EINSTEIN RELATION FOR BROWNIAN-MOTION OF MEMBRANES

We study form fluctuations of topologically nontrivial membranes, spon ges for instance, assuming that there is a principal mode described by Langevin's equation, which has a character of chaotic relaxation to t he equilibrium position. The influence of the ambient liquid being tak en into account by the relaxation coefficients and the source of noise . The chaotic change of the surface is characterized by the quantity D elta(2) (similar to the activity of rotational Brownian motion), which satisfies Einstein's equation Delta(2)/t = gammaIT, where t is the ti me, I a factor depending on the form of the membrane, and gamma the di ssipative 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 co rrelation functions, and develop the theory of perturbations by the cu rvature of the surface.