What is the relationship between variance and energy

Question

I was speaking with someone today who told me that variance, in the sense of probability theory, is equivalent mathematically to energy in physics. Can anyone elaborate on this relationship?

Answer 1

In the Maxwell-Boltzmann kinetic theory of gases, a gas molecule has velocity modeled as a random vector $V$ whose components $V_x, V_y, V_z$ along three orthogonal axes are independent zero-mean Gaussian (or normal) random variable with variance $\sigma^2$. The kinetic energy of the particle is thus $\frac{1}{2}mV^2$ where $V$ is a random variable. Note that the expected value of the kinetic energy is thus $$E\left[\frac{1}{2}m|V|^2\right] = \frac{1}{2}mE[V_x^2+ V_y^2+ V_z^2] = \frac{3}{2}m\sigma^2.$$