The Dictionary
Mechanics on the left, statistics on the right. Exact whenever energies are quadratic.
One identity runs under the whole table: energy is a negative log-likelihood. A Gaussian's log-density is a downward parabola, so its negative is the potential energy of a spring, and a belief is a spring. Statistical mechanics ran the trade the other way, writing probability as $e^{-E}$; take the log and energy becomes the negative log-probability. Every row below is one case.
| Mechanics | Statistics | |
|---|---|---|
| mass $k$ (the spring's stiffness) | precision $1/\sigma^2$ | |
| potential energy $\tfrac12 k (z-x)^2$ | negative log-likelihood | |
| force $k(x-z)$ | score $\tfrac{d}{dz}\log p(z)$ | |
| equilibrium | posterior mean (= mode) | |
| centre of mass | Kalman update $m' = \dfrac{P^{-1}m + R^{-1}y}{P^{-1}+R^{-1}}$ | demo |
| masses in parallel add | independent precisions add | demo |
| springs in series: reduced mass $\dfrac{m_1 m_2}{m_1+m_2}$ | variances add through a noisy channel; the Kalman prediction step | demo |
| rigid rod on vertical springs | least squares regression | demo |
| rod on frictionless sliding collars | PCA (total least squares) | demo |
| bead chain: measurement + neighbour springs | Kalman smoother; Whittaker graduation | demo |
| masses re-weighted by error correlations | GLS; bid-ask bounce as anti-redundant evidence | demo |
| bead between corner posts; repulsive spring | minimum-variance portfolio; short selling | demo |
| one body, mass shrinking then regrowing over time | the Kalman filter, tick by tick | demo |
| pinning a node in a spring network | conditioning on a Gaussian graph | demo |
| elastic membrane pinned at points | Gaussian-field interpolation; kriging | demo |
| constant-tension pulley vs. Hookean spring | Lasso vs. Ridge; soft-thresholding | demo |
| thermal jitter | posterior sampling (Langevin) |
The correspondence is exact only for Gaussians. A non-Gaussian likelihood is a non-Hookean spring. Equilibrium still finds the mode. But subsystems can no longer be swapped for one equivalent body.