Abstract
We present a Kullback–Leibler (KL) control treatment of the fundamental problem of erasing a bit. We introduce notions of reliability of information storage via a reliability timescale τ r , and speed of erasing via an erasing timescale τ e . Our problem formulation captures the tradeoff between speed, reliability, and the KL cost required to erase a bit. We show that rapid erasing of a reliable bit costs at least log 2 - log 1 - e - τ e τ r > log 2 , which goes to 1 2 log 2 τ r τ e when τ r > > τ e .
Highlights
Biological systems are remarkably ordered at multiple scales and dimensions, from the spatial order witnessed in the packing of DNA inside the nucleus, the arrangement of cells to form tissues, and organs, and whole organisms, to the temporal order witnessed in the execution of various cellular processes
From this point of view, understanding how much energy is required to create order becomes an instance of the investigation of the connection between information processing and thermodynamics
We find theoptimal control for rapid erasing of a reliable bit, and argue that it requires cost of at τe r least log 2 − log 1 − e− τr > log 2, which goes to 12 log 2τ τe when τr >> τe
Summary
Biological systems are remarkably ordered at multiple scales and dimensions, from the spatial order witnessed in the packing of DNA inside the nucleus, the arrangement of cells to form tissues, and organs, and whole organisms, to the temporal order witnessed in the execution of various cellular processes. To copy a single base, a DNA polymerase hydrolyzes a triphosphate molecule to a monophosphate, which provides close to 18k B T at temperature T = 300 K Note that this is still almost two orders of magnitude away from k B T log 2. Several groups [10,11,12] have recognized that rapid erasing requires entropy production which pushes up the cost of erasing beyond k B T log 2, and have obtained bounds for this problem. The bounds obtained by such considerations depend on technological parameters like the heat conductivity σ, and not just on fundamental constants of physics and the requirement specifications of the problem. Our answer does not depend on any technological parameters, but only on the requirement specifications τr and τe of the problem
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