Classical Correlation Function via Generalized Langevin Equation

This Demonstration calculates the position-position correlation function , where stands for "averaged over thermal equilibrium" for a system consisting of a harmonic oscillator with frequency , coupled to a harmonic bath whose spectral density is ohmic with cutoff frequency and friction , by solving the generalized Langevin equation numerically:

.

Here the friction kernel is given by

.

The result can be compared to the reference case of a bath-free system (no bath, red), as well as the limit where decays much faster than the system dynamics (Markovian, green), and the limit where changes much more slowly than the system dynamics (sluggish bath, blue), for all three of which can be obtained in closed forms. The three parameters , and can be manipulated to study the behavior of and the mentioned reference cases.

See [1] for a more detailed introduction to correlation functions.

Snapshot 1: the behavior of the system when the system-bath coupling is very small

Snapshot 2: the behavior of the system at the Markovian limit

Snapshot 3: the behavior of the system at the sluggish bath limit

Controls

Toggle the display for the three approximations at the lower left. Drag the sliders on top to change the value of the three parameters, and the plot changes accordingly. You can change the range of the sliders using the popup menus to the left of the sliders. After necessary adjustments to the parameters, click the "begin calculation" button at the bottom right to begin the numeric calculation, during which the parameters can no longer be changed. The calculation rate can be toggled or paused using the setter bar at the bottom; a setting of 10 is highly recommended. After the calculation, click "reset" to discard the numeric result, and the parameters can be altered again.

Mathematical Details

All positions and momentum are mass-weighted (), , .

The Hamiltonian of the system is given by

,

where are the position and momentum of the system and are the position and momentum of the bath mode. The system-bath coupling coefficient is given by the ohmic spectral density :

.

The position-position correlation function can be derived in closed form for the bath-free Hamiltonian:

.

Markovian limit:

.

,

where

can be imaginary.

The sluggish bath limit is

,

,

where

.

Reference

[1] A. Nitzan, Chemical Dynamics in Condensed Phases: Relaxation, Transfer, and Reactions in Condensed Molecular Systems, New York: Oxford University Press, 2006.

Submission from the Compute-to-Learn course at the University of Michigan.