  In the present note, we discuss a simple example of a macroscopic quantum
many-body system in which the approach to thermal equilibrium from an arbitrary
initial state in the microcanonical energy shell is proved without relying on
any unproven assumptions. The model, which is equivalent to a free fermion
chain, is designed to be a toy model for a weakly heat-conducting
one-dimensional solid. We take a phenomenological point of view and perceive
that the system is in thermal equilibrium when the measured coarse-grained
energy distribution is uniform.
  The result on thermalization reported here is a variation (and an
improvement) of our previous result on the irreversible expansion in a free
fermion chain. As far as we know, this is the first concrete and rigorous
realization of the philosophy on the foundation of equilibrium statistical
mechanics proposed by von Neumann in 1929, and further developed recently by
Goldstein, Lebowitz, Mastrodonato, Tumulka, and Zangh\`\i and the present
author, namely, to characterize thermal equilibrium from a macroscopic
viewpoint and to make use of the strong ETH to control the long-time dynamics.
  This note will be the most technical part of my longer article on
thermalization, "What is thermal equilibrium and how do we get there?". I am
making this document public at this stage since I have already announced (and
will announce) the results at some of my talks.
