Handy fluctuation-dissipation relation to approach generic noisy systems and chaotic dynamics.

Area of Science:

• Statistical Physics
• Nonlinear Dynamics
• Complex Systems

Background:

• Fluctuation-Dissipation Relations (FDRs) are crucial for understanding systems in equilibrium.
• Extending FDRs to far-from-equilibrium stochastic systems remains a significant challenge.
• Existing methods often require knowledge of stationary probability distributions, limiting their applicability.

Purpose of the Study:

• To develop a general formulation of fluctuation-dissipation relations (FDRs) valid for far-from-equilibrium stochastic dynamics.
• To create an FDR formulation that does not necessitate explicit knowledge of the stationary probability density function.
• To demonstrate the formula's applicability to systems with generic noise distributions and chaotic behavior.

Main Methods:

• Introduced a general formulation for fluctuation-dissipation relations (FDRs).
• Applied the formulation to Markov stochastic systems with various noise distributions (additive, Gaussian, multiplicative, non-Gaussian like Cauchy).
• Validated the results against numerical simulations and explored a small-noise limit.

Main Results:

• The generalized FDR holds for far-from-equilibrium stochastic dynamics.
• The formulation simplifies to known relations for additive Gaussian noise.
• Exact results were obtained for multiplicative and non-Gaussian noise, matching simulations.
• The method successfully reproduced response functions of nonlinear and chaotic dynamical models.

Conclusions:

• The developed FDR formulation offers a powerful tool for analyzing complex stochastic systems beyond equilibrium.
• Its independence from stationary distributions broadens its applicability, especially for systems with non-Gaussian noise.
• The ability to model chaotic systems opens avenues for practical applications in fields like geophysics and climate science, exemplified by the Lorenz '63 model.