Nontrivial Exponent for Simple Diffusion

Abstract

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

Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation

“…This implies that in the L → ∞ limit, p 0 (t) ≡ p 0 (t, L → ∞) ∼ t −θ(d) for large t. It was shown in Ref. [6] that the probability P 0 (T ) that a Gaussian stationary process (GSP) with zero mean and correlations [cosh(T /2)] −d/2 decays for large T as P 0 (T ) ∼ exp [−θ(d)T ] where θ(d) is the same as the persistence exponent in diffusion equation. This exponent θ(d) was measured in numerical simulations [6,7], yielding for instance θ sim (1) = 0.12050(5), θ sim (2) = 0.1875 (1).…”

“…In the literature, they are sometimes called SO(2) random polynomials because their m-point joint probability distribution of zeros is SO(2) invariant for all m [20]. We will show below that the gap probabilities for these classes of random polynomials (6,7) are closely related to the persistence probability for the diffusion equation in the limit of large dimension. Our main results, together with the layout of the paper, are summarized below.…”

“…In the scaling limit t ≫ 1, L ≫ 1 keeping the ratio t/L 2 fixed, it was found in Ref. [6] that p 0 (t, L) takes the scaling form…”

“…[6] that the probability P 0 (T ) that a Gaussian stationary process (GSP) with zero mean and correlations [cosh(T /2)] −d/2 decays for large T as P 0 (T ) ∼ exp [−θ(d)T ] where θ(d) is the same as the persistence exponent in diffusion equation. This exponent θ(d) was measured in numerical simulations [6,7], yielding for instance θ sim (1) = 0.12050(5), θ sim (2) = 0.1875 (1). The case of dimension d = 1 is particularly interesting because θ(1) was determined experimentally using NMR techniques to measure the magnetization of spin polarized Xe gas [5], yielding θ exp (1) = 0.12 in good agreement with numerical simulations.…”

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“…This problem falls in the class of the so-called "persistence phenomenon" discovered in a large variety of systems [11,12], and which specifies how long a relaxing dynamical system remains in a neighborhood of its initial configuration. For a Gaussian process, the persistence exponent x can be shown to be a functional of the two-point temporal correlator [11,12,13]. For Markovian or weakly non-Markovian random walk processes, the exponent x and therefore b is close to 1/2, as we find empirically.…”