## Stochastic variational principles for dissipative and conservative systems

published: Oct. 16, 2012, recorded: September 2012, views: 3868

# Slides

# Related content

# Report a problem or upload files

If you have found a problem with this lecture or would like to send us extra material, articles, exercises, etc., please use our**to describe your request and upload the data.**

__ticket system__*Enter your e-mail into the 'Cc' field, and we will keep you updated with your request's status.*

# Description

In the frame of the method of statistical ensembles, relaxation to thermodynamic equilibrium can be efficiently described by using stochastic differential equations. This scheme is intrinsically non-invariant with respect to time reversal. However, we can consider an additional dynamical variable, called the importance function, whose meaning and motivation arises from the neutron diffusion theory. The resulting scheme is now time reversal invariant, with a complete symplectic structure. The equations for the density and the importance function are Hamilton equations for a properly chosen Hamiltonian, and obey a stochastic variational principle. On the other hand, we can consider the formulation of quantum mechanics, according to the stochastic scheme devised by Edward Nelson. In this frame a stochastic variational principle can be easily introduced. The theory is intrinsically time reversal invariant. We have still a symplectic structure, and canonical Hamilton equations. Here the conjugated variables are the quantum mechanical probability density and the phase of the wave function. We give a synthetic description of the two schemes, by pointing out their structural similarity, and their deep physical difference.

# Link this page

Would you like to put a link to this lecture on your homepage?

Go ahead! Copy the HTML snippet !

## Write your own review or comment: