= Thermodynamic equilibrium
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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. <Statistical Physics by David Tong/4 Classical Thermodynamics>{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 (thermodynamics)>, 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.
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