Thermodynamic equilibrium

A thermodynamic system is in thermodynamic equilibrium when its macroscopic state has no spontaneous tendency to change through any exchange allowed by its constraints. For an isolated system, stable thermodynamic equilibrium maximizes entropy over the accessible states at fixed total energy and conserved material inventories. A wall can forbid an exchange, so always specify what can cross or move. Thermal, mechanical and chemical conditions answer different questions about that same constrained state. ^[ref]

For example, two compartments separated by a rigid, impermeable wall that transmits heat can reach thermal equilibrium while retaining different pressures and compositions. Allowing an unloaded piston to move adds a mechanical equilibrium condition. Allowing one chemical species to cross adds chemical potential equality for particle exchange; allowing a chemical reaction adds its chemical equilibrium condition. A fixed barrier can therefore support a constrained thermodynamic equilibrium that ceases to be an equilibrium when the barrier is removed.

Uniformity is not the definition of thermodynamic equilibrium. An isothermal atmosphere in hydrostatic equilibrium can have a common temperature while its pressure and mass density vary with height: the pressure gradient balances gravity. In an external field, the exchange condition must include the relevant potential energy; for charged matter this leads to the electrochemical potential. Relativistic gravitational corrections to thermal equilibrium are described separately by the Tolman temperature law.

A steady state only says that the macroscopic description is constant in time. A bar maintained between hot and cold thermal reservoirs has steady heat flux and produces entropy, so it is not in thermodynamic equilibrium. Even matching temperatures at one perfect-contact interface can coexist with heat transfer through the surrounding material; matching those two local values does not establish equilibrium of the whole apparatus. See thermal conduction for that spatial transport problem.

The useful simplification is that equilibrium conditions predict an endpoint without computing every collision or flow during relaxation. The first law of thermodynamics supplies the conserved quantities, and the Second law of thermodynamics supplies the direction of change. For a system held at fixed temperature and volume, the corresponding criterion is minimization of Helmholtz free energy; at fixed temperature and pressure, it is minimization of Gibbs free energy, with the material and exchange constraints still imposed.

Coexistence without a thermodynamic driving force for further exchange. Phases able to exchange heat and constituents have equal temperature and chemical potentials of the exchanging species. Mechanical balance is also needed; interface curvature can require a pressure jump rather than equal bulk pressures.

Mechanical equilibrium requires force balance for the mechanical displacements that are allowed. Two uniform fluids separated by a freely moving, unloaded, frictionless piston with equal face areas require equal pressures. A supported piston can sustain a pressure difference, and a fluid in gravity instead obeys hydrostatic equilibrium. For an upward height coordinate $z$, that balance is $dp/dz=-\rho g$, where $\rho$ is mass density and $g$ is the downward gravitational acceleration. Mechanical equilibrium alone does not establish thermal equilibrium or chemical equilibrium.

Two fluids at a stationary, freely movable interface require equal pressures when surface tension and other normal stresses are negligible. This local pressure equilibrium does not require equal temperatures or mass densities. Curvature with nonzero surface tension or an applied load changes the normal-stress balance. Comparing points at different heights also requires the hydrostatic equilibrium condition.

For a mechanical coordinate $x$ at fixed temperature, total volume and particle numbers, the relevant thermodynamic potential is the Helmholtz free energy $F(x)$. Its generalized force is $f=-F^{\prime }(x)$. At mechanical equilibrium, $f(x_{\ast })=0$, and a linearly restoring force has $f^{\prime }(x_{\ast })<0$, equivalently $F^{\prime \prime }(x_{\ast })>0$. A Taylor expansion then proves a strict local free-energy minimum. If $F$ is a strictly convex function throughout the accessible domain, the stationary state is the unique global minimum.