Scale invariance in acicular martensites emerges as a consequence of a competition between the front velocity $v$ and the rate of nucleation $I$ of the martensite grains. A renormalization group for the stationary probability distribution of the martensite grain size, $P\left(l\right)$, has two fixed points—an unstable fixed point at $v/I=0\mathrm{}$, associated with a gamma distribution, and a stable (critical) fixed point at $v/I=\infty$ with a Lévy distribution, $P\left(l\right)\sim l^{-\alpha }$. Crossover functions and exponents governed by the unstable fixed point describe $P\left(l\right)$ at intermediate $v/I$. Acicular martensites thus represent a novel example of a flow towards self-organized criticality. Our results provide a method to compute dynamic quantities ( $v$ or $I$) from static optical micrographs of martensite grains.

• Received 15 February 1995

DOI:https://doi.org/10.1103/PhysRevLett.75.2164