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The closed-box model of galactic chemical evolution has $M=M_0{e}^{-Z/y}$ and $S=M_0-M$, giving $dN/dZ=(\kappa M_0/y)e^{-Z/y}$. To compare shapes without confusing different total stellar counts, the figure normalizes both metallicity distribution functions to unit integral after complete gas exhaustion and uses $x=Z/y$:The right panel also shows the corresponding cumulative fractions, $1-e^{-x}$ and $\min (x/2,1)$.

Pristine galactic gas inflow replenishes the fuel and dilutes freshly produced metals, so enrichment per unit newly formed stellar mass is slower and asymptotes to $2y$. More stars are formed at later abundances; the proportional-infall model of galactic chemical evolution makes twice as many stars as the closed box for the same initial gas mass in this limit. Consequently the normalized low-abundance fraction is smaller: for $Z\ll y$, it is $Z/(2y)$ rather than approximately $Z/y$. Their unnormalized $dN/dZ$ actually agree at $Z=0$ for the same $M_0$, which is why stating the normalization matters. At a finite stopping abundance $Z_f$, use $p_Z=1/Z_f$ for infall and $p_Z=e^{-Z/y}/[y(1-e^{-Z_f/y})]$ for the closed box, each truncated at its own chosen stopping point.

Let $\psi (t)$ be the star formation rate, $M$ the remaining gas mass, and $S$ the mass put into stars. With zero returned bulk gas, pristine galactic gas inflow, and no galactic outflow, mass conservation givesA well-mixed gas reservoir loses existing metals at rate $Z\psi$ and gains newly synthesized metals at rate $y\psi$; the incoming gas brings none. ThusFor $0<\nu <1$, eliminate time and integrate:The proportional-infall model of galactic chemical evolution therefore gives, for initially pristine gas,The displayed form requires $Z_0=0$; more generally $Z=y/\nu +(Z_0-y/\nu )(M/M_0{)}^{\nu /(1-\nu )}$. Also $S=(M_0-M)/(1-\nu )$ for an initially star-free reservoir, so $M/M_0$ is not the gas fraction of a galaxy $M/(M+S)$. As a check, the $\nu \to 0$ limit is the closed-box model of galactic chemical evolution relation $Z=y\log (M_0/M)$; at $\nu =1$ the gas mass is constant and one must instead solve $\dot{Z}=(y-Z)\psi /M_0$. Retaining a nonzero stellar yield while setting bulk mass return to zero is the prescribed trace-metal approximation, not a literal claim that stellar ejecta have no mass.

The G-dwarf problem is the observed shortage of metal-poor, long-lived disc stars relative to a simple closed-box model of galactic chemical evolution. In the fully evolved comparison, the half-rate proportional-infall model of galactic chemical evolution lowers the low-$Z$ fraction from $1-e^{-Z/y}$ to $Z/(2y)$, and prevents the closed-box tail from extending to arbitrarily high stellar metallicity. It therefore reduces the relative weight of the earliest metal-poor generation and the very metal-rich tail.

This particular constant-proportional-infall model is an improvement, not a complete fit. It still forms a metal-poor population immediately from its initially pristine reservoir, with nonzero constant $dN/dZ$ all the way down to zero, and predicts a broad linear distribution followed by an abrupt upper cutoff instead of a generally smooth, concentrated observed distribution. Comparison with observed abundance histograms must first use the same coordinate: its logarithmic metallicity distribution rises in proportion to $Z$ up to that cutoff, rather than remaining flat in logarithmic bins. Local G-dwarf measurements exhibit a prominent logarithmic-abundance peak and a paucity of low-abundance stars; translating iron abundances into total $Z$ also requires an abundance-ratio model. More realistic histories can vary the galactic gas inflow and star formation rate, allow recycling and delayed enrichment, and account for migration and selection of different stellar populations; simply tuning a constant stellar yield does not remove all these shape deficiencies.