The expectation that scientific productivity follows regular patterns over a career underpins many scholarly evaluations, including hiring, promotion and tenure, awards, and grant funding. However, recent studies of individual productivity patterns reveal a puzzle: on the one hand, the average number of papers published per year robustly follows the "canonical trajectory" of a rapid rise to an early peak followed by a graduate decline, but on the other hand, only about 20% of individual researchers' productivity follows this pattern. We resolve this puzzle by modeling scientific productivity as a parameterized random walk, showing that the canonical pattern can be explained as a decrease in the variance in changes to productivity in the early-to-mid career. By empirically characterizing the variable structure of 2,085 productivity trajectories of computer science faculty at 205 PhD-granting institutions, spanning 29,119 publications over 1980--2016, we (i) discover remarkably simple patterns in both early-career and year-to-year changes to productivity, and (ii) show that a random walk model of productivity both reproduces the canonical trajectory in the average productivity and captures much of the diversity of individual-level trajectories. These results highlight the fundamental role of a panoply of contingent factors in shaping individual scientific productivity, opening up new avenues for characterizing how systemic incentives and opportunities can be directed for aggregate effect.

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