The diffusion equation \partial_t\phi = \nabla^2\phi is considered, with initial condition \phi( _x_ ,0) a gaussian random variable with zero mean. Using a simple approximate theory we show that the probability p_n(t_1,t_2) that \phi( _x_ ,t) [for a given space point _x_ ] changes sign n times between t_1 and t_2 has the asymptotic form p_n(t_1,t_2) \sim [\ln(t_2/t_1)]^n(t_1/t_2)^{-\theta}. The exponent \theta has predicted values 0.1203, 0.1862, 0.2358 in dimensions d=1,2,3, in remarkably good agreement with simulation results.Comment: Minor typos corrected, affecting table of exponents. 4 pages, REVTEX, 1 eps figure. Uses epsf.sty and multicol.st

Numerical and analytic results for the exponent θ describing the decay of the first return probability of an interface to its initial height are obtained for a large class of linear Langevin equations. The models are parametrized by the dynamic roughness exponent β, with 0 < β < 1; for β = 1/2 the time evolution is Markovian. Using simulations of solid-on-solid models, of the discretized continuum equations as well as of the associated zero-dimensional stationary Gaussian process, we address two problems: The return of an initially flat interface, and the return to an initial state with fully developed steady state roughness. The two problems are shown to be governed by different exponents. For the steady state case we point out the equivalence to fractional Brownian motion, which has a return exponent θS = 1 − β. The exponent θ0 for the flat initial condition appears to be nontrivial. We prove that θ0 → ∞ for β → 0, θ0 ≥ θS for β < 1/2 and θ0 ≤ θS for β > 1/2, and calculate θ0,S perturbatively to first order in an expansion around the Markovian case β = 1/2. Using the exact result θS = 1 − β, accurate upper and lower bounds on θ0 can be derived which show, in particular, that θ0 ≥ (1 − β) 2 /β for small β.

We study the thermodynamics and the collapse of a self-gravitating gas of Brownian particles. We consider a high-friction limit in order to simplify the problem. This results in the Smoluchowski-Poisson system. Below a critical energy or below a critical temperature, there is no equilibrium state and the system develops a self-similar collapse leading to a finite time singularity. In the microcanonical ensemble, this corresponds to a "gravothermal catastrophe" and in the canonical ensemble to an "isothermal collapse." Self-similar solutions are investigated analytically and numerically.