Free energy and the Fokker-Planck equation

✍ Scribed by Richard Jordan; David Kinderlehrer; Felix Otto

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 372 KB
• Volume: 107
• Category: Article
• ISSN: 0167-2789
• DOI: 10.1016/s0167-2789(97)00093-6

✦ Synopsis

We establish a new and intriguing connection between the Fokker-Planck equation with gradient drift term and an associated free energy functional. Namely, we demonstrate that such a Fokker-Planck equation may be interpreted as a gradient flux, or a steepest descent, of a free energy functional with respect to a certain metric. This is accomplished through the construction of a time-discrete iterative variational scheme whose solutions converge to the solution of the Fokker-Planck equation. The time step in this scheme is governed by the Wasserstein metric on probability measures.