Random walks and emergent properties

Random walks and diffusion provide our most basic example of how new laws emerge in statistical mechanics. Random walks have fractal properties, with fluctuations on all scales. The microscopic nature of the random steps is unimportant to the emergent fractal behavior. The diffusion equation is the law which emerges when many random walks are combined; it describes the evolution of any locally conserved density at macroscopic distances and times. The chapter introduces powerful Fourier and Green’s function methods for solving for the resulting behavior. Exercises explore this emergence, with applications to perfume, polymers, stock markets, the Sun, bacteria, and flocking of animals.

Multiplicative noise and the diffusion of conserved densities

Stochastic fluid dynamics governs the long time tails of hydrodynamic correlation functions, and the critical slowing down of relaxation phenomena in the vicinity of a critical point in the phase diagram. In this work we study the role of multiplicative noise in stochastic fluid dynamics. Multiplicative noise arises from the dependence of transport coefficients, such as the diffusion constants for charge and momentum, on fluctuating hydrodynamic variables. We study long time tails and relaxation in the diffusion of a conserved density (model B), and a conserved density coupled to the transverse momentum density (model H). Careful attention is paid to fluctuation-dissipation relations. We observe that multiplicative noise contributes at the same order as non-linear interactions in model B, but is a higher order correction to the relaxation of a scalar density and the tail of the stress tensor correlation function in model H.