Thermodynamic Theory of STRUCTURE, STABILITY AND FLUCTUATIONS

Transcription

1 Thermodynamic Theory of STRUCTURE, STABILITY AND FLUCTUATIONS P. GLANSDORFF and I. PRIGOGINE Universite" Libre de Bruxelles, Britssels, Belgium, and University of Texas, Austin, Texas / WILEY - INTERSCIENCE a division of John Wiley & Sons, Ltd. London - New York - Sydney - Toronto

2 TU Darmstadt FB Maschinenbau

3 Copyright 1971 John Wiley & Sons Ltd. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical photocopying, recording or otherwise, without the prior written permission of the Copyright owner. Library of Congress Catalog Card No ISBN Reprinted 1974 Reprinted 1977 Reprinted 1978 ffs)ch Chemie (FB 21) V^VBIBLIOTHEK Printed by Vnwin Brothers Limited, Old Wohing, Surrey

4 Contents Introduction XI PART I GENERAL THEORY CHAPTER I Conservation Laws and Balance Equations General form of a balance equation Conservation of mass Conservation of momentum and equation of motion Conservation of energy CHAPTER II CHAPTER III CHAPTER IV The Second Law of Thermodynamics and the Entropy Balance Equation The second law of thermodynamics Local equilibrium Entropy balance equation Basic thermodynamic relations Second order differential of entropy Use of complex variables...27 Linear Thermodynamics of Irreversible Processes Flows and forces Onsager's reciprocity relations Symmetry requirements on coupling of Irreversible processes Non-equilibrium steady states and theorem of minimum entropy production Chemical reactions Concluding remarks Gibbs-Duhem Stability Theory of Thermodynamic Equilibrium Introduction Gibbs-Duhem stability criterion Explicit form of the stability conditions. 46

5 VJ CHAPTER V CHAPTER VI CHAPTER VII Contents 4. Phase separation in binary mixtures Stability of chemical reactions Limitation of the Gibbs-Duhem theory. 51 General Stability Theory of Thermodynamic Equilibrium Thermodynamic stability and entropy balance equation Thermodynamic stability conditions Comparison with kinetic stability theory. 58 Thermodynamic and Hydrodynamic Stability Conditions for Non-Equilibrium States Introduction The definition of stability Liapounoff functions Stability of dissipative systems Theorems of moderation and the Le Chatelier-Braun principle Global stability conditions Characteristic properties of 8 2 s considered as a Liapounoff function Stability involving convective effects Comparison with kinetic stability theory Separate thermodynamic and hydrodynamic stability conditions Explicit Form of the Stability Conditions for Non-equilibrium States Introduction Thermal stability Helmholtz's theorem on the motion of viscous fluids Chemical reactions Excess balance equations Excess entropy balance equation Explicit stability criterion for dissipative processes Stability and linear thermodynamics Stability and entropy production Stability and equilibrium...89 H ' t

6 Contents 11. Comparison with the entropy balance equation Hydro-thennodynamic stability Explicit form of the separate thermodynamic and hydrodynamic stability criteria vii CHAPTER VIII Stability and Fluctuations. 1. Einstein's fluctuation formula 2. Chemical reactions 3. Fluctuations of temperature. 4. Regression of fluctuations 5. Causal description and fluctuations CHAPTER IX The General Evolution Criterion Introduction Evolution criterion for dissipative processes Evolution criterion and theorem of minimum entropy production Evolution criterion and steady-state conditions Rotation around steady states Kinetic potential Behaviour of normal modes around a steady state in dissipative systems Convective processes Time-dependent convection processes. 124 PART II: VARIATIONAL TECHNIQUES AND HYDRODYNAMIC APPLICATIONS CHAPTER X The Local Potential Conservation equations and variational calculus Local potential for the heat conduction problem Time-dependent heat conduction problem Relation with the Galerkin method Convergence of the self-consistent method The time-dependent problem The iteration method

7 VH1 CHAPTER XI CHAPTER XII Contents 8. General formulation of the local potential for a steady state General formulation of the local potential for the time-dependent processes The excess local potential Local potentials in kinetic theory Comparison with other variational techniques Stability Problems in Fluids at Rest Introduction Perturbation equations Stability conditions for a fluid layer B&iard instability and entropy production Thermodynamic interpretation and dissipative structure Neutral stability condition The principle of exchange of stabilities and the evolution criterion A variational free minimum principle for the critical Rayleigh number Normal mode approach to the Benard problem Approximate determination of the critical Rayleigh number by the free minimum method Onset of instability in the two-component Benard problem Stability of a vertical column of fluid. 182 Application of the Local Potential to Stability Problems of Laminar Flow Introduction The eigenvalue problem for hydrodynamic stability The excess local potential for hydrodynamic stability The excess local potential for stability of flow with a transverse temperature gradient The critical Reynolds number for the plane Poiseuille flow

8 Contents 6. The critical Rayleigh number for the Benard problem The Benard problem for laminar flow Influence of a transverse temperature gradient on turbulence CHAPTER Xm Stability of Finite Amplitude Waves Introduction Sound waves Compression and rarefaction waves. Riemann invariants Small disturbances of travelling waves Instability of the simple compression wave Stability of the simple rarefaction wave Reduction of P[Z] IX PART ID: CHEMICAL PROCESSES CHAPTER XIV Time Order in Chemical Reactions Introduction Thermodynamic threshold for chemical oscillations Sustained oscillations of the Lotka-Volterra type Chemical instabilities Time behaviour beyond the instability Limit cycle Comparison between the Lotka-Volterra model and limit cycle behaviour Fluctuations Example of oscillating systems The Zhabotinski reaction CHAPTER XV Space Order and Dissipation in Chemical Reactions Introduction Symmetry breaking instabilities Thermodynamic interpretation of symmetry breaking instabilities Thermodynamic threshold for symmetry breaking instabilities

9 x Contents 5. Dissipative space structures Examples of a dissipative space structure The Zhabotinski reaction Limit cycles and dissipative structures in multi-enzymatic reactions CHAPTER XVI Multiple Steady States Introduction Single independent variable Model with multiple steady states Membrane excitability the model Membrane excitability steady-state equations CHAPTER XVII Unity of Physical Laws and Levels of Description. 1. Introduction Biological structures Hierarchy of structures. 290 REFERENCES GLOSSARY OF INDEX. PRINCIPAL SYMBOLS