Thermodynamics of stochastic Turing machines

TL;DR

This paper models stochastic Turing machines as Markovian systems to analyze their thermodynamic properties, revealing that entropy production can be minimized locally but accumulates over computation.

Contribution

It introduces a thermodynamic framework for stochastic Turing machines using Markov processes, linking computation with entropy production analysis.

Findings

Steady-state entropy production rate can be made arbitrarily small.

Total entropy production grows logarithmically with the number of steps.

Master equation can be approximated by a Fokker-Planck equation in the stationary regime.

Abstract

In analogy to Brownian computers we explicitly show how to construct stochastic models, which mimic the behaviour of a general purpose computer (a Turing machine). Our models are discrete state systems obeying a Markovian master equation, which are logically reversible and have a well-defined and consistent thermodynamic interpretation. The resulting master equation, which describes a simple one-step process on an enormously large state space, allows us to thoroughly investigate the thermodynamics of computation for this situation. Especially, in the stationary regime we can well approximate the master equation by a simple Fokker-Planck equation in one dimension. We then show that the entropy production rate at steady state can be made arbitrarily small, but the total (integrated) entropy production is finite and grows logarithmically with the number of computational steps.