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, 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.
Zürcher, Ulrich, "Thermal Fluctuations in Systems with Continuous Symmetry" (1997). Physics Faculty Publications. 265.
Electronic version of an article published as Fractals, 5, 1, 1997 p. 87-93. 10.1142/S0218348X97000097] © 1997 World Scientific Publishing Company http://www.worldscientific.com/doi/abs/10.1142/S0218348X97000097