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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...
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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.
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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
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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.
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If a driven oscillator needs to resonate at a specific frequency, then very light damping is required. An example of light damping includes playing piano strings and many other musical instruments. Conversely, to achieve small-amplitude oscillations as in a car's suspension system, heavy damping is required. Heavy damping reduces the amplitude, but the tradeoff is that the system responds at more frequencies. Speed bumps and gravel roads prove that even a car's suspension system is not...
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Exponential damping induced by random and realistic perturbations.

Jonas Richter1, Fengping Jin2, Lars Knipschild1

  • 1Department of Physics, University of Osnabrück, D-49069 Osnabrück, Germany.

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A random-matrix perturbation to quantum systems causes dynamics to exponentially decay. This finding, relevant for realistic many-body models, was confirmed in spin-1/2 ladder systems.

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Area of Science:

  • Quantum mechanics
  • Condensed matter physics
  • Statistical mechanics

Background:

  • Understanding quantum many-body dynamics is crucial for many areas of physics.
  • Perturbations to a system's Hamiltonian can significantly alter its dynamics.
  • Random matrix theory provides a framework for studying complex quantum systems.

Purpose of the Study:

  • To investigate how Hamiltonian perturbations affect quantum many-body dynamics.
  • To determine if random-matrix-like perturbations lead to observable changes in dynamics.
  • To connect theoretical predictions to realistic quantum many-body models.

Main Methods:

  • Projection operator techniques were used to analyze the effect of perturbations.
  • Dynamical quantum typicality and numerical linked cluster expansions were employed.
  • The decay of current autocorrelation functions in spin-1/2 ladder systems was studied.

Main Results:

  • A perturbation with a random-matrix structure effectively causes exponential damping of the original dynamics.
  • Theoretical findings for random matrices show relevance for realistic quantum many-body models.
  • A convincing agreement was found between exact dynamics and theoretical predictions in spin-1/2 ladder systems.

Conclusions:

  • Random-matrix perturbations provide a mechanism for understanding dynamical decay in quantum systems.
  • The study validates the applicability of random matrix theory to realistic quantum phenomena.
  • The results offer insights into the behavior of quantum many-body systems under perturbation.