Heat flows from hot to cold. This is not a law you prove from first principles and then apply; it is the observation from which thermodynamics begins. The second law guarantees it in equilibrium. In steady-state transport between reservoirs, the Landauer-Büttiker formalism confirms it: heat conductance is positive, current flows down the temperature gradient.
Menczel, Flindt, and colleagues (arXiv:2602.21190) find that this can reverse during transients. In a quantum system coupled to reservoirs at different temperatures, the heat conductance can be transiently negative — heat flowing from cold to hot — depending on the initial preparation of the system. The effect requires strong coupling and non-Markovian dynamics, both captured exactly by their method.
The reversal is not a violation of thermodynamics. It is a transient powered by the initial state's correlations with the reservoirs. The system was prepared in a state that stores energy in system-reservoir correlations, and the relaxation of those correlations can drive heat temporarily against the gradient. Once the initial correlations decay, the conductance becomes positive and stays positive.
The distinction matters: steady-state heat conductance is always positive; transient heat conductance can be negative. The sign of heat flow is not a property of the temperature difference alone — it is a property of the temperature difference AND the initial conditions. In the Markovian, weak-coupling limit, the initial conditions don't matter and heat always flows correctly. Strong coupling and memory make history relevant.
The general observation: the direction of a macroscopic flow is fixed by gradients only in steady state. During transients, the initial preparation of the system can override the gradient. History competes with gradient, and for a time, history wins.