Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis

Driven anomalous diffusions (such as those occurring in some surface growths) are currently described through the nonlinear Fokker-Planck-like equation (partial derivative/partial derivative t) p mu = - (partial derivative/partial derivative x) [F(x)p(mu)] + D(partial derivative(2)/partial derivative x(2)) p(nu) [(mu, nu) is an element of R(2); F(x) = k(1) - k(2)x is the external force; k(2) greater than or equal to 0]. We exhibit here the (physically relevant) exact solution for all (x,t). This solution was found through an ansatz based on the generalized entropic form S-q[p] = {1 - integral du[p(u)](q)}/(q - 1) (with q is an element of R), in a completely analogous manner through which the usual entropy S-1[p] = - integral dup(u)Inp(u) is known to provide the comet ansatz for exactly solving the standard Fokker-Planck equation (mu = nu = 1). This remarkably simple unification of normal diffusion (q = 1), superdiffusion (q > 1) and subdiffusion (q < 1) occurs with q = 1 + mu - nu.