  We discuss the scaling of the interaction energy with particle numbers for a
harmonically trapped two-species mixture at thermal equilibrium experiencing
interactions of arbitrary strength and range. In the limit of long-range
interactions and weak coupling, we recover known results for the integrable
Caldeira-Leggett model in the classical limit. In the case of short-range
interactions and for a balanced mixture, numerical simulations show scaling
laws with exponents that depend on the interaction strength, its attractive or
repulsive nature, and the dimensionality of the system. Simple analytic
considerations based on equilibrium statistical mechanics and small
interspecies coupling quantitatively recover the numerical results. The
dependence of the scaling on interaction strength helps to identify a threshold
between two distinct regimes. Our thermalization model covers both local and
extended interactions allowing for interpolation between different systems such
as fully ionized gases and neutral atoms, as well as parameters describing
integrable and chaotic dynamics.
