Escape mechanism in bistable systems driven by strongly correlated noise

Abstract

This thesis focuses on the analysis of nonlinear dynamical systems influenced by colored noise, which appears in diverse fields such as physics, chemistry, natural sciences, and engineering. The core problem involves solving stochastic differential equations of the form: x(t)=f(x(t))+g(x(t))??(t)x(t) = f(x(t)) + g(x(t)) \, \xi(t)x(t)=f(x(t))+g(x(t))?(t) Here, f(x(t))f(x(t))f(x(t)) and g(x(t))g(x(t))g(x(t)) are nonlinear functions, and ?(t)\xi(t)?(t) represents Ornstein–Uhlenbeck noise with zero mean and correlation function: ??(t)?(t?)?=D?exp?(??t?t???)\langle \xi(t) \xi(t') \rangle = \frac{D}{\tau} \exp\left(-\frac{|t - t'|}{\tau}\right)??(t)?(t?)?=?D?exp(???t?t???) where DDD is the noise strength and ?\tau? is the correlation time. Key Focus: Bistable Systems Driven by Colored Noise (BSDCN) The thesis extends two major theoretical frameworks to the finite correlation time regime: Effective Fokker–Planck Equation (EFPE) Formalism Fluctuating Potential Theory (FPT) Using path integral methods, an EFPE is derived. The effective diffusion constant is obtained via the steepest descent approximation, resulting in an EFPE that is: Nonlocal in state space Local in time Positive-definite across the entire state space

The Stationary Probability Density Function (SPDF) and Mean First Passage Time (MFPT) for the BSDCN problem are computed for both small and large ?\tau?. In the small but finite ?\tau? regime, the MFPT predictions from this work outperform existing theories. Limitations of EFPE and FPT The EFPE formalism fails to accurately compute MFPT in the finite ?\tau? regime due to an ill-defined separatrix. Corrections are needed in both the drift and diffusion terms of EFPE for finite ?\tau?. FPT becomes invalid near the inflection points of the bistable potential, leading to underestimation of MFPT. Noise Spike Dynamics and Adiabatic Approach Transitions in the bistable potential at large but finite ?\tau? and non-zero DDD are driven by noise spikes with: Exponentially rising and falling edges (duration ~ ?\tau?) Sufficient peak amplitude to drive deterministic transitions Simulations using such noise spikes yield accurate MFPT values. In this regime, ?(t)\xi(t)?(t) can be viewed as a superposition of: A smooth exponential component (timescale ~ ?\tau?) A white noise component with strength D/?2D/\tau^2D/?2 This allows the use of an adiabatic approximation to compute escape rates, showing excellent agreement with numerical simulations. The thesis also highlights parallels between this approach and the Stochastic Resonance phenomenon.