TY - JOUR
T1 - Metastable states for the Becker-Döring cluster equations
AU - Penrose, Oliver
PY - 1989/12
Y1 - 1989/12
N2 - The Becker-Döring equations, in which cl(t) can represent the concentration of l-particle clusters or droplets in (say) a condensing vapour at time t, are {Mathematical expression} with {Mathematical expression} and either c1=const. ('case A') or {Mathematical expression}=const. ('case B'). The equilibrium solutions are cl=Qlzl, where {Mathematical expression}. The density of the saturated vapour, defined as {Mathematical expression}, where zs is the radius of convergence of the series, is assumed finite. It is proved here that, subject to some further plausible conditions on the kinetic coefficients al and bl, there is a class of "metastable" solutions of the equations, with c1-zs small and positive, which take an exponentially long time to decay to their asymptotic steady states. (An "exponentially long time" means one that increases more rapidly than any negative power of the given value of c1-zs (or, in case B, ?-?s) as the latter tends to zero). The main ingredients in the proof are (i) a time-independent upper bound on the solution of the kinetic equations (this upper bound is a steady-state solution of case A of the equations, of the type used in the Becker-Döring theory of nucleation), and (ii) an upper bound on the total concentration of particles in clusters greater than a certain critical size, which (with suitable initial conditions) remains exponentially small until the time becomes exponentially large. © 1989 Springer-Verlag.
AB - The Becker-Döring equations, in which cl(t) can represent the concentration of l-particle clusters or droplets in (say) a condensing vapour at time t, are {Mathematical expression} with {Mathematical expression} and either c1=const. ('case A') or {Mathematical expression}=const. ('case B'). The equilibrium solutions are cl=Qlzl, where {Mathematical expression}. The density of the saturated vapour, defined as {Mathematical expression}, where zs is the radius of convergence of the series, is assumed finite. It is proved here that, subject to some further plausible conditions on the kinetic coefficients al and bl, there is a class of "metastable" solutions of the equations, with c1-zs small and positive, which take an exponentially long time to decay to their asymptotic steady states. (An "exponentially long time" means one that increases more rapidly than any negative power of the given value of c1-zs (or, in case B, ?-?s) as the latter tends to zero). The main ingredients in the proof are (i) a time-independent upper bound on the solution of the kinetic equations (this upper bound is a steady-state solution of case A of the equations, of the type used in the Becker-Döring theory of nucleation), and (ii) an upper bound on the total concentration of particles in clusters greater than a certain critical size, which (with suitable initial conditions) remains exponentially small until the time becomes exponentially large. © 1989 Springer-Verlag.
UR - http://www.scopus.com/inward/record.url?scp=0001477522&partnerID=8YFLogxK
U2 - 10.1007/BF01218449
DO - 10.1007/BF01218449
M3 - Article
SN - 0010-3616
VL - 124
SP - 515
EP - 541
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
IS - 4
ER -