4 - Free entropy and equilibrium

Summary

Abstract

In this chapter we are mainly concerned with one-dimensional electrostatic problems; that is, with measures on the circle or the real line that represent charge distributions subject to logarithmic interaction and an external potential field. First we consider configurations of electrical charges on the circle and their equilibrium configuration. Then we review some classical results of function theory and introduce the notion of free entropy for suitable probability densities on the circle; these ideas extend naturally to spheres in Euclidean space. The next step is to introduce free entropy for probability distributions on the real line, and show that an equilibrium distribution exists for a very general class of potentials. For uniformly convex potentials, we present an effective method for computing the equilibrium distribution, and illustrate this by introducing the semicircle law. Then we present explicit formulæ for the equilibrium measures for quartic potentials with positive and negative leading term. Finally we introduce McCann's notion of displacement convexity for energy functionals, and show that uniform convexity of the potential implies a transportation inequality.

Logarithmic energy and equilibrium measure

Suppose that N unit positive charges of strength β > 0 are placed upon a circular conductor of unit radius, and that the angles of the charges are 0 ≤ θ₁ < θ₂ < … < θ_N < 2π.