Shankar P. Das

School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India.

In this set of three lectures, I will discuss the continuum description for a many-particle system using a set of coarse-grained densities or slow modes. The equations of motion for the slow modes are stochastic partial differential equations and constitute fluctuating nonlinear hydrodynamic (FNH) description. The continuum models of the fluid using such approaches have been used to study its time-dependent properties, like relaxation behaviour of correlation of fluctuations and dynamical phase transitions. A brief sketch of the topics covered in the respective lectures are:

Lecture I: Formulation of the equations of FNH for a system of particles following Hamiltonian dynamics. The reversible and dissipative parts of the equations of motion for the slow modes. These nonlinear stochastic partial differential equations are treated as plausible generalizations of the hydrodynamic description for the system. The role of the nonlinear coupling of modes in the dynamical equations in describing cooperative behaviour in a dense liquid will be discussed. The Fokker-Planck description and stationary states. The free energy is functional in terms of coarse-grained variables. Density-functional models. Static and dynamic nonlinearities.

Lecture II: We will discuss the implications of the nonlinearities in FNH equations for a simple liquid. In particular, we will explain how the coupling between density fluctuations can give rise to an ergodicity-nonergodicity (ENE) transition at high density. Here we will discuss a field-theoretic model which is particularly useful for obtaining a self-consistent formulation of the renormalized transport coefficients, which gives rise to a feedback mechanism essential for the ENE transition in the supercooled liquid. We will also briefly discuss the implications of the fluctuation-dissipation constraints in the field theory. Here the full consequences of the nonlinearities in the FNH equations on the asymptotic dynamics will be indicated.

Lecture III: I will consider systems where the microscopic dynamics are Brownian and, therefore, intrinsically dissipative. The corresponding equations of coarse-grained collective densities follow through a coarse-graining procedure. Here I will demonstrate how a simplified description of the microscopic dynamics obtains the coarse-grained form of the so- called Dean-Kawasaki equation, often used in dynamic density functional studies. Finally, we will discuss how the same approach of starting from microscopic Brownian dynamics can be generalized to obtain the coarse-grained equations of the hydrodynamics of flocking systems consisting of active particles and breaking of Galilean invariance in such systems.