An Introduction to Stochastic Processes and Nonequilibrium Statistical Physics

Horacio S Wio, Roberto R Deza, Juan M López

• Publisher: WSPC
• Year: 2012
• ISBN:
  9789814434638
• Topic: Physical Sciences
• Subtopic: Thermodynamics

This book aims to provide a compact and unified introduction to the most important aspects in the physics of non-equilibrium systems. It first introduces stochastic processes and some modern tools and concepts that have proved their usefulness to deal with non-equilibrium systems from a purely probabilistic angle. The aim is to show the important role played by fluctuations in far-from-equilibrium situations, where noise can promote order and organization, switching among non-equilibrium states, etc. The second part adopts a more historical perspective, retracing the first steps taken from the purely thermodynamic as well as from the kinetic points of view to depart (albeit slightly) from equilibrium. The third part revisits the path outlined in the first one, but now undertakes the mesoscopic description of extended systems, where new phenomena (patterns, long-range correlations, scaling far from equilibrium, etc.) are observed.

This book is a revised and extended version of an earlier edition published in 1994. It includes topics of current research interest in far-from-equilibrium situations like noise-induced phenomena and free energy-like functionals, surface growth and roughening, etc. It can be used as an advanced textbook by graduate students in physics. It also covers topics of current interest in other disciplines and interdisciplinary approaches in engineering, biophysics, and economics, among others.

The level of detail in the book is enough to capture the interest of the reader and facilitate the path to more learning by exploring the modern research literature provided. At the same time, the book is also complete enough to be self-contained for those readers who just need an overview of the subject.

Contents:

• The Stochastic Approach:
• • Stochastic Processes and the Master Equation
  • The Fokker–Planck Equation
  • The Ω–Expansion
• Thermodynamics and Kinetics Near Equilibrium:
• • Distributions, BBGKY–Hierarchy, Balance Equations, and the Density Operator
  • Linear Nonequilibrium Thermodynamics and Onsager Relations
  • Linear Response Theory, Fluctuation–Dissipation Theorem
• The Mesoscopic Approach Far from Equilibrium: Non-Extended Systems:
• • Stability of Dissipative Dynamical Systems
  • Noise-Induced Phenomena in Non-Extended Dynamical Systems
• The Mesoscopic Approach Far from Equilibrium: Extended Systems:
• • Formation and Propagation of Patterns in Far-from-Equilibrium Systems
  • Noise-Induced Phenomena in Extended Systems
  • Surface Growth and Kinetic Roughening
  • Final Comments

Readership: Graduate students and researchers interested not only in statistical physics, but engineering, biophysics and economics.

PART 2

Thermodynamics and kinetics near equilibrium

Those are my principles, and if you don't like them …
well, I have others.
Groucho Marx

Provando e riprovando
Galileo Galilei

Chapter 4

Distributions, BBGKY—hierarchy,
balance equations, and the density
operator

Time, that mathematical abstraction, that twister of fools' minds,
fools who flaunt the badge of learning. Baby, time is real.
Roberto Arlt

4.1 Introduction

In the previous chapters we studied the temporal evolution of probability distributions for Markov processes, where the dynamics is determined by a transition probability (that very often is phenomenologically determined). The resulting equations, master or Fokker–Planck equations, might be considered to be justified semi-phenomenologically, and to show the kind of behavior needed to describe the irreversible decay toward equilibrium states. We can ask ourselves if it is possible to derive similar (irreversible) kinetic equations but starting from a (reversible) rigorous microscopic theoretical point of view (i.e. Newton’s or Schrödinger equations).

In order to answer such a question we will concentrate on a classical situation with a large number of degrees of freedom, for instance composed of a large number, say N, of interacting particles in a box, or interacting objects in a lattice. The behavior of such objects will be described by Newton’s laws or by Hamiltonian dynamics. In a tridimensional system there are 3N degrees of freedom (assuming there are no internal degrees like spin) and classically the state of the system is determined when we specify 6N independent coordinates (3N corresponding to position and 3N to the -conjugate- momentum variables). That is specifying a point in the 6N-dimensional phase space. Such a point will evolve according to Hamiltonian dynamics, corresponding to the exact knowledge of the state of the system. But it is clear that in general we only know a certain probability of being at a given point (or rather, in a small region). Therefore, we assign a probability distribution on the phase space.

The N-body probability density for a classical (or quantal) system contains far more information than what we really need. In practical situations the main use of such a density is to obtain expectation values, or correlation functions for diverse observables, as these are the quantities measured experimentally. In general the observables that are usually treated correspond to one- and two-body operators, and then it is necessary to use some reduced distributions instead of the complete one. We will see how it is possible to obtain evolution equations for these reduced distributions through the famous BBGKY hierarchy. We will also present microscopic balanc...