A LAGRANGIAN FORMULATION OF THEORY OF RANDOM MOTION

The motion of a particle is studied under the action of two types of forces, some slowly varying and others rapidly fluctuating. The slowly varying forces are assumed to be given in every particular problem. The rapidly fluctuating forces, which are unknown, are assumed to have a random character. It is shown that the motion of the particle depends on the random forces only through the diffusion effect that they produce. The theory is statistical in character and only the evolution of a probability density can be determined. A Lagrangian formalism is developed and from it the general equations of motion are derived. These turn out to be formally equivalent to the Schrödinger equation. The generalization to a system of particles is straightforward if it is assumed that the random forces act independently and with like intensity on every elementary particle. The expectation values of the fundamental dynamical variables are obtained. The theory proves to be very similar to, but not fully identical with, nonrelativistic quantum mechanics without spin. © 1969 Società Italiana di Fisica.