Dynamics from iterated averaging

TL;DR

This work shows that iterated averaging via conditional expectations on $L^ty(X)$ is dynamically rich: the strong operator closure of the semigroup they generate contains the full group of measure-preserving automorphisms on a standard Lebesgue space. The water-tank puzzle provides a concrete analogue, leading to an optimal equilibration strategy whose final red-tank content is $q_{n,n} = \frac{n}{4^n}\binom{2n}{n}$ and asymptotically $\sqrt{\frac{n}{\pi}}$. The authors prove that any measure-preserving automorphism can be approximated by finite compositions of conditional expectations, via a reduction to involutions and a partition-refinement construction. In the finite setting, they relate the dynamics to Dalton transfers and majorization, presenting a constructive characterization and algorithm, with relevance to areas such as quantum thermodynamics through continuous thermomajorization.

Abstract

We prove that for a standard Lebesgue space $X$, the strong operator closure of the semigroup generated by conditional expectations on $L^\infty(X)$ contains the group of measure-preserving automorphisms. This is based on a solution to the following puzzle: given $n$ full water tanks, each containing one unit of water, and $n$ empty ones, how much water can be transferred from the full tanks to the empty ones by repeatedly equilibrating the water levels between pairs of tanks?

Figures (3)

• Figure 1: Pseudocode for the first water tank strategy.
• Figure 2: Schematic of a countercurrent heat exchanger. Hot fluid enters in the left tube on top and cold fluid in the right tube at the bottom. Where the tubes touch, heat is exchanged (squiggly lines). As each fluid exits its tube, its temperature is close to the temperature of the other fluid as it enters, and matches it exactly in the ideal case.
• Figure 3: Pseudocode for a non-deterministic generalization of the two for loops of \ref{['pseudocode1']} to arbitrary initial water levels.

Theorems & Definitions (13)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Remark 2.1
• Theorem 2.2: nitrodon
• proof
• proof : Proof of \ref{['thm:optimaltransport']}
• Lemma 3.1
• proof