Memory-induced slow relaxation in the generalized Langevin equation

Area of Science:

• Statistical Mechanics
• Physical Chemistry
• Non-linear Dynamics

Background:

• The generalized Langevin equation (GLE) describes systems with memory effects, crucial for understanding complex dynamics.
• Non-Markovian dynamics, where past events influence future states, are challenging to analyze.
• Understanding relaxation processes is key in various physical and chemical systems.

Purpose of the Study:

• To investigate the relaxation dynamics of the GLE with an exponential memory kernel.
• To elucidate the relationship between memory decay rate and relaxation time.
• To explain the physical origin of memory-induced slow relaxation.

Main Methods:

• Analytical investigation of the generalized Langevin equation (GLE) with an exponential memory kernel.
• Analysis of escape dynamics from a potential well.
• Reformulation of the GLE using an auxiliary variable to embed non-Markovian dynamics into a higher-dimensional Markovian system.

Main Results:

• The relaxation time (τrelax) scales inversely with the square of the memory decay rate (λ), specifically τrelax ∝ λ-2.
• The mean escape time from a potential well exhibits the same λ-2 dependence.
• Memory-suppressed fluctuations were identified as the cause of delayed escape events and slow relaxation.

Conclusions:

• The study reveals a precise scaling law for relaxation time in non-Markovian systems governed by the GLE.
• Memory effects significantly influence system dynamics by suppressing fluctuations and delaying relaxation.
• Reformulating the GLE with an auxiliary variable provides a tractable framework for understanding non-Markovian phenomena.