Macroscopic Irreversibility in Quantum Systems: Free Expansion in a Fermion Chain (2401.15263v4)

Abstract: We consider a free fermion chain with uniform nearest-neighbor hopping and let it evolve from an arbitrary initial state with a fixed macroscopic number of particles. We then prove that, at a sufficiently large and typical time, the measured coarse-grained density distribution is almost uniform with (quantum mechanical) probability extremely close to one. This establishes the emergence of irreversible behavior, i.e., a ballistic diffusion, in a system governed by quantum mechanical unitary time evolution. It is conceptually important that irreversibility from any initial state is proved here without introducing any randomness to the initial state or the Hamiltonian, while the known examples, both classical and quantum, rely on certain randomness or apply to limited classes of initial states. The essential new ingredient in the proof is the large deviation bound for every energy eigenstate, which is reminiscent of the strong ETH (energy eigenstate thermalization hypothesis).

• The paper demonstrates that a free fermion chain exhibits macroscopic irreversibility despite unitary evolution, challenging traditional equilibrium assumptions.
• It employs rigorous mathematical proofs, notably Theorems 2 and 3, to show that almost all time evolution leads to equilibrium with high probability.
• The results extend ETH by illustrating that deterministic quantum systems can thermalize naturally, without requiring randomness in initial conditions or Hamiltonians.

Macroscopic Irreversibility in Quantum Systems: ETH and Equilibration in a Free Fermion Chain

In the paper "Macroscopic Irreversibility in Quantum Systems: ETH and Equilibration in a Free Fermion Chain," the author examines the emergence of macroscopic irreversibility within quantum systems governed by unitary time evolution. The premise of this paper challenges traditional views by demonstrating irreversible behavior in a free fermion chain without relying on randomness in the initial state or Hamiltonian. This paper primarily addresses the energy eigenstate thermalization hypothesis (ETH) and its implications on macroscopic quantum systems as they approach equilibrium.

Key Concepts and Theorems

The paper considers a chain of free fermions with uniform nearest-neighbor hopping. The focus is to analyze the system's evolution from an arbitrary initial state with a fixed number of particles. Through rigorous proofs, the paper asserts that such a system will exhibit macroscopic irreversibility—embodied as ballistic diffusion—with quantum mechanical probability close to one. This conclusion is achieved without introducing stochastic elements, which is conceptually pivotal.

Two significant results, Theorems 2 and 3, encapsulate the findings:

1. Theorem 2: For any initial state of a free fermion chain with $N$ particles, the time-averaged probability that the system remains in a nonequilibrium state is insignificantly small as time approaches infinity. This theorem underscores that such irreversibility can be observed uniformly without relying on randomness in the configuration.
2. Theorem 3: This theorem extends Theorem 2, indicating that for almost any sufficiently large time $T$, the measurement outcome for a coarse-grained density distribution will reflect an equilibrium state. This affirms that the system's behavior over time leads to a state where particle distribution is typically uniform.

Theorems are supported by Lemmas that establish critical properties of the fermion chain, particularly the non-degeneracy of energy eigenvalues and a strong version of ETH in a large-deviation setting. These elements are vital for establishing the validity of the presented irreversibility.

Implications and Discussion

The implications of this paper are manifold, both theoretically and practically. It lays down a cornerstone for understanding macroscopic irreversibility without recourse to randomness in quantum systems, potentially influencing fields concerned with quantum thermodynamics and statistical mechanics. The contrast between classical and quantum systems is illuminated: while classical irreversibility often requires randomness or specific probabilistic initial conditions, this paper shows that quantum systems can exhibit irreversible behavior deterministically.

Practically, the results hold promise for elucidating thermalization processes in quantum systems, providing a framework that does not depend on stochastic initial state distributions. This has potential relevance for developing quantum technologies where control over entropy production and state predictability at a macroscopic level is crucial.

Future Directions

The paper identifies several areas for future research. One significant challenge is determining the specific time scales over which equilibration and irreversibility occur. Addressing this would enhance the practical utility of the theoretical results. Moreover, expanding the framework to include more complex, non-integrable quantum systems could broaden the applicability of the findings beyond free fermion chains. Another intriguing extension would be to investigate other observables beyond density distributions, considering their potential impact on understanding irreversible processes in quantum systems.

In conclusion, this paper contributes to the ongoing discourse on quantum system behavior, challenging conventional wisdom by demonstrating inherent irreversible processes without stochastic preconditions. It also extends foundational ETH concepts within the context of macroscopic equilibration, marking a notable development in quantum statistical mechanics and its many-body applications.