Scaling for first-order phase transitions in thermodynamic and finite systems

Scaling behavior for first-order phase transitions can be derived alternatively but consistently from renormalization-group, phenomenological, or finite-size considerations. A general analysis of densities at a renormalization-group fixed point demonstrates that if the coexistence of $p$ distinct phases is possible, then $p$ distinct eigenvalue exponents must equal the spatial dimensionality. This basic eigenvalue (or scaling) exponent condition can also be derived phenomenologically by various arguments not depending on detailed renormalization-group considerations. A scaling description of first-order phase transitions is presented and extended to finite systems with linear dimensions $L$, leading to a rounding proportional to $L^{-d}$, response-function maxima varying as $L^d\propto N$, and boundary-condition-dependent shifts which may be as large as $\sim L^{-1\mathrm{}}$.