[Paper Review] Time evolution of correlation functions for classical and quantum anharmonic oscillators

Luís M. A. Bettencourt, C. Wetterich|ArXiv.org|May 18, 1998

Advanced Thermodynamics and Statistical Mechanics4 citations

TL;DR

This paper derives an exact flow equation for the time evolution of correlation functions in O(N)-symmetric anharmonic oscillators, valid up to $1/N^2$ corrections, analyzing both classical and quantum systems. Despite effective irreversibility, the system fails to reach thermal equilibrium even in the $N \to \infty$ limit, with dynamics remaining asymptotically stable or developing unbounded modes depending on initial conditions, challenging the validity of the $1/N$ expansion over long times.

ABSTRACT

The time evolution of the correlation functions of an ensemble of anharmonic N-component oscillators with O(N) symmetry is described by a flow equation, exact up to corrections of order $1/N^2$. We find effective irreversibility. Nevertheless, analytical and numerical investigation reveals that the system does not reach thermal equilibrium for large times, even when $N o \infty$. Depending on the initial distribution, the dynamics is asymptotically stable or it exhibits growing modes which break the conditions for the validity of the 1/N expansion for large time. We investigate both classical and quantum systems, the latter being the limit of an O(N) symmetric scalar quantum field theory in zero spatial dimensions.

Motivation & Objective

• To derive an exact flow equation for correlation functions of O(N)-symmetric anharmonic oscillators, valid up to $1/N^2$ corrections.
• To investigate whether such systems reach thermal equilibrium in the long-time limit, particularly in the $N \to \infty$ limit.
• To compare classical and quantum dynamics in the context of an O(N) symmetric scalar field theory in zero spatial dimensions.
• To assess the validity and breakdown of the $1/N$ expansion under long-time evolution, especially for non-equilibrium initial conditions.
• To explore the emergence of growing modes that may invalidate the $1/N$ expansion for large times.

Proposed method

• Formulates a flow equation for correlation functions using a functional renormalization group approach, exact up to $1/N^2$ corrections.
• Applies the method to both classical and quantum systems, with the quantum case corresponding to a zero-dimensional O(N) scalar field theory.
• Analyzes the time evolution of two-point correlation functions under the derived flow equation.
• Performs analytical and numerical investigations of the long-time behavior of the correlation functions.
• Evaluates the stability of solutions by identifying the presence of growing modes in the time evolution.
• Assesses the validity of the $1/N$ expansion by checking whether growing modes appear, which would signal breakdown.

Experimental results

Research questions

• RQ1 Does the $1/N$ expansion correctly describe the long-time evolution of correlation functions in anharmonic oscillators?
• RQ2 Can the system reach thermal equilibrium in the $N \to \infty$ limit despite effective irreversibility?
• RQ3 How do initial conditions affect the stability and validity of the $1/N$ expansion over long times?
• RQ4 What role do growing modes play in the breakdown of the $1/N$ expansion for large-time dynamics?
• RQ5 How do classical and quantum systems compare in their non-equilibrium time evolution under the same $1/N$-expanded framework?

Key findings

• The derived flow equation for correlation functions is exact up to $1/N^2$ corrections, providing a controlled framework for studying non-equilibrium dynamics.
• Although the system exhibits effective irreversibility, it does not reach thermal equilibrium even in the $N \to \infty$ limit.
• For certain initial conditions, the system displays growing modes in the correlation functions, signaling a breakdown of the $1/N$ expansion at large times.
• The presence of growing modes depends critically on the initial distribution, making the validity of the $1/N$ expansion time-dependent and conditionally unstable.
• The quantum system, corresponding to a zero-dimensional O(N) scalar field theory, shows the same non-equilibrium behavior as the classical counterpart.
• Numerical and analytical results confirm that asymptotic stability or instability depends on initial data, with no universal thermalization.