[Paper Review] Comment on "Foundation of Statistical Mechanics under Experimentally Realistic Conditions"

Zhexuan Gong, Luming Duan|arXiv (Cornell University)|Sep 22, 2011

Advanced Thermodynamics and Statistical Mechanics3 citations

TL;DR

This paper challenges a key claim in Reimann (2008) that isolated quantum many-body systems equilibrate under generic conditions by showing a critical flaw in the proof of the time-averaged variance bound. The authors demonstrate that for experimentally realistic systems, the assumption that only diagonal terms ($j=n, k=m$) contribute to the time average is invalid due to exponentially many near-resonant terms in the dense energy spectrum, rendering the claimed exponential suppression of fluctuations unjustified.

ABSTRACT

Reimann [Phys. Rev. Lett. 101, 190403 (2008)] claimed that generic isolated macroscopic quantum system will equilibrate under experimentally realistic conditions by proving a theorem. We here show that the proof is invalid for most many-body systems and is unable to demonstrate equilibration in realistic experiment.

Motivation & Objective

• To identify and expose a critical flaw in the proof of equilibration in Reimann (2008) for isolated quantum many-body systems.
• To demonstrate that the assumption that only diagonal terms ($j=n, k=m$) contribute to the time-averaged variance is invalid under experimentally realistic conditions.
• To show that the energy spectrum of many-body systems is exponentially dense, leading to a vast number of near-resonant terms with non-vanishing time-averaged matrix elements.
• To argue that the claimed exponential suppression of fluctuations in observables is therefore unjustified due to the unaccounted contribution of exponentially many off-diagonal terms.
• To clarify that the two conditions in Reimann’s proof—broadly spread initial energy populations and bounded observable range—are insufficient to guarantee equilibration without additional constraints.

Proposed method

• Analyzes the time-averaged variance expression $\sigma_A^2 = \overline{[\text{Tr}(\rho(t)A) - \text{Tr}(\rho_{\text{eq}}A)]^2}$ in Reimann’s Eq. (11), focusing on the phase factor $\overline{e^{i(E_j - E_k + E_m - E_n)t}}$.
• Identifies the critical error: Reimann assumes only terms with $j=n$ and $k=m$ survive the time average, discarding all others.
• Demonstrates that for quadratic Hamiltonians (e.g., non-interacting quasi-particles), the condition $E_j - E_k + E_m - E_n = 0$ can be satisfied via $j_i + m_i = k_i + n_i$, leading to exponentially many non-diagonal solutions.
• Shows that for generic many-body systems, the energy spectrum is exponentially dense, so for any finite time window $T$, there are exponentially many energy differences $|E_j - E_k + E_m - E_n| < \epsilon \sim 1/T$ with non-vanishing time-averaged matrix elements.
• Argues that resolving such energy differences requires time scales far exceeding the age of the universe, making the time average physically irrelevant for realistic systems.
• Concludes that the suppression of $\sigma_A^2$ by $\sum_n \rho_{nn}^2(0)$—claimed to be exponentially small—is an artifact of dropping these resonant terms.

Experimental results

Research questions

• RQ1 Does the time-averaged variance of an observable in an isolated quantum many-body system truly vanish exponentially under Reimann’s two conditions?
• RQ2 Are the off-diagonal terms in the time-averaged variance expression, with $j \neq n$ or $k \neq m$, negligible for realistic many-body systems?
• RQ3 Can the energy spectrum of a many-body system be resolved with finite time averaging, given its exponential density?
• RQ4 Is the assumption that $\overline{e^{i(E_j - E_k + E_m - E_n)t}}$ vanishes for non-diagonal terms valid under experimentally realistic conditions?
• RQ5 Does the claimed exponential suppression of fluctuations in Reimann’s proof hold when all resonant terms are properly accounted for?

Key findings

• The proof of Eq. (1) in Reimann (2008) is flawed because it unjustifiably discards exponentially many off-diagonal terms in the time-averaged variance expression.
• For quadratic Hamiltonians, the condition $E_j - E_k + E_m - E_n = 0$ admits exponentially many solutions beyond the diagonal terms $j=n, k=m$, such as $j_i + m_i = k_i + n_i$.
• In generic many-body systems, the energy spectrum is exponentially dense, leading to an exponentially large number of near-resonant terms with $|E_j - E_k + E_m - E_n| < \epsilon \sim 1/T$ for any finite time window $T$.
• These near-resonant terms contribute non-vanishingly to the time average, invalidating the assumption that only diagonal terms matter.
• The required time to resolve energy differences in the spectrum exceeds the age of the universe for systems with hundreds of particles, making the time average physically meaningless in practice.
• Therefore, the claimed exponential suppression of $\sigma_A^2$ by $\sum_n \rho_{nn}^2(0)$ is an artifact of an incorrect simplification and does not hold under experimentally realistic conditions.