Evidence for simple arrow of time functions in closed chaotic quantum systems

Area of Science:

• Statistical Mechanics
• Quantum Chaos
• Theoretical Physics

Background:

• Autocorrelation functions are crucial for understanding system dynamics and approach to equilibrium.
• The concept of 'time' in statistical mechanics is often linked to irreversibility and the approach to equilibrium.
• Investigating nonchaotic regimes is essential for a complete understanding of dynamical systems.

Purpose of the Study:

• To introduce and define a new set of functions, "arrows of time functions" (AOTFs), derived from autocorrelation functions.
• To explore the conditions under which AOTFs exist and their relationship to chaotic and nonchaotic regimes.
• To establish a connection between the existence of AOTFs and the approach to thermodynamic equilibrium.

Main Methods:

• Explicit construction of AOTFs (αⁿ(t)) from infinite temperature autocorrelation functions (C(t)).
• Calculation requires the first 2n temporal derivatives of C(t) at times 0 and t.
• Numerical analysis of AOTFs for few-body observables, focusing on monotonically decreasing functions.

Main Results:

• AOTFs are found to exist for autocorrelation functions unless the system is near a nonchaotic regime.
• All AOTFs provide upper bounds to the autocorrelation functions (αⁿ(t) ≥ C²(t)).
• The existence of an AOTF implies a directed approach to equilibrium, analogous to the H theorem.

Conclusions:

• AOTFs offer a new perspective on the directionality of time and approach to equilibrium in physical systems.
• The presence of AOTFs is linked to the chaotic nature of a system.
• Numerical findings are potentially explainable through the operator growth hypothesis within the recursion method framework.