Statistical mechanics of elastica on a plane: origin of the MKdV hierarchy

Abstract

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.

On density of state of quantized Willmore surface - a way to a quantized extrinsic string in

“…While the Willmore surface is related to the modified Novikov-Veselov (MNV) equation, the elastica is related to the modified KdV equation [1][2][3][4][5][6][7][25][26][27]. Recently I exactly quantized the elastica of the Bernoulli-Euler functional (1-4) preserving its local length [25]. Then I found that its moduli is completely represented by the MKdV equation and closely related to the two-dimensional quantum gravity [28][29][30].…”

“…Then I found that its moduli is completely represented by the MKdV equation and closely related to the two-dimensional quantum gravity [28][29][30]. The quantized elastica obeys the MKdV equation and at a critical point, the Painlevé equation of the first kind appears [25] while in the quantized two-dimensional gravity which is defined at a critical point of the discrete tiling model, there appears the Painlevé equation of the first kind with the KdV hierarchy [28][29][30].…”

“…Section 2 reviews the argument of the quantized elastica following to that in ref. [25]. In §3, I will quantize the Willmore surface and then the density of states of the Willmore functional is given as the volume of the MNV equation.…”

“…I will fix the metric of the curve C induced from the natural metric of C; ds = √ dXdX. The Frenet-Serret relations are expressed as [6,[25][26][27] …”

“…I will consider quantization of a closed elastica whose local length preserves in the quantization process. The partition function of the elastica is given as [25],…”