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

This paper proves that a free fermion chain exhibits macroscopic irreversibility through ballistic diffusion toward a uniform density distribution at typical times, establishing this emergence from unitary quantum evolution without relying on randomness or restricted initial states by utilizing a large deviation bound for energy eigenstates.

The Gist

The Big Question: Why Does Time Only Move Forward?

Imagine you drop a drop of red ink into a glass of clear water. Over time, the ink spreads out until the whole glass is a uniform pink. This is irreversibility. You never see the pink water spontaneously un-mix itself back into a single red drop.

In the world of classical physics (like billiard balls or gas molecules), we know this happens because of chaos and randomness. If you have trillions of particles moving randomly, it's statistically impossible for them all to accidentally line up to go back to the start.

But quantum physics is weird. At the smallest level, the laws of physics are reversible. If you played a movie of two quantum particles bouncing off each other backward, it would look perfectly normal. So, how does a universe made of reversible quantum rules create a world where time only moves forward?

The Paper's Discovery: A Magic Trick with Quantum Particles

Physicist Hal Tasaki has proven a surprising fact: You don't need randomness to get irreversibility in quantum systems.

Usually, scientists say, "If you start with a random mess of particles, they will spread out." But Tasaki says, "No, even if you start with a perfectly ordered, non-random setup, and even if the particles follow strict, predictable quantum rules, they will still spread out and never go back."

He proved this using a specific model: a chain of free fermions (a type of quantum particle).

The Analogy: The "Quantum Dance Floor"

Imagine a long, circular dance floor (the chain) with $L$ spots. There are $N$ dancers (the fermions) on it.

1. The Setup: At the start ($t=0$), all the dancers are crowded into one small corner of the room. They are standing perfectly still in a specific formation. This is a very "ordered" state.
2. The Rules: The dancers are quantum particles. They don't bump into each other (they are "free"). They move according to a strict, unchangeable song (the Hamiltonian). The song is deterministic; there is no DJ changing the beat randomly.
3. The Dance: As time passes, the dancers move. Because of quantum mechanics, they don't just walk in straight lines; they spread out like waves.
4. The Result: Tasaki proved that if you wait long enough, the dancers will be spread out so evenly across the dance floor that if you took a snapshot, it would look like a uniform crowd.

The Magic Part: In classical physics, if you arranged the dancers perfectly and told them to move in a specific pattern, they might eventually all return to the corner (like a clock ticking backward). But Tasaki proved that for this quantum system, no matter how you arrange the dancers initially, they will almost certainly spread out and stay spread out. They won't spontaneously gather back into a corner.

The "No-Guessing" Rule

Here is the most important part of the paper:

• Old View: To get irreversibility, you usually have to assume the starting state is "random" (like shuffling a deck of cards).
• Tasaki's View: You don't need to shuffle the cards. Even if the deck is perfectly ordered (Ace to King), the quantum rules will force the cards to spread out and look random on their own.

He proved that for any starting arrangement, the system will evolve into a state that looks "equilibrium" (uniform) at almost any random time you check.

How Did He Prove It? (The "Strong ETH" Secret)

To prove this, Tasaki used a concept called the Eigenstate Thermalization Hypothesis (ETH).

Think of the quantum system as having a library of "energy songs" (energy eigenstates).

• The Old Idea: We thought only a mix of these songs would look random.
• Tasaki's New Ingredient: He proved that every single song in the library, by itself, already looks like a uniform, spread-out crowd.

It's like saying that even if you play just one specific note on a piano, it sounds like a full orchestra. Because every possible state of the system is already "spread out," any combination of them will also be spread out.

Why Does This Matter?

1. It Solves a Paradox: It explains how a world built on reversible quantum laws creates the "Arrow of Time" (the fact that we remember the past but not the future) without needing to cheat by adding randomness.
2. It's a Blueprint: While this specific example uses a simplified model (a chain of particles), the math suggests this behavior is a fundamental feature of quantum systems. It hints that even in complex, real-world materials, order naturally turns into disorder.
3. It's Not "Thermalization" (Yet): The paper notes that while the particles spread out, they don't necessarily reach "thermal equilibrium" in the strict sense (where they forget everything about how they started). They reach a "steady state" that looks uniform, but they still hold onto some quantum memory. However, this is the first step toward understanding full thermalization.

The "Time Travel" Paradox (The Conclusion)

The paper addresses a classic objection: "If the laws are reversible, why can't we just reverse time?"

Tasaki explains it like this:
Imagine you take a photo of the dancers when they are perfectly spread out (Time $T$). If you hit "rewind" on the universe, they would go back to the corner.
BUT, to hit "rewind," you would need to set up the universe in a state that is impossibly specific and rare.

• It is easy to start with a corner crowd and let them spread (Forward).
• It is statistically impossible to start with a spread-out crowd and have them accidentally gather into a corner (Backward).

The "Arrow of Time" exists not because the laws of physics change, but because the "spread out" state is the typical state, and the "crowded" state is the rare state. The universe naturally flows from the rare to the typical.

Summary

Hal Tasaki has shown that in the quantum world, order naturally dissolves into uniformity without needing any external randomness. Even if you start with a perfectly arranged line of quantum particles, the strict rules of quantum mechanics will eventually force them to spread out evenly, creating a one-way street for time.

Technical Summary

1. Problem Statement

The paper addresses a fundamental problem in statistical mechanics: how macroscopic irreversibility (the arrow of time) emerges from microscopic laws that are time-reversible (unitary quantum evolution).

• Context: In classical deterministic systems, irreversibility is typically observed only for "typical" initial states drawn from a random probability distribution. Deterministic rules (e.g., all particles having identical velocities) often fail to produce irreversibility.
• The Gap: While the Eigenstate Thermalization Hypothesis (ETH) suggests that isolated quantum systems thermalize, rigorous proofs of macroscopic irreversibility for arbitrary initial states in deterministic quantum systems have been scarce. Most existing rigorous examples rely on random Hamiltonians or random initial states.
• Goal: To rigorously prove that a non-random, integrable quantum many-body system (a free fermion chain) exhibits irreversible expansion from any arbitrary initial state, without introducing randomness into the Hamiltonian or the initial conditions.

2. Model and Methodology

The Physical Model

The author considers a system of $N$ spinless free fermions on a 1D chain of length $L$ with periodic boundary conditions.

• Hamiltonian: A standard nearest-neighbor hopping model with an artificial flux $\theta$:
  $$\hat{H} = \sum_{x=1}^L \left( e^{i\theta} \hat{c}^\dagger_x \hat{c}_{x+1} + e^{-i\theta} \hat{c}^\dagger_{x+1} \hat{c}_x \right)$$
• Constraints:
  • $L$ is chosen to be a large prime number.
  • $N < L$ (density $\rho_0 = N/L$).
  • The flux $\theta$ is carefully chosen (e.g., $\theta \neq 0$ and small) to ensure the absence of energy degeneracy in the many-body spectrum.

Mathematical Framework

The proof relies on three main pillars:

1. Non-degeneracy: Proven via Lemma 1, ensuring that for the chosen $L$ and $\theta$, all many-body energy eigenvalues $E_k$ are distinct. This eliminates quantum recurrences that could prevent equilibration over long times.
2. Coarse-Grained Observables: The system is divided into $m$ intervals ($I_1, \dots, I_m$). The observable of interest is the particle density $\hat{\rho}_j = \hat{N}_j / \ell$ in each interval.
3. Equilibrium Definition: A state is considered "in equilibrium" if the measured densities $\rho_j$ satisfy $|\rho_j - \rho_0| \leq \rho_0 \delta$ for all $j$, where $\delta$ is a small tolerance. The "non-equilibrium" projection operator $\hat{P}_{\text{neq}}$ projects onto states violating this condition.

Key Technical Tool: Large Deviation Bound

The core of the methodology is Lemma 4, which establishes a "Strong ETH-like" bound for every single energy eigenstate.

• Lemma 4: For any energy eigenstate $|\Psi_k\rangle$, the probability of finding the system out of equilibrium is exponentially small:
  $$\langle \Psi_k | \hat{P}_{\text{neq}} | \Psi_k \rangle \leq e^{-\frac{\delta^2}{3(m-1)} N}$$
• Proof Strategy: The proof utilizes a large deviation bound derived from the properties of free fermions. It involves bounding the expectation value of the exponential of the particle number operator $\langle e^{\lambda \hat{N}_j} \rangle$ within an eigenstate, showing that fluctuations are suppressed as $e^{-cN}$.

3. Key Results

The paper presents two main theorems establishing irreversibility:

Theorem 2: Ergodic Version (Time-Averaged)

For any initial state $|\Phi(0)\rangle$ with $N$ particles, the long-time average of the probability of being in a non-equilibrium state is exponentially small:
$$\lim_{T \to \infty} \frac{1}{T} \int_0^T dt \, \langle \Phi(t) | \hat{P}_{\text{neq}} | \Phi(t) \rangle \leq e^{-\frac{\delta^2}{3(m-1)} N}$$

• Significance: This holds for every initial state, not just a random ensemble. It proves that the system spends the vast majority of its time in equilibrium.

Theorem 3: Instantaneous Version (Typical Time)

For any initial state and sufficiently large $T$, there exists a subset of "atypical" times $A \subset [0, T]$ with a very small measure ($l(A)/T \leq e^{-cN}$) such that for all $t \in [0, T] \setminus A$:
$$\langle \Phi(t) | \hat{P}_{\text{neq}} | \Phi(t) \rangle \leq e^{-\frac{\delta^2}{8(m-1)} N}$$

• Interpretation: At a "sufficiently large and typical" time $t$, a single quantum measurement of the coarse-grained density will yield a uniform distribution with probability extremely close to 1.
• Irreversibility: If the system starts in a localized state (e.g., all particles in $I_1$), it evolves to a uniform distribution. The reverse process (uniform to localized) is statistically impossible for typical times, establishing the arrow of time.

4. Distinction from Thermalization

The author explicitly clarifies that while the system exhibits irreversibility, it does not necessarily exhibit thermalization in the strict sense (approaching a Gibbs state).

• Integrability: The free fermion chain is integrable. The long-time stationary state is the "diagonal ensemble" $\hat{\rho}_{\text{diag}} = \sum_k |\alpha_k|^2 |\Psi_k\rangle\langle\Psi_k|$, which retains memory of the initial momentum distribution.
• Contrast: In non-integrable systems, the Strong ETH implies that all macroscopic observables thermalize. Here, only the coarse-grained density relaxes to uniformity, while other observables (like momentum distribution) do not.
• Significance: This demonstrates that macroscopic irreversibility (relaxation to a uniform density) can occur even in the absence of full thermalization.

5. Significance and Impact

1. Arbitrary Initial States: This is the first rigorous proof of macroscopic irreversibility in a deterministic quantum system that holds for any initial state. Previous rigorous results required random initial conditions or random Hamiltonians.
2. ETH Connection: The proof introduces a "Strong ETH" bound (Lemma 4) applied to every eigenstate, rather than just a typical one. This suggests that the mechanism for irreversibility is deeply rooted in the structure of individual energy eigenstates.
3. Resolution of Time-Reversal Paradox: The paper addresses the apparent paradox of time-reversal symmetry. While the equations are reversible, the set of initial states that evolve from uniform to localized is "atypical" (measure zero in the thermodynamic limit). The theorem shows that for any specific initial state, the "return" to the initial localized state occurs only at atypical times.
4. Future Directions: The author posits that this mechanism is robust and likely extends to non-integrable systems, where full thermalization would occur. The paper suggests that breaking integrability (e.g., via interactions) should preserve the irreversibility of the density distribution while restoring full thermalization.

Conclusion

Hal Tasaki provides a rigorous mathematical demonstration that a free fermion chain evolves from an arbitrary non-equilibrium state to a uniform density distribution with probability approaching unity. By leveraging the non-degeneracy of the spectrum and a large-deviation bound for eigenstates, the work establishes that macroscopic irreversibility is an intrinsic property of unitary quantum dynamics for large systems, independent of randomness in the initial conditions.