Nontrivial Exponent for Simple Diffusion

Satya N. Majumdar (1) , Clément Sire (2) , Alan J. Bray (3) , Stephen J. Cornell (3)

1 AT&T Bell Laboratories
2 GPT-LPQ - Groupe de Physique Théorique (LPQ)
3 Department of Theoretical Physics [Manchester]

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.

Mots clés: diffusion equation; random processes; coarsening; spin system; phase separation; Ising model; critical exponent; phase transition; persistance; équation de la diffusion; processus aléatoire; retour à l'équilibre; séparation de phase; modèle d'Ising; exposant critique; transition de phase; Persistence

Domaines: Autre [cond-mat.other]

Satya N. Majumdar, Clément Sire, Alan J. Bray, Stephen J. Cornell. Nontrivial Exponent for Simple Diffusion. Physical Review Letters, 1996, 77, pp.2867-2870. ⟨10.1103/PhysRevLett.77.2867⟩. ⟨hal-00004855⟩