INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER J. Phys.: Condens. Matter 19 (2007) 065145 (12pp) doi:10.1088/0953-8984/19/6/065145 Asymptotics of rare events in birth–death processes bypassing the exact solutions Charles R Doering 1 , Khachik V Sargsyan 1 , Leonard M Sander 2 and Eric Vanden-Eijnden 3 1 Department of Mathematics and Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109-1043, USA 2 Department of Physics and Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109-1120, USA 3 Courant Institute, New York University, New York, NY 10012, USA E-mail: doering@umich.edu, ksargsya@umich.edu, lsander@umich.edu and eve2@cims.nyu.edu Received 4 September 2006 Published 22 January 2007 Online at stacks.iop.org/JPhysCM/19/065145 Abstract We investigate the near-continuum asymptotics of mean first passage times in some one-variable birth–death processes. The particular problem we address is how to extract mean first passage times in the near-continuum limit from their defining finite-difference equations alone. For the simple class of processes we consider here, exact closed-form solutions for the mean first passage time between any two states are available and the near-continuum expansion of these formulae defines the correct limiting behaviour and is used to check the results of asymptotic analysis of the difference equations. We find that in some cases the asymptotic approach does not lead unequivocally to the proper result. 1. Introduction The theoretical transition from discrete to continuous dynamical descriptions of particle systems is a classical problem of continuing—even increasing—interest in view of current developments in materials scence, biology and nanotechnology. Among the fundamental phenomena of importance is the emergence of long ‘macroscopic’ timescales from an ensemble of particles evolving on relatively fast ‘microscopic’ timescales. Relaxation rates and mean first passage times are familiar examples of such quantities [1] that enter into the next level of modelling of the bulk kinetics, often in terms of deterministic rate equations. These quantities are of interest in other contexts as well, for example in population biology and epidemiology where the evolution of a large collection of individuals or, say, an infection within the group, is often more easily studied by modelling the population as a continuum. These issues have a long history in physics, chemistry and biology. For example Van Kampen [2] introduced the 1/ expansion to recover some aspects of the dynamics and 0953-8984/07/065145+12$30.00 © 2007 IOP Publishing Ltd Printed in the UK 1