Towards optimal nonlinear fluctuation-response tests of Markovianity

• Applicant: Professor Dr. Benjamin Lindner
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 566258001

Nonequilibrium systems with pronounced fluctuations are encountered in soft matter physics and cell biology, e.g. in the dynamics of colloidal particles in optical tweezers, the active motion of Janus particles, or the stochastic oscillations of sensory hair bundles. Current experimental techniques permit to observe the spontaneous fluctuations but also the response to perturbations of such nonequilibrium systems. Their theoretical description, however, remains challenging. Fluctuation-response relations (FRR), also known as fluctuation-dissipation theorems, link a system's spontaneous fluctuations to its response to external perturbations. A nonequilibrium FRR has been used to assess whether a system exhibits Markovian dynamics, which is a fundamental characteristic of systems described by standard Langevin, Fokker-Planck, and Master equations. However, the standard nonequilibrium FRR often proves impractical in experiments due to the large number of trials required to test the theorem with reasonable statistical accuracy. This limitation arises from the need for a weak stimulus to determine the system's linear response, necessitating extensive averaging across many experiments. To address this challenge, we have recently developed a nonlinear version of the nonequilibrium FRR. While this approach is tied to a specific experimental protocol, it requires fewer experimental data points and thus serves as a more versatile tool for testing a system for Markovianity. The suggested nonlinear FRR can be extended to encompass perturbations across various parameters and observables, leading to entire FRR families. While all of these relations are theoretically exact, their efficacy as a test of Markovianity using finite data sets may be very different. In our project we would like to identify i) optimal nonlinear FRRs for confirmation in a Markovian setting and ii) optimal sets of nonlinear FRRs displaying characteristic violations in a non-Markovian setting. To this end, we will study a number of theoretical models: i) overdamped Brownian motion in a potential, ii) underdamped noisy harmonic oscillator, iii) limit-cycle systems with noise, iv) equilibrium and non-equilibrium systems with memory damping and colored noise. We will also investigate two experimental systems: i) colloidal particles in a laser trap with time-dependent forcing in a viscous or visco-elastic solution and ii) a critically damped/underdamped tracer particle in a macroscopic active heat bath of bristle robots. In all these systems the characteristic step-like perturbation required for the nonlinear FRR can be applied in different parameters of the system, the perturbation strength can be varied, and different observables can be used. Analytical calculations, extensive numerical simulations, and systematic experimental testing will provide criteria for the optimality of nonlinear FRRs as Markovianity tests as well as fingerprints of different forms of non-Markovianity.

• DFG Programme: Research Grants
• International Connection: Israel
• Cooperation Partner: Professorin Yael Roichman