Mixed Entropy Power Inequalities and Log-Concavity of Equilibrium Distribution with application to the Physical Limits of Computation

Landauer's bound establishes a quantitative relationship between logically irreversible manipulations of information and the associated energy consumption. Explicitly, it asserts that the amount of energy required to erase one bit of information is kT ln 2, and thus proportional to the decrease in entropy of the system. A Brownian particle in a bi-stable potential is a commonly used model for a single bit of memory. We show that the equilibrium distribution of the Brownian particle before and after erasure can be modeled as a weighted mixture of mixed random variables(product of a continuous log concave and Bernoulli random variable), where the discreteness is associated with the success of the erasure and the continuous log concave aspect is associated to the potential being a continous convex function. Additionally, we prove that the equilibrium distribution obeys an Entropy Power Inequality analogous to Shannon's. In this framework of log concave distributions, we present bounds on the decrease in entropy associated with erasure and show convergence to the Landauer's bound when the log concave distributions for the two states of a memory do not overlap.