Coarse-grained Shannon entropy of random walks with shrinking steps

In one-dimensional random walks with discrete steps whose magnitude decays geometrically, the usual Gaussian broadening familiar from Brownian motion is replaced by bounded probability distributions over particle positions that are often characterized by multi-scale fractal structures. In this work, we study random walks with shrinking steps (known as Bernoulli convolutions), focusing on their behavior in the vicinity of the dyadic contraction ratio 1/2, where the length of each subsequent step decreases by a factor of two. Our analytical and numerical results show that the coarse-grained Shannon entropy of particle distributions induced by Bernoulli convolutions exhibits a local maximum at the dyadic ratio, arising from the competition between diffusive spreading, which increases entropy, and emergent fine structure, which tends to decrease it. This entropy maximum is generally expected to arise in autoregressive processes driven by bounded discrete noise with a geometrically decaying amplitude. We discuss potential implications of Bernoulli convolution dynamics for protocell self-replication and vesicle proliferation, establishing a link between our information-theoretic approach and biophysical models of early-life cell division.