Barrier crossing driven by Levy noise: Universality and the Role of Noise Intensity

Aleksei V. Chechkin; Oleksii Yu. Sliusarenko; Ralf Metzler; Joseph Klafter

paper

Barrier crossing driven by Levy noise: Universality and the Role of Noise Intensity

arXiv:cond-mat/0612618

Abstract

We study the barrier crossing of a particle driven by white symmetric Levy noise of index $α$ and intensity $D D f or t h r ee d i f f er e n t g e n er i c t y p eso f p o t e n t ia l s : (a) abi s t ab l e p o t e n t ia l; (b) am e t a s t ab l e p o t e n t ia l; an d (c) a t r u n c a t e d ha r m o ni c p o t e n t ia l . F or t h e l o w n o i se in t e n s i t y r e g im e w er eco v er t h e p r e v i o u s l y p r o p ose d a l g e b r ai c d e p e n d e n ceo n$ D $o f t h ec ha r a c t er i s t i cesc a p e t im e,$ T_{\mathrm{esc}}\simeq C(α)/D^{μ(α)} $, w h er e$ C(α) $i s a coe f f i c i e n t . I t i ss h o w n t ha tt h ee x p o n e n t$ μ(α) $r e main s a pp r o x ima t e l y co n s t an t,$ μ\approx 1 $f or$ 0<α<2 $; a t$ α=2 $t h e p o w er - l a w f or m o f$ T_{\mathrm{esc}} $c han g es in t o t h e k n o w n e x p o n e n t ia l d e p e n d e n ceo n 1/ D; i t e x hibi t s a d i v er g e n ce - l ik e b e ha v i or a s$ α $a pp r o a c h es 2. I n t hi sr e g im e w eo b ser v e am o n o t o n o u s in cr e a seo f t h eesc a p e t im e$ T_{\mathrm{esc}} $w i t hin cr e a s in g$ α $(k ee p in g t h e n o i se in t e n s i t y$ D $co n s t an t) . T h e p r o babi l i t y d e n s i t y o f t h eesc a p e t im e d ec a y se x p o n e n t ia l l y . I na dd i t i o n, f or l o w n o i se in t e n s i t i es t h eesc a p e t im escor r es p o n d t o ba r r i er cr oss in g b y m u l t i pl e L e v y s t e p s . F or hi g hn o i se in t e n s i t i es, t h eesc a p e t im ec u r v esco l l a p se f or a l l v a l u eso f$ α $. A t in t er m e d ia t e n o i se in t e n s i t i es, t h eesc a p e t im ee x hibi t s n o n - m o n o t o ni c d e p e n d e n ceo n t h e in d e x$ α$, while still retaining the exponential form of the escape time density.

12 pages, 8 figures, REVTeX4