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Forced Oscillations

Forced Oscillations

When an oscillator is forced with a periodic driving force, the motion may seem chaotic. The motions of such oscillators are known as transients. After the transients die out, the oscillator reaches a steady state, where the motion is periodic, and the displacement is determined.

Damped Oscillations

Damped Oscillations

In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
Although friction and other non-conservative...

Oscillations about an Equilibrium Position

Oscillations about an Equilibrium Position

Stability is an important concept in oscillation. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable equilibrium point will cause the object to oscillate around that point. For an unstable equilibrium point, if the object is disturbed slightly, it will not return to the equilibrium point. There are three conditions for equilibrium points—stable, unstable, and half-stable. A half-stable equilibrium point is also unstable, but is named so because...

Types of Damping

Types of Damping

If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about...

Simple Harmonic Motion

Simple Harmonic Motion

Simple harmonic motion is the name given to oscillatory motion for a system where the net force can be described by Hooke's law. If the net force can be described by Hooke's law and there is no damping (by friction or other non-conservative forces), then a simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position. To derive an equation for period and frequency, the equation of motion is used. The period of a simple harmonic oscillator is given...

The de Broglie Wavelength

The de Broglie Wavelength

In the macroscopic world, objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle; it will continue traveling in a straight line unless it collides with another ball, or it is acted on by some other force, such as friction. The ball has a well-defined position and velocity or well-defined momentum, p = mv, which is defined by mass m and velocity v at any given moment. This is the typical...

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Updated: Jun 25, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

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Published on: December 4, 2017

Exact dynamics of driven Brownian oscillators.

Rui-Xue Xu¹, Bao-Ling Tian, Jian Xu

¹Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China. rxxu@ustc.edu.cn

The Journal of Chemical Physics

|February 26, 2009

Summary

We developed an exact quantum master equation for a driven Brownian oscillator, revealing how external fields enhance polarization through effective field corrections. This cooperative effect is crucial in low-frequency driving and non-Markovian dynamics.

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Area of Science:

Quantum optics
Condensed matter physics
Statistical mechanics

Background:

Brownian oscillators are fundamental in quantum mechanics.
Understanding driven systems with dissipation is key to quantum technologies.
Non-Markovian effects are crucial in realistic quantum systems.

Purpose of the Study:

Construct an exact quantum master equation for a driven Brownian oscillator.
Investigate the interplay between external fields and dissipation.
Analyze non-Markovian effects on system dynamics.

Main Methods:

Wigner phase-space Gaussian wave packet approach.
Construction of an exact quantum master equation.
Analysis of linear response and nonlinear dynamics.

Main Results:

An effective field correction enhances polarization due to external field and dissipation.
This cooperative property arises from effective bath response.
Non-Markovian effects were demonstrated in system dynamics.

Conclusions:

The study provides an exact quantum master equation for driven Brownian oscillators.
Effective field corrections and bath response are key to enhanced polarization.
Non-Markovian dynamics are significant in specific driving and memory regimes.