State function

A state function is a function $X$ defined on the set of admissible thermodynamic states such that, for any process taking the system from state A to state B, the change $\Delta X = X(B)-X(A)$ depends only on the endpoints and not on the particular process path .

Equivalently, the differential $dX$ is an exact differential: its integral along a path in state space is path-independent.

Physical interpretation. State functions represent properties that can be assigned to the system “at an instant” in equilibrium—stored energy, entropy, or thermodynamic potentials encoding equilibrium constraints. They are contrasted with path functions such as heat and work , which describe energy transfer during a process.

Key properties

• Path independence: for any two paths from A to B, $\int_{\text{path 1}} dX = \int_{\text{path 2}} dX = X(B)-X(A)$.
• Cyclic integral vanishes: for any cycle , $\oint dX = 0$.
• Exactness generates Maxwell-type relations: when a thermodynamic potential is written in terms of its natural variables, equality of mixed partial derivatives yields Maxwell relations .

Common thermodynamic state functions. Important examples include internal energy $U$, entropy $S$, enthalpy $H$, Helmholtz free energy $F$, Gibbs free energy $G$, and (for open systems ) the grand potential $\Omega$. Many thermodynamic potentials are related by a Legendre transform that exchanges an extensive variable for its conjugate intensive variable.

Connection to statistical mechanics. In the canonical ensemble (fixed $T,V,N$), the Helmholtz free energy is linked to the partition function $Z$ by

with $k_B$ the Boltzmann constant . This makes $F$ a generating object for equilibrium averages and response functions.