic variables, which then are supposed to obey the macroscopic equations of motion. A general formula for the anomalous part of the transport coefficient is given, restricting ourselves to the quadratic terms in this expansion. The general theory is illustrated for the shear viscosity of critical mixtures, choosing local concentration and local temperature as macroscopic variables. The anomaly is attributed to the cooperation of the two effects: (1) anomalous increase in certain large-scale fluctuations of macroscopic variables contained in the flux, and (2) the anomalous increase in the lifetimes associated with these fluctuations. If we ignore the local temperature fluctuations, Fixman's result of the anomalous viscosity is obtained. Generalizing these results, the large frequency and wave-vector dependence which is expected for the anomalous transport coefficients near the critical point is studied explicitly for the shear viscosity. The thermal conductivity of the critical mixture is also examined, and is found to have no anomaly in the same approximation, in agreement with the existing experiments. In an appendix, a more rigorous and systematic treatment of the general theory is given with the help of Mori's general theory of Brownian motion for macroscopic variables, and we indicate a possibility of obtaining a self-consistent set of equations for general nonlocal transport coefficients.