Stochastic processes, hydrodynamics and active motion
Abstract:
Our everyday experience is determined by classical mechanics and statistical physics, since the physical processes, which we are witnessing consciously or unconsciously, typically involve a huge number of interacting atoms and molecules. The air of the earth's atmosphere, for example, is comprised of simple gas molecules, which can display complex patterns, turbulence, and chaotic behavior with severe consequences on our lives. Similarly, biological processes such as DNA replication, cell division, and the transport of blood through our body is determined by a multitude of particles and interactions. The understanding of the diversity of phenomena over the broad spectrum of length and time scales calls for a specific description. A suitable approach is stochastic dynamics, where the dynamics of the objects of interest is studied under the influence of random noise, i.e., the details of environmental degrees of freedom are treated as external noise. The lecture will provide an introduction to the field of stochastic dynamics from the physics point of view. The basic dynamical equations will be derived and analytical tools be provided to solve important problems. In addition, examples illustrate consequences of . Specifically, the following aspects will be addressed:
- - Langevin equation: Heuristic derivation, Markov process, Gaussian white noise, solution for various scenarios, e.g., Brownian motion, harmonic oscillator, additive and multiplicative noise
- - Fokker-Planck equation: Derivation from the Langevin equation, solutions, boundary conditions, detailed balance, fluctuation-dissipation theorem
- - Applications: dynamics of biopolymers
- - Hydrodynamic interactions: Navier-Stokes equations, fluctuating hydrodynamics, Oseen tensor, long-time tail
- - Application: dynamics of colloids and polymers
- - Active Matter: active Brownian particles, violation of detailed balance and fluctuation-dissipation theorem, colored noise