The time dependent solution of the Smoluchowski equation: Kinetics of escaping from the well for different dimensionalities

A recently developed method of analytical solution of the Smoluchowski equation is applied to the investigation of the kinetics of diffusional escape from a potential well for different space dimensionalities n. The kinetics is described by the time dependence of the well occupation probability Nn(t). The formulas derived are valid for times longer than the time t0=a 2/D of relaxation within the well U(r) [a is the Onsager’s radius determined by U(a)=kT ]. In the absence of absorption (or reaction) within the well the kinetics for low n=1,2 is found to be nonexponential at any time from the very beginning and for any depth of the well. The characteristic escaping time, however, significantly depends on the well depth. At longer times we obtain long time tails Nn(t)∼t−n/2 (n=1,2), typical for corresponding free diffusion processes. A simple general analytical expression for N1(t) is derived. In the limit of high reactivity within the well N1(t) appears to be exponential at moderately short t, but at long times it changes to the asymptotic function ∼t−3/2 (unlike the weak reactivity tail ∼t−1/2 ). In the two-dimensional (2D) case in the high reactivity limit the kinetics is also exponential at short t. The long time asymptotics somewhat differs from the weak reactivity one (∼t−1):N2(t)∼1/t ln2 t. It is shown that the three-dimensional (3D) escaping process is described by the same expression as the one-dimensional (1D) one in the high reactivity limit, i.e., at moderately short t, N3(t)∼exp(−Wt), while at long times N3(t)∼t−3/2. Some possible extensions of the method are discussed.