Analytical models describing the motion of colloidal particles in given
velocity fields are presented. In addition to local approaches, leading to well
known master equations such as the Langevin and the Fokker-Planck equations, a
global description based on path integration is reviewed. This shows that under
very broad conditions, during its evolution a dissipative system tends to
minimize its energy dissipation in such a way to keep constant the Hamiltonian
time rate, equal to the difference between the flux-based and the force-based
Rayleigh dissipation functions. At steady state, the Hamiltonian time rate is
maximized, leading to a minimum resistance principle. In the unsteady case, we
consider the relaxation to equilibrium of harmonic oscillators and the motion
of a Brownian particle in shear flow, obtaining results that coincide with the
solution of the Fokker-Planck and the Langevin equations
