Boltzmann Energy Distribution in Potential Well

The local minimum of a potential should look roughly like a parabola (), so the equilibrium distribution of a gas sitting in a potential well should be .

There is a direct proof (Lagrange multipliers) that says that a Gaussian probability distribution has the highest entropy of any with a fixed mean and standard deviation, which (since that's fixing and if X is the random variable) is just a quadratic potential well again.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.