= Solution
This relates an <thermodynamic equilibrium> <variance> to the local stiffness of an appropriate <thermodynamic free energy>. At fixed bath <temperature>, suppose the effective <thermodynamic free energy> $A(x)$ includes the constraints and the measure for a chosen coordinate. Its <thermodynamic equilibrium> <probability density function> is proportional to $e^{-A(x)/(k_BT)}$. Near a stable minimum $x_0$, write
$$
A(x)\simeq A(x_0)+\frac12A''(x_0)(x-x_0)^2,\qquad A''(x_0)>0.
$$
The resulting <normal distribution> gives the <Gaussian fluctuation approximation>
$$
\boxed{\langle(x-x_0)^2\rangle\simeq\frac{k_BT}{A''(x_0)}.}
$$
This is exact for a quadratic <thermodynamic free energy> with the stated flat coordinate measure; in general it is a small-fluctuation result. A smaller curvature means a softer restoring response and larger fluctuations. Near vanishing curvature, higher-order terms become important, so the quadratic approximation cannot alone predict an actual infinite variance.
For example <optical trap Langevin calibration> uses the Brownian displacement of a trapped bead: $A=\kappa(x-x_0)^2/2$ gives $\kappa=k_BT/\langle\Delta x^2\rangle$. A camera or position detector records the fluctuations. Finite exposure averages rapid motion and can bias the measured variance; the measurement protocol matters as well as the <thermodynamic equilibrium> identity.
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