Thibault Bertrand (Imperial College)

Diffusion in a noisy trap: dynamics and thermodynamics of the OU^2 process

The Ornstein–Uhlenbeck (OU) process is the standard model for confined Langevin dynamics and is used throughout theoretical biology. However, in a number of scenarios including in active cellular environments and optical tweezer experiments, the OU process fails to capture the fluctuating nature of the confining potential itself. To capture these inherently out-of-equilibrium dynamics, we recently introduced the OU2 process, a natural extension where the stiffness of the harmonic trap undergoes its own OU-like fluctuations. In this talk, we explore the dynamics and thermodynamics of this model through a combination of analytical and numerical methods. First, we show that this dynamic probe-controller coupling fundamentally alters the system's dynamics. Unlike the Gaussian decay of the standard OU process, the OU2 probability density exhibits heavy power-law tails. We discuss how these asymptotics qualitatively change the system's trapping behavior, extreme value statistics, and crucially its first passage time (FPT) statistics. Secondly, we analyze the thermodynamic consequences of external driving. When the trap's fluctuations violate detailed balance, we reveal a breakdown of quasistatic (infinite-time) optimality. Instead, continuous work exchange between the probe and controller drives the emergence of finite-time optimal protocols. Ultimately, we hope that the introduction of the OU2 model provides a refined analytical framework for exploring the interplay of anomalous FPT statistics and non-equilibrium control in fluctuating biological systems.