Large Scale Dynamics of Interacting Particles

ISBN-13: 9783642843730 / Angielski / Miękka / 2011 / 342 str.

Herbert Spohn

Opis

The core of the material on large scale dynamics of interacting particles grew out of courses I taught at the Katholieke Universiteit Leuven, Rutgers Universi ty, and the Ludwig-Maximilians-Universitat Munchen and out of lectures I gave at the workshop "Hydrodynamical Behavior of Microscopic Systems" at the Universita dell'Aquila. I had the good luck of being helped through difficult ground by many friends. Amongst them I am deeply indebted to Joel L. Lebowitz. He got me started. Relatively little would have been achieved without his never-ending curiosity and insistence on clarity. Furthermore, I gratefully acknowledge the cooperation of Michael Aizenman, Henk van Beijeren, Carlo Boldrighini, Jean Bricmont, Paola Calderoni, Brian Davies, Anna DeMasi, Roland Dobrushin, Detlef Durr, Gregory Eyink, Mark Fannes, Pablo Ferrari, Alberto Frigerio, Joseph Fritz, Antonio Galves, Shelly Goldstein, Vittorio Gorini, Reinhard Illner, Claude Kipnis, Joachim Krug, Oscar Lanford, Reinhard Lang, Joel Lebowitz, Christian Maes, Stefano Olla, George Papanicolaou, Errico Presutti, Mario Pulvirenti, Fraydoun Rezakhanlou, Hermann Rost, Yasha Sinai, Yuri Suhov, Domo Szasz, Ragu Varadhan, Andre Verbeure, David Wick, and Horng-Tzer Yau. The list is somewhat lengthy, perhaps, but besides thanks I want to make clear that what I will describe is the outcome of a common scientific enterprise. I thank Henk van Beijeren and Detlef Durr for careful reading of and com ments on a previous version. Paola Calderoni and Detlef Durr supplied me with the proof in Part I, Chapter 8. 4 which is most appreciated. Munchen, May 1991 Herbert Spohn Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Spis treści

Scales.- Outline.- I Classical Particles.- 1. Dynamics.- 1.1 Newtonian Dynamics.- 1.2 Boundary Conditions.- 1.3 Dynamics of Infinitely Many Particles.- 2. States of Equilibrium and Local Equilibrium.- 2.1 Equilibrium Measures, Correlation Functions.- 2.2 The Infinite Volume Limit.- 2.3 Local Equilibrium States.- 2.4 Local Stationarity.- 2.5 The Static Continuum Limit.- 3. The Hydrodynamic Limit.- 3.1 Propagation of Local Equilibrium.- 3.2 Hydrodynamic Equations.- 3.3 The Hard Rod Fluid.- 3.4 Steady States.- 4. Low Density Limit: The Boltzmann Equation.- 4.1 Low Density (Boltzmann—Grad) Limit.- 4.2 BBGKY Hierarchy for Hard Spheres and Collision Histories.- 4.3 Convergence of the Scaled Correlation Functions.- 4.4 The Boltzmann Hierarchy.- 4.5 Time Reversal.- 4.6 Law of Large Numbers, Local Poisson.- 4.7 The H-Function.- 4.8 Extensions.- 5. The Vlasov Equation.- 6. The Landau Equation.- 7. Time Correlations and Fluctuations.- 7.1 Fluctuation Fields.- 7.2 The Green—Kubo Formula.- 7.3 Transport for the Hard Rod Fluid.- 7.4 The Fluctuating Boltzmann Equation.- 7.5 The Fluctuating Vlasov Equation.- 8. Dynamics of a Tracer Particle.- 8.1 Brownian Particle in a Fluid.- 8.2 The Stationary Velocity Process.- 8.3 Brownian Motion (Hydrodynamic) Limit.- 8.4 Large Mass Limit.- 8.5 Weak Coupling Limit.- 8.6 Low Density Limit.- 8.7 Mean Field Limit.- 8.8 External Forces and the Einstein Relation.- 8.9 Self-Diffusion.- 8.10 Corrections to Markovian Limits.- 9. The Role of Probability, Irreversibility.- II Stochastic Lattice Gases.- 1. Lattice Gases with Hard Core Exclusion.- 1.1 Dynamics.- 1.2 Stochastic Reversibility.- 1.3 Invariant Measures, Ergodicity, Domains of Attraction.- 1.4 Driven Lattice Gases.- 1.5 Standard Models.- 2. Equilibrium Fluctuations.- 2.1 Density Correlations and Bulk Diffusion.- 2.2 The Green—Kubo Formula.- 2.3 Currents.- 2.4 The Gradient Condition.- 2.5 Linear Response, Conductivity.- 2.6 Steady State Transport.- 2.7 State of Minimal Entropy Production.- 2.8 Bounds on the Conductivity.- 2.9 The Field of Density Fluctuations.- 2.10 Scaling Limit for the Density Fluctuation Field (Proof).- 2.11 Critical Dynamics.- 3. Nonequilibrium Dynamics for Reversible Lattice Gases.- 3.1 The Nonlinear Diffusion Equation.- 3.2 Hydrodynamic Limit (Proof).- 3.3 Low Temperatures.- 3.4 Weakly Driven Lattice Gases.- 3.5 Nonequilibrium Fluctuations.- 3.6 Local Equilibrium States and Minimal Entropy Production.- 3.7 Large Deviations.- 4. Nonequilibrium Dynamics of Driven Lattice Gases.- 4.1 Hyperbolic Equation of Conservation Type.- 4.2 Asymmetric Exclusion Dynamics.- 4.3 Fluctuation Theory.- 5. Beyond the Hydrodynamic Time Scale.- 5.1 Navier-Stokes Correction for Driven Lattice Gases.- 5.2 Local Structure of a Shock.- 5.2.1 Macroscopic Equation with Fluctuations.- 5.2.2 Shock in a Random Frame of Reference.- 5.2.3 Shock in Higher Dimensions.- 6. Tracer Dynamics.- 6.1 Two Component Systems.- 6.2 Tracer Diffusion.- 6.3 Convergence to Brownian Motion.- 6.4 Nearest Neighbor Jumps in One Dimension: The Case of Vanishing Self-Diffusion.- 7. Stochastic Models with a Single Conservation Law Other than Lattice Gases.- 7.1 Lattice Gases Without Hard Core/Zero Range Dynamics.- 7.2 Interacting Brownian Particles.- 7.3 Ginzburg—Landau Dynamics.- 8. Non-Hydrodynamic Limit Dynamics.- 8.1 Kinetic Limit.- 8.2 Mean Field Limit.- References.- List of Mathematical Symbols.