Excitation Dynamics in Random One-Dimensional Systems

In a number of recent publications, [1] – [5], we have discussed the asymptotic form of the dynamics of a general type of random one-dimensional chains. The equations we discuss are of the form $${{\rm{C}}_n}\left( {\frac{{{\rm{d}}{{\rm{V}}_n}}}{{{\rm{dt}}}}} \right) = {{\rm{W}}_{n + 1}}\left( {{{\rm{V}}_{n,n + 1}} - {{\rm{V}}_n}} \right) + {{\rm{W}}_{n - 1}}\left( {{{\rm{V}}_{n,n - 1}} - {{\rm{V}}_n}} \right),\,\,\,\,{{\rm{W}}_{n,n + 1}} = {{\rm{W}}_{n,n + 1}},$$ (1) are independent positive random variables. Equations of this type arise in a variety of physical contexts. They can represent a master equation for hopping-type transport over random barriers (the Cn=1, the Wn,n+1 random hopping rates); a master equation for excitation transfer along a one-dimensional array of traps of random depth (the Cn random Boltzmann factors, the Wn,n+1=1); an electric transmission line (the Cn random capacitors or the Wn,n+1 random conductances); a random Heisenberg ferromagnetic chain at low temperatures (the Cn=1, Vn representing spin wave destruction operators, and the Wn,n+1 random near-neighbor exchange integrals). Replacing dVn/dt in (1) by d2Vn/dt2, they represent a harmonic chain with random masses (the Cn) or random force constants (the Wn,n+1).