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We show, without relying on any unproven assumptions, that a low-density freefermion chain exhibits thermalization in the following (restricted) sense. Wechoose the initial state as a pure state drawn randomly from the Hilbert spacein which all particles are in half of the chain. This represents anonequilibrium state such that the half chain containing all particles is inequilibrium at infinite temperature, and the other half chain is a vacuum. Welet the system evolve according to the unitary time evolution determined by theHamiltonian and, at a sufficiently large typical time, measure the particlenumber in an arbitrary macroscopic region in the chain. In this setup, it isproved that the measured number is close to the equilibrium value withprobability very close to one. Our result establishes the presence ofthermalization in a concrete model in a mathematically rigorous manner. Themost important theoretical ingredient for the proof of thermalization is thedemonstration that a nonequilibrium initial state generated as above typicallyhas a sufficiently large effective dimension. Here, we first give general proofof thermalization based on two assumptions, namely, the absence of degeneracyin energy eigenvalues and a property about the particle distribution in energyeigenstates. We then justify these assumptions in a concrete free-fermionmodel, where the absence of degeneracy is established by using number-theoreticresults. This means that our general result also applies to any lattice gasmodels in which the above two assumptions are justified. To confirm thepotential wide applicability of our theory, we discuss some other models forwhich the essential assumption about the particle distribution is easilyverified, and some non-random initial states whose effective dimensions aresufficiently large.

Nature abhors a vacuum: A simple rigorous example of thermalization in an isolated macroscopic quantum system