[Paper Review] Time dependence of occupation numbers and thermalization time in closed chaotic many-body systems

V. V. Flambaum, F. M. Izrailev|ArXiv.org|Aug 23, 2001

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This paper proposes a simple theoretical framework to describe the time evolution of occupation numbers in closed, chaotic many-body systems of interacting Fermi-particles. Using the survival probability of the initial state and a diagonal approximation, it shows that occupation numbers relax to equilibrium values—such as the Fermi-Dirac distribution—on a time scale τ, with excellent agreement between theory and numerical simulations across weak (Breit-Wigner) and strong (Gaussian) interaction regimes.

ABSTRACT

We study the time evolution of occupation numbers for interacting Fermi-particles in the situation when exact compound states are "chaotic". This situation is generic for highly excited many-particles states in heavy nuclei, complex atoms, quantum dots, spin systems and quantum computer models. Numerical data show perfect agreement with a simple theory for the onset of thermalization in close systems of interacting particles.

Motivation & Objective

• To understand the time evolution of occupation numbers in closed, chaotic many-body systems of interacting Fermi-particles.
• To identify the characteristic time scale τ for the onset of thermalization of occupation numbers.
• To develop a simple analytical model that captures the relaxation dynamics of occupation numbers toward equilibrium.
• To validate the model against numerical simulations using a random two-body interaction Hamiltonian.
• To clarify the role of the F-function and level spacing statistics in determining thermalization dynamics.

Proposed method

• The time evolution of the system is modeled using a superposition of exact eigenstates, with the initial state expressed as a sum over these eigenstates.
• The occupation number $ n_{\alpha}(t) $ is computed from the expectation value of the number operator $ \hat{n}_{\alpha} $, decomposed into diagonal and off-diagonal terms.
• The off-diagonal terms $ S_{q}^{(fl)} $ are assumed to average to zero for large times, leaving the diagonal term $ S_{q}^{(d)} $ as the dominant contribution.
• The diagonal term is approximated using the average $ \overline{S_{q}^{(d)}} $, which involves the F-function $ F(E_i, E) $ and the density of states $ \rho(E) $.
• The F-function is modeled using a Breit-Wigner or Gaussian form depending on the interaction strength, with parameters derived from the Fermi golden rule and level spacing statistics.
• The time dependence of the occupation numbers is modeled as $ n_{\alpha}(t) = n_{\alpha}(0)W_0(t) + n_{\alpha}(\infty)(1 - W_0(t)) $, where $ W_0(t) $ is the survival probability of the initial state.

Experimental results

Research questions

• RQ1 What is the time scale τ for the initial thermalization of occupation numbers in a closed chaotic many-body system?
• RQ2 How does the shape of the F-function (Breit-Wigner vs. Gaussian) affect the relaxation dynamics of occupation numbers?
• RQ3 To what extent can the time evolution of occupation numbers be described by a simple analytical model based on the survival probability of the initial state?
• RQ4 How do quantum oscillations and fluctuations differ between weak and strong interaction regimes in the thermalization process?
• RQ5 What is the relationship between the number of principal components in the eigenstate and the observed thermalization time?

Key findings

• The time evolution of occupation numbers is well described by a simple model involving the survival probability $ W_0(t) $, with excellent agreement between theory and numerical simulations.
• In the weak interaction regime (Breit-Wigner F-function), the relaxation exhibits damped oscillations due to a smaller number of principal components, leading to larger fluctuations.
• In the strong interaction regime (Gaussian F-function), the relaxation is monotonic and fast, with small fluctuations, due to a larger number of principal components.
• The thermalization time τ is identified as the characteristic time scale for the onset of thermalization of occupation numbers, distinct from the longer time scale for full statistical equilibrium.
• The equilibrium occupation numbers $ n_{\alpha}(\infty) $ are consistent with the infinite-temperature limit, yielding $ n_{\alpha}(\infty) = n/m = 1/2 $ in the symmetric case studied.
• Numerical simulations with $ n=6 $ Fermi-particles in $ m=12 $ orbitals confirm the theoretical predictions across both interaction regimes, validating the model's robustness.