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We present a systematic computation of the heat conductivity of the Markovjump process modeling the energy exchanges in an array of locally confined hardspheres at the conduction threshold. Based on a variational formula [Sasada M.2016, {\it Thermal conductivity for stochastic energy exchange models},arXiv:1611.08866], explicit upper bounds on the conductivity are derived, whichexhibit a rapid power-law convergence towards an asymptotic value. We therebyconclude that the ratio of the heat conductivity to the energy exchangefrequency deviates from its static contribution by a small negative correction,its dynamic contribution, evaluated to be $-0.000\,373$ in dimensionless units.This prediction is corroborated by kinetic Monte Carlo simulations which weresubstantially improved compared to earlier results.

Dynamical contribution to the heat conductivity in stochastic energy exchanges of locally confined gases