Statistical Description of a Many-Body System

Abstract

Molecules in a gas or a liquid, men in a metropolis and stars in a galaxy are in a state of incessant motion resulting from the action of various forces : internal, due to other molecules or (wo)men or stars, and external, due to the rest of the universe. It would be a frustrating task if we had to know the position and velocity of every molecule or star at every instant of time, in order to deal with the gas or the galaxy. The saving grace is that we need not know every move of every molecule or star and will still be able to manipulate the system to our advantage. For most purposes, it suffices to know the average properties like density, momentum, pressure, and energy, transport properties like thermal and electrical conductivities, and mechanical and electromagnetic stresses of a large system. In this chapter, we set up a mathematical framework to describe the behaviour of a macroscopic system. After establishing a transport equation for a discrete system in the phase space, continuum limits are taken to facilitate the study of fluids. The mass, momentum and energy conservation laws are derived for each species of particles in a fluid. This multifluid description is further simplified to a single fluid description with a two component fluid of electrons and ions taken as an example. This chapter contains the entire set of mathematical tools needed to investigate ‘a single particle’, ‘a multifluid’ and ‘a single fluid’ characteristics of an electrically conducting as well as an electrically nonconducting system. In the subsequent chapters, these descriptions will be further explored under various simplifying and tractable circumstances.