Spatial distribution of thermal energy in equilibrium

The equipartition theorem states that in equilibrium, thermal energy is equally distributed among uncoupled degrees of freedom that 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 or 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 $\frac{1}{2}{k}_{B}T$. 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 nontrivial spatial distributions emerge. Some implications are discussed.

Figure 1

Model system of two masses and three linear springs, connected in series between fixed walls. Here $x_i$ measures the deviation of the $i\mathrm{th}$ particle from its equilibrium position.

Figure 2

Average thermal energy $\langle {\epsilon }_{\alpha }\rangle$ (in units of $k_{B}T)$ as a function of $k_{\alpha }/\langle \langle k\rangle \rangle$ for a hexagonal portion of a two-dimensional triangular lattice. The data are partitioned into bulk (blue) and boundary springs (yellow) (cf. the inset). Spring constants are distributed normally with mean 1 and variance $0.3$. Each side of the hexagon consists of 20 springs, which means $N=2522$ (a smaller system, with 3 springs on each side and $N=74$, is shown in the inset for illustration). No quantitative change was observed with increasing $N$. The points show data from 30 realizations and the solid lines are guides to the eye. The dashed line shows the asymptotic value $\frac{1}{3}$, corresponding to $\langle {\epsilon }_{\alpha }\rangle$ in the bulk of a homogeneous triangular lattice in the thermodynamic limit.