Brownian motion is a Gaussian process described by the central limit theorem. How- ever, exponential decays of the positional probability density function $P(X, t)$ of packets of spreading random walkers, were observed in numerous situations that include glasses, live cells, and bacteria suspensions. We show that such exponential behavior is generally valid in a large class of problems of...

Brownian motion is widely used as a model of diffusion in equilibrium media throughout the physical, chemical and biological sciences. However, many real-world systems are intrinsically out of equilibrium owing to energy-dissipating active processes underlying their mechanical and dynamical features. The diffusion process followed by a passive tracer in prototypical active media, such as...

We derive exact expressions for the finite-time statistics of extrema (maximum and minimum) of the spatial displacement and the fluctuating entropy flow of biased random walks. Our approach captures key features of extreme events in molecular motor motion along linear filaments. For one-dimensional biased random walks, we derive exact results which tighten bounds for entropy production extrema...

In biophysics, the single-particle tracking (SPT) is broadly used to quantify the kinetics of fluorescently-labelled molecules/proteins in live cells. This powerful tool in microscopy permits ones to resolve modes of motion of individual molecules for better understanding their role in such systems. The motion is not limited to pure (Brownian motion) diffusion, but directed, confined,...

The KPZ equation is - as known - a stochastic field theory that describes the kinetic roughening of surfaces and interfaces, plus many other nonequilibrium processes that belong to its universality class. For $\lambda\neq 0$, the systems described are intrinsically unstable and "lack of stationary probability distribution". This can be readily inferred from the structure of the functional from...