Coarse-grained Shannon entropy of random walks with shrinking steps

Abstract

In one-dimensional diffusive processes with discrete steps characterized by geometrically decaying magnitudes, the usual Gaussian broadening familiar from Brownian motion is replaced by bounded probability distributions over particle positions that are 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. 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 a general property of systems driven by non-Gaussian discrete noise, whose dynamics near stable fixed points can be viewed as an autoregressive process - an approximation that is mathematically equivalent to unbiased random walks with shrinking steps. 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 cell division.

Figures (6)

• Figure 1: Probability distributions for random walks with shrinking steps (Bernoulli convolutions) near $\boldsymbol{\lambda = 2}$. Coarse-grained densities $P(x;\lambda,l)$ at $\lambda=1.9$ (blue), $\lambda=2.0$ (red), and $\lambda=2.1$ (green) as functions of the coordinate $x$, computed using the boundary-tracking method with $l = 20$ (Methods; Appendix \ref{['app:numerics']}). At $\lambda=2$, the distribution becomes uniform in the $l \rightarrow \infty$ limit, maximizing the entropy on its support. For $\lambda=1.9$ (i.e., $\lambda<2$), the distribution broadens and develops complicated internal structure. For $\lambda=2.1$ (i.e., $\lambda>2$), the distribution contracts and becomes increasingly disconnected due to the gaps between adjacent regions with non-zero probability density.
• Figure 2: Coarse-grained structure of the probability density. Shown is $P(x;\lambda,l)$ for $\lambda=1.95$ and $l=2$ (blue lines). Each of the $2^2=4$ paths contributes a rectangular kernel of width $2 a_l$ (Eq. \ref{['eq:tail_halfwidth']}; red lines with bars). For $\lambda<2$, adjacent kernels overlap, creating regions with density $\approx 1/2$ (single coverage, total length $\ell_1$, green lines with bars) and $\approx 1$ (double coverage, total length $\ell_2$, magenta lines with bars). The half-width of the full distribution is $a_0=(\lambda-1)^{-1}$ (black arrow with bars). The piecewise structure of $P(x;\lambda,l)$ enables analytical entropy calculation via Eqs. \ref{['eq:length_constraints']}--\ref{['eq:entropy_below']} and is also exploited in the boundary-tracking numerical method (Methods; Appendix \ref{['app:numerics']}).
• Figure 3: Fractal entropy in the vicinity of $\lambda=2$. Coarse-grained entropy $S(\lambda,l)$ was computed using BT (Methods; Appendix \ref{['app:numerics']}) and plotted versus $(\lambda-2)l$ for several resolution levels $l \in \{3,5,10,15,20,25\}$. A local maximum emerges for $l \geq 6$. Black lines show the asymptotic slopes given by Eqs. \ref{['eq:limit_left']} and \ref{['eq:limit_right']} -- see Eq. \ref{['eq:taylor_expansion']}. Each numerical curve is based on $101$ independent entropy calculations: $(\lambda-2)l = (-1.00, -0.98, \dots, 0.98, 1.00)$.
• Figure 4: Comparison of numerically computed fractal entropy curves. Shown are entropies of Bernoulli convolutions in the vicinity of $\lambda=2$, computed using BT ($l=5,15$), FFT ($l_\text{FFT}=10,20$), and GBI ($l_\text{GBI}=10,20$). The emergence of the local maximum at $\lambda=2$ is corroborated by all three numerical methods. Each numerical curve is based on $101$ independent entropy calculations: $(\lambda-2)l = (-1.00, -0.98, \dots, 0.98, 1.00)$.
• Figure 5: Cell growth and division model with binary volume increments as a dyadic Bernoulli convolution. Shown are the spectra of $2^n$ cell volumes after $n=\{ 1,2,3,4,\infty \}$ cell division events (Eq. \ref{['eq:recursion']}), with $\Delta v=2$ and $V_0=0$. At a given $n$, each point in the spectrum corresponds to an endpoint of a random walk with $n$ steps. In the $n \to \infty$ limit, the horizontal black bar shows the support of $V_\infty$ in Eq. \ref{['V:inf']}.