La relation d'Einstein pour le mouvement brownien des membranes

We study form fluctuations of topologically nontrivial membranes, sponges for instance, assuming that there is a principal mode described by Langevin's equation, which has a character of chaotic relaxation to the equilibrium position. The influence of the ambient liquid being taken into account by the relaxation coefficients and the source of noise. The chaotic change of the surface is characterized by the quantity Δ2 (similar to the activity of rotational Brownian motion), which satisfies Einstein's equation Δ2/t = γIT, where t is the time, I a factor depending on the form of the membrane, and γ the dissipative constant. Fluctuations are studied by using the chiral field that is obtained from the Gauss–Weingarten local frame, usual in the classical theory of surfaces. Thus, we determine the effective action for a chiral field having a supersymmetric structure, we derive the correlation functions, and develop the theory of perturbations by the curvature of the surface.[Journal translation]