Thermal Fluctuations in Systems with Continuous Symmetry

Abstract

We investigate relaxation and thermal fluctuations in systems with continuous symmetry in arbitrary spatial dimensions. For the scalar order parameter ζ(r, t) with r∈ℛd, the deterministic relaxation is caused by hydrodynamic modes η∂ζ(r, t)/∂t= K∇2ζ(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→∞. We find that the second moment is well defined for dimensions d≥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" relaxation time. For dimensions d<4, these two properties suggest scale-invariant properties of the scalar field in the limit V→∞. We show that thermal fluctuations are described by fractional Brownian motion for d ≤ 3 and by ordinary Brownian motion for d ≥ 4. The spectral density of the stochastic force follows 1/f for d=1 and d=2, [Formula: see text] for d=3, and "white noise," f0 for d≥4. We find explicit representation of the equilibrium distribution of the conserved scalar field. For d≥4 it is a Gaussian distribution, while for d=1 and d=2, it is the Cauchy distribution.