Thermal equilibrium

Two bodies are in thermal equilibrium when, placed in thermal contact, no net heat flows between them — equivalently, when they share the same temperature. It is the defining relation behind temperature itself: temperature is the quantity that equalises at equilibrium. An isolated pair reaches a shared value equal to the heat-capacity-weighted mean of their starting temperatures, since the heat lost by one equals the heat gained by the other.

The approach to equilibrium is governed by Newton's law of cooling, an exponential relaxation T(t) = T_eq + (T₀ − T_eq) e^(−kt) in which the temperature gap shrinks by a fixed fraction in each time interval. The rate constant k depends on the thermal coupling between the bodies — fast through a metal bridge, slow across a vacuum gap.

Thermal equilibrium is the simplest case of thermodynamic equilibrium, in which all bulk flows (of heat, matter, and momentum) have ceased. It is the reference state from which the entire formalism of thermodynamics is built.

Implicit in eighteenth-century calorimetry and made precise by the zeroth law in the twentieth century; the transitivity of equilibrium (A~B, B~C ⟹ A~C) is what licenses a thermometer to compare bodies that never touch.