Variational method applied to Brownian motion

Question

Is it possible to apply the variational method to the Brownian motion? I mean, one of the requisites on $y(t)$ is that it must be continuous and $\partial_t{y(t)}$ too, and in this case, $\partial_t{y(t)}$ is not.

Answer 1

expanding a bit on Lubos' answer: Brownian motion as a stochastic process in the sense of mathematics does not describe a trajectory (or many trajectories or a probability measure on trajectories) of point particles in the sense of classical mechanics. When this mathematical tool is used to model physical systems some - severe - assumptions about these systems are usually made. For more details have a look at the references listed here:

• Azimuth project: stochastic differential equations,

especially the reference

• William T. Coffey, Yuri P. Kalmykov, and John T. Waldron: The Langevin equation. With applications to stochastic problems in physics, chemistry and electrical engineering.

Brownian motion reappears in quantum mechanics as Lubos has explained, a classic reference is

• Barry Simon: Functional Integration and Quantum Physics

That's the same Simon that wrote the 4 part classic book about analysis and operator theory with Michael Reed.

Stochastic processes of continuous time live in function spaces and have therefore an infinite amount of degress of freedom. A calculus of variations has therefore to be a calculus of variations on infinite dimensional spaces. Such a calculus exists, it is usually called Malliavin or white noise calculus. There has been limited success to provide a rigorous construction of the Feynman path integral using white noise calculus (the buzzword is "Feynman path integral as a Hida distribution"), see for example

• Zhi-yuan Huang and Jia-an Yan: "Introduction to Infinite Dimensional Stochastic Analysis"

and especially chapter V paragraph 3.3, "Application to Feynman integrals", and paragraph 4, "applications to quantum physics".

P.S.: It is usually considered helpful if one explains what one already knows about the topic one is asking about and what the reasons for asking are. This is helpful for people in order to taylor the answer to the needs of the one who asked :-)

Answer 2

Dear gsAllan, in the real world, the Brownian motion is the result of collisions of the moving object we study - a pollen particle - with small (water) molecules that have some positions and that may be described by the variational method - either in classical mechanics or in quantum field theory (where I mean the Feynman path integral by the variational method), at least in principle.

Once you do so with all the real-world degrees of freedom of all the water molecules that are unmeasurable in practice, you will find out that the system is described by continuous functions and the pollen particle moves continuously all the time. There will exist a typical "mean path" and "mean time between collisions" and during this time, the velocity of the pollen particle will be finite (and piecewise linear, with the turning points that may be either sharp or smoothened into a finite acceleration, depending on the description of the collisions: whether you repel them by an infinite potential wall or a finite repulsive potential).

What you're apparently talking about, however, is a simplified or idealized description of the Brownian motion where the force is "random", i.e. when one doesn't claim to be able to keep track of the positions of all the water molecules and where one only wants to compute the statistical averages (e.g. of the distance how far the pollen particle gets).

And moreover, the randomness of the force is extrapolated to arbitrarily short distance scales. In that case, the velocities are not continuous and it would make no sense to describe the motion by the variational principle because the action itself would have to contain some "random" parameters as well. Because the energy of the pollen particle may be always transferred from/to the water molecules, the energy conservation is inconsequential, too. Systems where "nothing useful is conserved" are usually not usefully described by the actions.

The very purpose of the variational principle is to find the single one unique canonical correct solution for the trajectory. But if there are random forces, there is no single correct trajectory.

Quantum mechanics

However, there is an interesting relationship of the Brownian motion and the quantum version of the variational principle. In quantum mechanics, the action may also be declared to be the "primary object" defining all of dynamics. The approach is called the Feynman path integral definition of quantum mechanics. One sums the phase $\exp(iS/\hbar)$ over all trajectories, where $S$ is the action of the trajectory, $i$ is the imaginary unit, and $\hbar$ is the reduced Planck's constant, and interprets the sum as the probability amplitude for a transition.

Interestingly enough, the typical trajectories that contribute to the Feynman path integral look much like the trajectories of the Brownian motion! So the Brownian motion paths have a profound importance for physics formulated in terms of the action - but only at the quantum level.