= Negative entropy production over finite durations
{title2=$\Pr(S_n<0)$}
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:
$$
\Pr(S_n<0)=\sum_{k=0}^{\lfloor(n-1)/2\rfloor}\binom nkp^kq^{n-k}.
$$
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 $np$.
\Image[/negative-entropy-window-duration.png]
{title=Exact biased-step model: negative totals become rare with duration while negative increments remain frequent}
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 $\Pr(S_n<0)\le\langle e^{-S_n/2}\rangle=(2\sqrt{pq})^n$. Since $2\sqrt{pq}<1$ for $p\ne q$, negative totals are exponentially suppressed even though the negative-step fraction stays $q$.
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