Volume 136, number 7,8 PHYSICS LETTERS A 17 April 1989 MEAN EXIT TIMES OVER FLUCTUATING BARRIERS D.L. STEIN Department ofPhisles, University ofAri:ona, Tucson, AZ 85721, USA R.G. PALMER Department of Phi’sics, Duke University, Durham, NC 27706, USA LL. VAN HEMMEN Universität Heidelberg, Sonderforschungbereich 123, 6900 Heidelberg 1, ERG and Charles R. DOERING Department ofPhvsics, Clarkson University, Potsdam, NY 13676, USA Received 22 January 1989; accepted for publication 1 February 1989 Communicated by A.R. Bishop We investigate the problem of thermal activation over a fluctuating barrier. Three regimes are considered: the fluctuations slow compared to the mean crossing time tA of the average barrier height, fluctuations on roughly the same timescale as ~. and fluc- tuations extremely fast compared to ~A• In the latter two cases, the mean barrier crossing time is reduced. The relevance of these results to a variety of problems in complex systems is discussed. The problem of the dynamical evolution of highly these systems — their dynamics is governed by many constrained, strongly interacting many-body systems relevant timescales, from very fast to very slow, and with multiple locally stable states occurs in a wide often no obvious timescale gaps exist so that one can variety of contexts, and has often served as a unify- separate the “fast” from the “slow” degrees of free- ing framework for studying the behavior of physi- dom. It therefore makes sense to expect that during cally diverse systems. Some examples include slow a typical barrier crossing event the barrier itself does relaxation in glasses [1—4],chemical kinetics in large not remain static; it will likely be modulated by some biomolecules [5—7], and biological evolution [8— strongly coupled degrees of freedom which are 121. All involve barrier crossing in some form, and changing on a timescale comparable to the crossing many display complex behavior such as nonexpo- time itself. In such cases one must consider the prob- nential relaxation in time or non-Arrhenius temper- lem of activation over a fluctuating barrier. ature dependence of crossing rates. In some sense this Some simple examples may clarify the picture we is not surprising in view of the multiple barrier cross- have in mind. In many cases, the difficulty arises ings needed to achieve relaxation of some pertur- from constraints [131 — e.g., ina glassy liquid, a given bation; however, theories based on multiple hops atom A may be blocked from relaxing or diffusing over static barriers have not satisfactorily explained because of the presence of other atoms B, C, ... “in may of the observed phenomena. the way”. When enough of these atoms happen to There is an additional complication, which in fact move away, opening up a channel or lowering a bar- is crucial in determining the complex behavior of rier, atom A is able to move. This can actually be 0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. 353 (North-Holland Physics Publishing Division)