Temporal fluctuations and hurst exponents of goldstone and massive modes

We elucidate the connection of the Hurst exponent H and power spectra observed in recent experimental and numerical studies of liquid crystals and the Heisenberg model with the more familiar Langevin description. In agreement with empirical results, H=1 for Goldstone modes and H=12 for massive modes. We discuss the crossover phenomenon when a symmetry-breaking field is introduced. We relate the Hurst exponent at the critical point to the static and dynamic critical exponents. We confirm our conclusions by an explicit calculation for an exactly solvable model and discuss possible applications.