Clausius inequality ($\oint \delta Q/T\le 0$)

For a thermodynamic cycle, with heat positive into the system and each transfer divided by the absolute temperature of the relevant thermal boundary or reservoir, the cyclic integral is nonpositive. Equality holds for a reversible cycle. Closing an arbitrary process by a reversible return path yields $\Delta S\ge \int \delta Q/T$, because the reversible integral defines the entropy difference. The difference is the nonnegative entropy production; entropy transferred with heat need not itself be positive.