Thermal equilibrium

Two systems are in thermal equilibrium if, when placed in contact through a diathermal wall (allowing energy transfer as heat) while otherwise isolated from external influences, there is no net energy transfer as heat and no macroscopic change of state occurs.

In equilibrium thermodynamics this condition is summarized by equality of the temperature : if systems 1 and 2 are in thermal equilibrium, then $T_1 = T_2$.

For a single extended system, “internal” thermal equilibrium means there are no sustained temperature gradients driving heat currents (a prerequisite for thermodynamic equilibrium ).

Physical interpretation

Thermal equilibrium is the state in which energy exchange has no preferred direction: neither system can increase total entropy by sending a small amount of energy to the other purely as heat. A thermal reservoir is an idealized system that remains at effectively fixed temperature despite exchanging finite heat; bringing a system into contact with a reservoir drives it toward the reservoir’s temperature.

Key relation: entropy maximization and equality of temperature

Consider two subsystems that can exchange energy but not particles or volume, with total energy fixed. Writing $U_{\text{tot}} = U_1 + U_2$ and $U_2 = U_{\text{tot}} - U_1$, equilibrium corresponds to an extremum of the total entropy:

Using the equilibrium definition $1/T = (\partial S/\partial U)_{V,N}$ (from the fundamental entropy relation ), the condition becomes $1/T_1 = 1/T_2$, i.e. $T_1=T_2$.

It is often convenient to package temperature as the inverse temperature $\beta = 1/(k_B T)$, with $k_B$ the Boltzmann constant .

Connection to the zeroth law

The empirical content that makes temperature a consistent state variable is the zeroth law of thermodynamics ; it is often phrased as the transitivity of thermal equilibrium and formalized via zeroth-law equivalence .