  The equipartition theorem states that in equilibrium thermal energy is
equally distributed among uncoupled degrees of freedom which appear
quadratically in the system's Hamiltonian. However, for spatially coupled
degrees of freedom --- such as interacting particles --- one may speculate that
the spatial distribution of thermal energy may differ from the value predicted
by equipartition, possibly quite substantially in strongly
inhomogeneous/disordered systems. Here we show that for systems undergoing
simple Gaussian fluctuations around an equilibrium state, the spatial
distribution is universally bounded from above by $\frac{1}{2}k_BT$. We further
show that in one-dimensional systems with short-range interactions, the thermal
energy is equally partitioned even for coupled degrees of freedom in the
thermodynamic limit and that in higher dimensions non-trivial spatial
distributions emerge. Some implications are discussed.
