Detailed Balance in Ehrenfest Mixed Quantum-Classical Dynamics

Area of Science:

• Quantum mechanics
• Statistical mechanics
• Chemical physics

Background:

• Mixed quantum-classical dynamics are essential for describing systems with both quantum and classical behavior.
• The Ehrenfest method is a common approach for approximating these dynamics.
• Understanding energy exchange between quantum and classical subsystems is crucial.

Purpose of the Study:

• To investigate the equilibrium energy limits in self-consistent field (Ehrenfest) mixed quantum-classical dynamics.
• To derive an analytical expression for the equilibrium mean energy of a quantum oscillator coupled to a classical bath.
• To determine how system properties influence energy transfer and equilibrium states.

Main Methods:

• Derivation of an analytical expression for equilibrium mean energy.
• Analysis of a multistate quantum oscillator coupled to a classical bath.
• Numerical simulations to verify theoretical predictions.

Main Results:

• The mean energy of the quantum subsystem consistently exceeds the classical bath temperature at equilibrium for ergodic systems.
• This energy difference increases with the number of quantum states.
• The energy diverges as the number of quantum levels approaches infinity.

Conclusions:

• The Ehrenfest approximation exhibits a tendency to overestimate energy in the quantum subsystem at equilibrium.
• This overestimation is more pronounced in systems with a larger number of quantum states.
• The findings highlight limitations of the Ehrenfest method for long-time equilibrium properties.