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Linear Approximation in Time Domain

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...

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...

Linear Approximation in Frequency Domain

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.

Classification of Systems-I

Classification of Systems-I

Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:

Second Order systems II

Second Order systems II

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.
If ζ...

State Space Representation

State Space Representation

The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...

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

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

Published on: December 4, 2017

Nonlinear semiclassical dynamics of open systems.

A M Ozorio de Almeida¹, O Brodier

¹Centro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil. ozorio@cbpf.br

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences

|December 15, 2010

Summary

This study reviews a semiclassical approximation for quantum systems, simplifying complex dynamics using the chord function. It presents generalized asymptotic equilibrium solutions, offering new insights into quantum system evolution.

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Area of Science:

Quantum Mechanics
Statistical Physics
Theoretical Chemistry

Background:

The evolution of quantum systems is often described by master equations, which can become complex for open systems.
Semiclassical approximations offer a way to simplify these descriptions by bridging quantum and classical mechanics.

Purpose of the Study:

To review and extend a semiclassical approximation for density operator evolution in open quantum systems.
To introduce a novel approach using the chord function and double phase space for analyzing quantum dynamics.

Main Methods:

The study reviews a semiclassical approximation based on the chord function, the Fourier transform of the Wigner function.
It employs a classical double Hamiltonian to interpret semiclassical formulae in a real double phase space.
The theory is extended to include dissipative Markovian evolution via non-Hermitian Lindblad operators.

Main Results:

The semiclassical approximation provides an exact solution to the Lindblad master equation for specific quadratic Hamiltonians and linear Lindblad operators.
Decoherence effects are analyzed, showing how they narrow the relevant phase space region.
A propagator is defined in both Wigner and chord function representations to overcome difficulties associated with decoherence.

Conclusions:

Generalized asymptotic equilibrium solutions for open quantum systems are presented for the first time.
The developed semiclassical approach offers a powerful tool for studying quantum dynamics in the presence of dissipation and decoherence.