 Negative entropy production over finite durations

A single step has negative entropy production with probability $q$, independent of how long an observation lasts. A whole window is negative only if negative steps outnumber positive ones:For even $n$, the tie $K=n/2$ has zero entropy production and is excluded. At $p=3/4$, the negative-window probabilities for $n=1,9,21$ are approximately $0.25$, $0.04893$, and $0.00642$. Longer windows still contain negative increments: their expected number is $nq$. Their total is usually positive because the expected number of positive increments is $np$.

Only odd window lengths are plotted to avoid alternating tie effects. The left panel gives the exact negative-window probability and the upper bound $(2\sqrt{pq}{)}^n$; the right gives expected counts $np$ and $nq$. For steps of fixed duration $\delta t$, physical duration is $n\delta t$. These are counts and probabilities predicted by the model, not bead traces. The bound follows from $\mathrm{Pr}(S_n<0)\le ⟨e^{-S_n/2}⟩=(2\sqrt{pq}{)}^n$. Since $2\sqrt{pq}<1$ for $p\ne q$, negative totals are exponentially suppressed even though the negative-step fraction stays $q$.