[Paper Review] Universal Bound on Dynamical Relaxation Time from Condition for Relaxing Quantity to be Classical

K. Ropotenko|ArXiv.org|May 24, 2007

Advanced Thermodynamics and Statistical Mechanics3 citations

TL;DR

This paper derives Hod's universal bound on relaxation time, τ ≥ ℏ/(πT), from fundamental principles of quantum mechanics and fluctuation theory, showing it arises naturally from the condition that a relaxing quantity must behave classically—i.e., quantum fluctuations are negligible when ℏω ≪ kT. The bound emerges from Landau and Lifshitz’s classicality criterion, offering a more foundational derivation than information-theoretic approaches.

ABSTRACT

It is shown that the Hod's universal bound on the relaxation time of a perturbed system \cite{hod} can be derived from a well-known condition for a relaxing quantity to be classical in the fluctuation theory.

Motivation & Objective

• To re-derive Hod's universal bound on relaxation time using quantum mechanics and thermodynamics instead of information theory.
• To show that the condition for a relaxing quantity to be classical—derived from fluctuation theory—directly implies Hod's bound.
• To clarify the physical basis of the bound by linking it to the dominance of classical over quantum fluctuations.
• To demonstrate that the bound holds universally, regardless of whether equilibrium is reached via quantum or thermodynamic fluctuations.

Proposed method

• Uses Landau and Lifshitz’s criterion that a fluctuating quantity x is classical if τ ≫ ℏ/T, ensuring quantum effects are negligible.
• Applies the condition ℏω ≪ kT to matrix elements of x between energy eigenstates, requiring negligible transitions for large energy differences.
• Analyzes relaxation dynamics in thermal equilibrium, focusing on how fast a system returns to equilibrium after perturbation.
• Compares relaxation timescales in macroscopic systems (e.g., sound waves with τ ∼ R/c) to black hole quasinormal modes, which exhibit similar oscillatory relaxation.
• Establishes that τ need not be the full relaxation time to equilibrium, but can be a characteristic timescale of oscillatory decay.
• Uses dimensional analysis and physical consistency to reinforce the universality of the bound.

Experimental results

Research questions

• RQ1 Can Hod's universal relaxation time bound be derived without invoking information theory?
• RQ2 What is the fundamental physical condition that ensures a relaxing quantity behaves classically?
• RQ3 How does the classicality condition ℏω ≪ kT relate to the relaxation time bound τ ≥ ℏ/(πT)?
• RQ4 Why does the bound hold even for systems like black holes where τ ≈ ℏ/T?
• RQ5 Can oscillatory relaxation processes, such as quasinormal modes, still satisfy the classicality condition?

Key findings

• The universal bound τ ≥ ℏ/(πT) follows directly from the classicality condition in fluctuation theory, specifically τ ≫ ℏ/T.
• The condition ℏω ≪ kT ensures that only low-energy transitions contribute significantly to the dynamics of x, making it classical.
• When τ is too small or T too low, quantum fluctuations dominate, and the system cannot be treated classically.
• The bound is more fundamental than previously thought, as it arises from quantum statistical mechanics rather than information-theoretic reasoning.
• Even for black holes with τ ≈ ℏ/T, the relaxation process can still be considered classical because τ may represent an oscillatory decay timescale rather than full equilibration time.
• Quasinormal modes of black holes, which are oscillatory, can still satisfy the classicality condition if their frequency components obey ℏω ≪ kT.