Abstract
We study response of liquid to a scale transformation, which generates a change of the liquid density, and obtain a set of differential equations for correlation functions. The set of equations, which we call density renormalization group equations (DRGEs), is similar to the BBGKY hierarchy as it relates different multiple-point correlation functions. In particular, we derive DRGEs for one-particle irreducible vertex functions of liquid by performing Legendre transformations, which enables us to calculate properties of liquid at higher density in terms of correlation functions at lower density.
Highlights
Thermodynamical properties of gas and liquid are described by the equation of state, and the microscopic derivation is one of the most important issues in the liquid theory
Liquid is microscopically described by a collection of interacting particles and the thermodynamical quantities such as pressure, internal energy, and isothermal compressibility can be calculated from density correlation functions [1, 2]
In order to understand the evolution of the thermodynamical quantities as an increase of the liquid density, we notice a similarity to the renormalization group (RG) in quantum field theories (QFTs)
Summary
Thermodynamical properties of gas and liquid are described by the equation of state, and the microscopic derivation is one of the most important issues in the liquid theory. Such analogies between statistical mechanics and quantum field theories have been occasionally pointed out, but our present study is strongly stimulated by Nambu’s seminar paper [24] in which an analogy between the renormalization group (RG) equation in gauge theories and thermodynamic equation of state was discussed. The purpose of the present paper is to make the analogy more concrete and to propose a set of (density) differential equations for correlation functions by studying response of the liquid to a scale transformation.
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