  We show, without relying on any unproven assumptions, that a low-density free
fermion chain exhibits thermalization in the following (restricted) sense. We
choose the initial state as a pure state drawn randomly from the Hilbert space
in which all particles are in half of the chain. This represents a
nonequilibrium state such that the half chain containing all particles is in
equilibrium at infinite temperature, and the other half chain is a vacuum. We
let the system evolve according to the unitary time evolution determined by the
Hamiltonian and, at a sufficiently large typical time, measure the particle
number in an arbitrary macroscopic region in the chain. In this setup, it is
proved that the measured number is close to the equilibrium value with
probability very close to one. Our result establishes the presence of
thermalization in a concrete model in a mathematically rigorous manner. The
most important theoretical ingredient for the proof of thermalization is the
demonstration that a nonequilibrium initial state generated as above typically
has a sufficiently large effective dimension. Here, we first give general proof
of thermalization based on two assumptions, namely, the absence of degeneracy
in energy eigenvalues and a property about the particle distribution in energy
eigenstates. We then justify these assumptions in a concrete free-fermion
model, where the absence of degeneracy is established by using number-theoretic
results. This means that our general result also applies to any lattice gas
models in which the above two assumptions are justified. To confirm the
potential wide applicability of our theory, we discuss some other models for
which the essential assumption about the particle distribution is easily
verified, and some non-random initial states whose effective dimensions are
sufficiently large.
