Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- Part I Reaction-Diffusion Systems and Models of Catalysis
- Part II Kinetic Ising Models
- Part III Ordering, Coagulation, Phase Separation
- Part IV Random Adsorption and Relaxation Processes
- Part V Fluctuations in Particle and Surface Systems
- 13 Microscopic models of macroscopic shocks
- 14 The asymmetric exclusion model: exact results through a matrix approach
- 15 Nonequilibrium surface dynamics with volume conservation
- 16 Directed-walk models of polymers and wetting
- Part VI Diffusion and Transport in One Dimension
- Part VII Experimental Results
- Index
- Abbreviations
13 - Microscopic models of macroscopic shocks
Published online by Cambridge University Press: 18 December 2009
- Frontmatter
- Contents
- Contributors
- Preface
- Part I Reaction-Diffusion Systems and Models of Catalysis
- Part II Kinetic Ising Models
- Part III Ordering, Coagulation, Phase Separation
- Part IV Random Adsorption and Relaxation Processes
- Part V Fluctuations in Particle and Surface Systems
- 13 Microscopic models of macroscopic shocks
- 14 The asymmetric exclusion model: exact results through a matrix approach
- 15 Nonequilibrium surface dynamics with volume conservation
- 16 Directed-walk models of polymers and wetting
- Part VI Diffusion and Transport in One Dimension
- Part VII Experimental Results
- Index
- Abbreviations
Summary
We present some rigorous and computer-simulation results for a simple microscopic model, the asymmetric simple exclusion process, as it relates to the structure of shocks.
Introduction
In this chapter our concern is the underlying microscopic structure of hydrodynamic fields, such as the density, velocity and temperature of a fluid, that are evolving according to some deterministic autonomous equations, e.g., the Euler or Navier-Stokes equations. When the macroscopic fields described by these generally nonlinear equations are smooth we can assume that on the microscopic level the system is essentially in local thermodynamic equilibrium. What is less clear, however, and is of particular interest, both theoretical and practical, is the case where the evolution is not smooth—as in the occurrence of shocks. Looked at from the point of view of the hydrodynamical equations these correspond to mathematical singularities—at least at the compressible Euler level—possibly smoothed out a bit by the viscosity, at the Navier-Stokes level. But what about the microscopic structure of these shocks? Is there really a discontinuity, or at least a dramatic change in the density, at the microscopic scale or does it look smooth at that scale?
It is clear that this question cannot be answered by the macroscopic equations.
- Type
- Chapter
- Information
- Nonequilibrium Statistical Mechanics in One Dimension , pp. 263 - 276Publisher: Cambridge University PressPrint publication year: 1997