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arxiv: 2401.15263 · v4 · submitted 2024-01-27 · ❄️ cond-mat.stat-mech · math-ph· math.MP· quant-ph

Macroscopic Irreversibility in Quantum Systems: Free Expansion in a Fermion Chain

Hal Tasaki This is my paper

Pith reviewed 2026-05-24 04:40 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MPquant-ph
keywords free fermion chainquantum irreversibilityunitary evolutionlarge deviation boundenergy eigenstateballistic diffusioncoarse-grained densitystatistical mechanics
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The pith

Unitary quantum evolution in a free fermion chain produces almost uniform coarse-grained density at typical large times from any initial state.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that in a free fermion chain with uniform hopping, any initial state with a fixed macroscopic particle number evolves under unitary dynamics so that the measured coarse-grained density becomes almost uniform at sufficiently large and typical times, with quantum probability extremely close to one. This shows irreversible ballistic diffusion emerging directly from reversible quantum evolution without randomness in the initial state or Hamiltonian. A sympathetic reader cares because the result applies to arbitrary initial states, unlike earlier examples that required randomness or restrictions, and it addresses how macroscopic irreversibility can arise in closed quantum systems. The proof introduces a large deviation bound that holds for every energy eigenstate.

Core claim

For a free fermion chain with uniform nearest-neighbor hopping evolving from an arbitrary initial state with fixed macroscopic particle number, 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, specifically ballistic diffusion, in a system governed by quantum mechanical unitary time evolution from any initial state without randomness.

What carries the argument

The large deviation bound for every energy eigenstate, which bounds fluctuations of the coarse-grained density within each eigenstate and permits the uniformity result to hold for arbitrary superpositions representing any initial state.

If this is right

  • Irreversible macroscopic behavior appears in a quantum system without any randomness.
  • Ballistic diffusion is realized as the concrete form of the irreversibility.
  • The uniformity result holds for every initial state with fixed particle number.
  • The measured coarse-grained density is the observable that becomes uniform.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar large-deviation bounds per eigenstate could be sought in other integrable chains to obtain parallel irreversibility proofs.
  • Cold-atom realizations of one-dimensional fermions offer a direct experimental test of the predicted uniformity after long evolution times.
  • The approach separates the question of eigenstate fluctuations from the need for chaotic Hamiltonians.

Load-bearing premise

A large deviation bound on the coarse-grained density holds inside every energy eigenstate.

What would settle it

An exact diagonalization or numerical time evolution for a concrete initial state on a moderate-size chain that finds the probability of non-uniform density staying order-one at large typical times.

read the original 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).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper considers a free fermion chain with uniform nearest-neighbor hopping evolving unitarily from an arbitrary initial state with fixed macroscopic particle number. It claims to prove that at sufficiently large and typical times the coarse-grained density measurement yields an almost uniform distribution with quantum probability exponentially close to 1, establishing ballistic diffusion and macroscopic irreversibility without any randomness in the initial state or Hamiltonian. The essential new ingredient is asserted to be a large deviation bound that holds for every energy eigenstate (reminiscent of strong ETH), which is then used to control the time-evolved superposition.

Significance. If the large-deviation bound for literally every eigenstate can be established with explicit scaling, the result would be significant: it would demonstrate emergence of irreversible behavior from unitary dynamics alone for arbitrary (not merely typical) initial states, going beyond standard ETH applications that rely on typicality. The approach avoids introducing randomness and targets a concrete observable (coarse-grained density), which strengthens its conceptual interest in the foundations of statistical mechanics.

major comments (2)
  1. [Abstract; §3 (proof outline)] The central claim rests on the large deviation bound holding for every energy eigenstate (not merely typical ones). The manuscript asserts this bound as the essential new ingredient but supplies neither an explicit probability estimate nor a scaling with system size L, nor a proof that rules out counter-example eigenstates such as Slater determinants whose occupied momenta produce atypically large bin-to-bin fluctuations. Because the argument expands an arbitrary initial state in the energy basis and transfers the bound to the time-evolved state, failure of the bound for even one eigenstate would invalidate the step that converts the eigenstate property into a statement for generic superpositions.
  2. [§4 (time averaging)] The measure of 'typical time' is not shown to be robust against possible degeneracies or commensurate energies in the spectrum of the free-fermion Hamiltonian. Without an explicit lower bound on the measure of times where the bound applies, the statement that irreversibility emerges 'at a sufficiently large and typical time' remains formally incomplete.
minor comments (2)
  1. [§2] Notation for the coarse-graining bins and the precise definition of the measured density operator should be stated once in a dedicated paragraph rather than introduced piecemeal.
  2. [Abstract vs. §2] The abstract states the result holds 'from any initial state'; the body should explicitly list the technical assumptions (fixed particle number, finite support in momentum space, etc.) that are actually used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for recognizing the potential significance of proving macroscopic irreversibility from unitary dynamics alone for arbitrary initial states. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract; §3 (proof outline)] The central claim rests on the large deviation bound holding for every energy eigenstate (not merely typical ones). The manuscript asserts this bound as the essential new ingredient but supplies neither an explicit probability estimate nor a scaling with system size L, nor a proof that rules out counter-example eigenstates such as Slater determinants whose occupied momenta produce atypically large bin-to-bin fluctuations. Because the argument expands an arbitrary initial state in the energy basis and transfers the bound to the time-evolved state, failure of the bound for even one eigenstate would invalidate the step that converts the eigenstate property into a statement for generic superpositions.

    Authors: We agree that the large-deviation bound must be stated with an explicit probability estimate and L-scaling for the argument to be complete. In the revised §3 we will formulate the bound explicitly and supply the direct calculation, based on the plane-wave structure of the free-fermion eigenstates, that establishes the bound for every energy eigenstate. This calculation also rules out the suggested counter-examples: any momentum-space Slater determinant has a uniform occupation of the Brillouin zone, so the variance of the coarse-grained density is O(1/L) and the large-deviation probability is exponentially small in L. We will add a short paragraph making this explicit. revision: yes

  2. Referee: [§4 (time averaging)] The measure of 'typical time' is not shown to be robust against possible degeneracies or commensurate energies in the spectrum of the free-fermion Hamiltonian. Without an explicit lower bound on the measure of times where the bound applies, the statement that irreversibility emerges 'at a sufficiently large and typical time' remains formally incomplete.

    Authors: We acknowledge that the present treatment of typical times in §4 does not explicitly address spectral degeneracies or commensurate frequencies. For the free-fermion dispersion the energies are E_k = 2 cos(2π k/L); degeneracies occur only for isolated system sizes and can be absorbed into degenerate subspaces. In the revision we will add an explicit lower bound showing that the measure of times satisfying the large-deviation inequality tends to 1 as the averaging window tends to infinity. The argument uses the quasi-periodic nature of the time evolution and notes that degeneracies merely reduce the number of independent frequencies without preventing equidistribution on the torus. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on model-specific large-deviation bound proved within the paper

full rationale

The paper derives macroscopic irreversibility for the free-fermion chain by expanding an arbitrary initial state in the energy eigenbasis, applying a large-deviation bound asserted to hold for every eigenstate of this solvable model, and showing that typical-time evolution yields near-uniform coarse-grained density with probability exponentially close to 1. This bound is presented as the essential new ingredient and is established directly from the model's single-particle spectrum and fermionic statistics rather than by fitting, self-definition, or reduction to prior self-citations. No load-bearing step equates the target result to its inputs by construction; the argument remains self-contained against the unitary dynamics and the explicit eigenstate property.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the large deviation bound for energy eigenstates (introduced as the essential new ingredient) and the domain assumption of the free fermion chain model with uniform hopping; no free parameters or invented entities are indicated.

axioms (2)
  • domain assumption The system is a free fermion chain with uniform nearest-neighbor hopping
    Explicitly stated as the model under consideration in the abstract.
  • ad hoc to paper A large deviation bound holds for every energy eigenstate
    Presented as the essential new ingredient reminiscent of the strong ETH that enables the proof.

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