2nd law

Abstract

**Abstract** We present a simple two-compartment model of CO₂ mixing with air to illustrate the second law of thermodynamics through three complementary propositions. Proposition 1 recalls the standard monotonic relaxation of local concentration (with possible overshoot). Proposition 2 introduces a local Boolean probe—a small fixed-radius sphere at an arbitrary point—and tracks the Shannon binary entropy of the event “at least one CO₂ molecule is present.” This entropy rises during equilibration and can exhibit a transient maximum. Proposition 3 interprets the irreversible growth of the union volume of small spheres centred on each CO₂ molecule as a geometric/topological signature of the second law. The constructions are elementary, rely only on standard Poisson statistics, binary entropy, and excluded-volume ideas, yet appear novel in their specific framing. They are offered primarily as pedagogical devices that make the increase of microscopic uncertainty and the “claiming of spatial degrees of freedom” concrete. An appendix containing the original propositions and independent assessments by several large language models is supplied for transparency.