Time reversal and the heat equation

Abstract According to standard thermodynamics, heat spontaneously propagates from hot to cold reservoirs as time progresses towards the future. Consequently, one might assume that for the purpose of describing thermodynamic machines, a world in which heat spontaneously propagates from cold to hot (i.e., a world in which the heat equation has a different sign) would be simply the time reversal of our world. In this article, we explain why this is not the case. Thermodynamics is characterized by the universal approach to equilibrium states from arbitrary initial conditions, such that the final state of a thermodynamic equilibration process does not contain the information necessary to recreate the initial state. In particular, the initial state cannot be recreated by evolving the final state via the rule “heat always propagates from cold to hot”, as this rule would lead to temperature gradients on arbitrarily small scales (which is not a feature that the initial state will generally have had). This toy problem illustrates some general features of the role of the direction of time in thermodynamics. These features are discussed mathematically using the Mori-Zwanzig projection operator formalism.