THERMAL FLUCTUATIONS IN SYSTEMS WITH CONTINUOUS SYMMETRY - BROWNIAN-MOTION AND LEVY-FLIGHT DESCRIPTION

We investigate relaxation and thermal fluctuations in systems with con tinuous symmetry in arbitrary spatial dimensions. For the scalar order parameter zeta(r, t) with r is an element of R-d, the deterministic r elaxation is caused by hydrodynamic modes eta partial derivative zeta( r, t)/partial derivative t = K del(2) zeta(r, t). For a finite volume V, we expand the scalar field in a discrete Fourier series and then we study the behavior in the limit V --> infinity. We find that the seco nd moment is well defined for dimensions d greater than or equal to 3, while it diverges for d = 1, 2. Furthermore, we show that for d < 4, the decay of the scalar field does not define an ''effective'' relaxat ion time. For dimensions d < 4, these two properties suggest scale-inv ariant properties of the scalar field in the limit V --> infinity. We show that thermal fluctuations are described by fractional Brownian mo tion for d less than or equal to 3 and by ordinary Brownian motion for d greater than or equal to 4. The spectral density of the stochastic force follows 1/f for d = 1 and d = 2, 1/root f for d = 3, and ''white noise,'' f(0) for d greater than or equal to 4. We find explicit repr esentation of the equilibrium distribution of the conserved scalar fie ld. For d greater than or equal to 4 it is a Gaussian distribution, wh ile for d = 1 and d = 2, it is the Cauchy distribution.