 Metallicity distribution in a leaky box

For initially pristine gas, zero bulk mass return, a constant stellar yield $y$ and a well-mixed galactic outflow of mass-loading factor $\nu$, gas conservation gives $d{M}_g=-(1+\nu )d{M}_s$ and metal accounting gives $M_{g}dZ=yd{M}_s$. Consequently $M_g=M_{g,0}{e}^{-Z/y_e}$ with $y_e=y/(1+\nu )$. Since $d{M}_s/dZ=M_g/y$, complete depletion gives a normalized mass-weighted metallicity distribution function $p(Z)=e^{-Z/y_e}/y_e$ and mean stellar metallicity $y_e$. A time-independent initial mass function makes the distribution of surviving tracers proportional to this distribution; a sample selected by light need not have the same weights.