Statistical Mechanics of Elastica on Plane as a Model of Supercoiled DNA ——Origin of the MKdV hierarchy——-

In this article, I have investigated statistical mechanics of a non-stretched elastica in two dimensional space using path integral method. In the calculation, the MKdV hierarchy naturally appeared as the equations including the temperature fluctuation.I have classified the moduli of the closed elastica in heat bath and summed the Boltzmann weight with the thermalfluctuation over the moduli. Due to the bilinearity of the energy functional,I have obtained its exact partition function.By investigation of the system,I conjectured that an expectation value at a critical point of this system obeys the Painlevé equation of the first kind and its related equations extended by the KdV hierarchy.Furthermore I also commented onthe relation between the MKdV hierarchy and BRS transformationin this system. Elastica has sometimes appeared in the history of mathematical physics according to refs.[1-4]. The problem of elastica, an ideal thin elastic rod, was proposed by James Bernoulli in 1691. By investigation on the behavior of an elastica, Bernoulli's family and their related people, Euler, d'Alembert and so on, discovered many non-trivial mathematical and physical facts, e.g., classical field theory, minimal principle, elliptic function, mode analysis, non-linear differential equation and others [1-4]. In fact, James Bernoulli derived the elliptic integral related to the lemniscate function in 1694, before Fagnano found his lemniscate function [1-3], and found that the force of elastica is proportional to inverse of its curvature radius [1]. His nephew, Daniel Bernoulli followed James's discoveries and discovered the energy functional of elastica and its minimal principle around 1738. Succeeding Daniel's and James's discoveries, Euler derived elliptic integral of general modulus as a shape of elastica using Daniel's minimal principle and numerically integrated it. Then he completely classified shapes of static elasticas by numerical sketch [1], which are, nowaday, known as special cases of loop soliton [5]. In the computations, Euler used the static sine-Gordon equation. These computations essentially imply discovery of the integrable nonlinear differential equation, the investigation of its moduli and first application of algebro-geometrical functions to physics. It should be noted that they came from the studies on the elastica. Furthermore it is well-known that the elastica problem is the simplest prototype σ-model S 1 → SO(2) or SO(3) [6,7], which was found in 18th century and investigated by Kirchhoff in last century [4]. Thus the elastica problem sounds to be legacy before last century but I believe that its properties are not completely understood and its role in …