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Related Concept Videos

Linear Approximation in Frequency Domain01:26

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.
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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...
Poisson's And Laplace's Equation01:25

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RLC Circuit as a Damped Oscillator01:30

RLC Circuit as a Damped Oscillator

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Damped Oscillations01:07

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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
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Published on: December 4, 2017

Fluctuation-dissipation relation for nonlinear Langevin equations.

V Kumaran1

  • 1Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India. kumaran@chemeng.iisc.ernet.in

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 24, 2011
PubMed
Summary

The fluctuation-dissipation theorem holds for nonlinear Langevin equations with specific conditions on kinetic coefficients. This ensures thermodynamic consistency in systems with constant susceptibility and field-dependent dynamics.

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Area of Science:

  • Statistical Mechanics
  • Nonlinear Dynamics
  • Physical Chemistry

Background:

  • The fluctuation-dissipation theorem (FDT) is a fundamental principle connecting microscopic fluctuations and macroscopic dissipation in physical systems.
  • Nonlinear Langevin equations are used to model complex systems where response is not proportional to the driving force.
  • Understanding the conditions under which FDT holds in nonlinear systems is crucial for accurate modeling.

Purpose of the Study:

  • To investigate the validity of the fluctuation-dissipation theorem (FDT) for a general class of nonlinear Langevin equations.
  • To identify the specific conditions on kinetic coefficients required for FDT satisfaction.
  • To analyze the role of quadratic free-energy functionals and field-dependent kinetic coefficients.

Main Methods:

  • Employing a perturbation expansion of nonlinear terms in the Langevin equations.
  • Utilizing functional integral calculations to determine correlation and response functions.
  • Verifying the satisfaction of the fluctuation-dissipation relation at each order of the expansion.

Main Results:

  • Demonstrated that the fluctuation-dissipation theorem is satisfied by solutions of nonlinear Langevin equations.
  • Established that satisfaction requires kinetic coefficients to meet Onsager reciprocal relations (irreversible terms) and antisymmetry relations (reversible terms).
  • Showed that a quadratic free-energy functional (constant susceptibility) is compatible with FDT under these conditions.

Conclusions:

  • The study confirms the applicability of the fluctuation-dissipation theorem to a broad range of nonlinear Langevin systems.
  • Provides precise criteria for the kinetic coefficients ensuring thermodynamic consistency in these models.
  • Highlights the importance of Onsager reciprocal and antisymmetry relations for nonlinear dynamics governed by quadratic free-energy functionals.