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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.
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Related Experiment Video

Updated: Jun 3, 2026

Universal Molecular Retention with 11-Fold Expansion Microscopy
10:31

Universal Molecular Retention with 11-Fold Expansion Microscopy

Published on: October 6, 2023

System size expansion for systems with an absorbing state.

Francesca Di Patti1, Sandro Azaele, Jayanth R Banavar

  • 1Dipartimento di Fisica Galileo Galilei, Università degli Studi di Padova, Padova, Italy.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 17, 2011
PubMed
Summary

The van Kampen expansion fails for systems with absorbing states. A new method introduces non-Gaussian fluctuations, accurately modeling system dynamics and lifetime distributions, especially for the voter model with speciation.

Related Experiment Videos

Last Updated: Jun 3, 2026

Universal Molecular Retention with 11-Fold Expansion Microscopy
10:31

Universal Molecular Retention with 11-Fold Expansion Microscopy

Published on: October 6, 2023

Area of Science:

  • Statistical Physics
  • Theoretical Chemistry
  • Computational Biology

Background:

  • The van Kampen system size expansion is a widely used method for analyzing stochastic processes.
  • This expansion has limitations in accurately describing systems with absorbing states, particularly their long-term dynamics.
  • Non-Gaussian fluctuations are crucial for understanding complex system behaviors beyond simple approximations.

Purpose of the Study:

  • To address the shortcomings of the van Kampen expansion for systems exhibiting absorbing states.
  • To develop a generalized expansion that captures non-Gaussian fluctuations.
  • To validate the new expansion against exact solutions for a relevant model.

Main Methods:

  • Generalization of the van Kampen ansatz to incorporate non-Gaussian fluctuations.
  • Explicit comparison of the original and generalized expansions using the infinite range voter model with speciation.
  • Numerical implementation using the Gillespie algorithm to obtain exact solutions.

Main Results:

  • The standard van Kampen expansion fails to reproduce qualitative features of time evolution near absorbing states.
  • The generalized van Kampen expansion successfully captures non-Gaussian fluctuations.
  • The new expansion converges to exact solutions for finite and infinite system sizes and finite times.
  • The predicted lifetime distribution exhibits correct asymptotic behavior.

Conclusions:

  • The generalized van Kampen expansion provides a more accurate description of systems with absorbing states.
  • This improved method is essential for understanding phenomena like speciation and other complex dynamics.
  • The approach offers a powerful tool for analyzing stochastic processes in various scientific domains.