Viscosity from Newton to Modern Non-Equilibrium Statistical Mechanics

Viscosity from Newton to Modern Non-equilibrium Statistical Mechanics Sébastien Viscardy Belgian Institute for Space Aeronomy, 3, Avenue Circulaire, B-1180 Brussels, Belgium Abstract In the second half of the 19th century, the kinetic theory of gases has probably raised one of the most impassioned de- bates in the history of science. The so-called reversibility paradox around which intense polemics occurred reveals the apparent incompatibility between the microscopic and macroscopic levels. While classical mechanics describes the motionof bodies such as atoms and moleculesby means of time reversible equations, thermodynamics emphasizes the irreversible character of macroscopic phenomena such as viscosity. Aiming at reconciling both levels of description, Boltzmann proposed a probabilistic explanation. Nevertheless, such an interpretation has not totally convinced gen- erations of physicists, so that this question has constantly animated the scientific community since his seminal work. In this context, an important breakthrough in dynamical systems theory has shown that the hypothesis of microscopic chaos played a key role and provided a dynamical interpretation of the emergence of irreversibility. Using viscosity as a leading concept, we sketch the historical development of the concepts related to this fundamental issue up to recent advances. Following the analysis of the Liouville equation introducing the concept of Pollicott-Ruelle resonances, two successful approaches — the escape-rate formalism and the hydrodynamic-mode method — establish remarkable relationships between transport processes and chaotic properties of the underlying Hamiltonian dynamics. Keywords: statistical mechanics, viscosity, reversibility paradox, chaos, dynamical systems theory Contents 1 Introduction 2 2 Irreversibility 3 2.1 Mechanics. Energyconservationand reversibility . ........................ 3 2.2 Thermodynamics. Energy conservation and irreversibility........................ 5 3 Hydrodynamics and viscosity 8 3.1 Euler’sequations ................................ ............. 8 3.2 Navier-Stokesequations . ............... 10 4 Kinetic theory of gases and statistical mechanics 12 4.1 Atomsandtheearlykinetictheoriesofgases . .................... 12 4.2 Kinetictheoryofgasesandthemean-freepath. ..................... 14 4.3 Boltzmann’s statistical interpretation of irreversibility.......................... 16 4.4 Gibbs’statisticalmechanics . ................. 21 4.5 Ergodichypothesisandmixing . ................ 24 arXiv:cond-mat/0601210v2 [cond-mat.stat-mech] 2 Nov 2010 4.6 Victoryoftheatomichypothesis . ................. 27 4.7 Developmentof theoriesfrom Boltzmann’sequation . ...................... 28 4.8 Irreversibility and microscopic fluctuations at equilibrium........................ 30 Email address: [email protected] (Sébastien Viscardy) Preprint submitted to Studies in History and Philosophy of Modern Physics November 3, 2010 5 Theoryofchaos 33 5.1 FromPoincarétoSmale,Lorenz,andRuelle . .................... 33 5.2 Characterizationofchaos . ................ 36 5.3 Chaosandfractals................................ ............. 39 6 Irreversibility and microscopic chaos 40 6.1 Statisticalmechanicsofsmallsystems . .................... 40 6.2 Existence of transport coefficientsinfew-degree-of-freedomsystems . ... 41 6.3 Hydrodynamicmodesandstatisticalmechanics . ..................... 42 6.3.1 Macroscopicapproachto relaxationprocesses . .................... 42 6.3.2 Microscopicapproachtorelaxationprocesses . .................... 43 6.4 Thermostatted-systemapproach . ................. 47 6.5 Escape-rateformalism . .............. 48 6.5.1 Escape-rate formalism and diffusion............................... 48 6.5.2 Escape-rateformalismandviscosity . ................. 51 6.6 Hydrodynamic-modemethod. .............. 52 7 Conclusions 55 Appendix A Phenomenological approach to viscosity 58 Appendix B Probability measure 60 Appendix C Microscopic expression of the viscosity 61 Appendix D Proof of the equivalence between Green-Kubo and Einstein-Helfand formulas 63 Appendix E Anosov systems and Axiom-A systems 64 Appendix F Escape rate and escape-rate formula 64 AppendixF.1 Closedsystems . ............ 66 AppendixF.2 Opensystems............................. ........... 66 1. Introduction The remarkable advances made in the engineering of steam engines over the 18th century induced a deep transfor- mation of the economic activities, which is usually called industrial revolution. The critical importance of optimizing the engine efficiency resulted in the development of a new science: thermodynamics, which emphasized the irreversible character of macroscopic phenomena (such as heat dissipation). On the other hand, the development of chemistry and, later, of the kinetic theory of gases, transformed the concept of matter from a continuous to a discrete description. In other words, matter should be considered as made of very small particles: the atoms and the molecules. However, the motion of these entities is governed by the reversible equations of Newton. As a result appeared the paradox opposing macroscopic and microscopic descriptions, which has given rise after the seminal works of Maxwell and Boltzmann to a large number of contributions addressing this paradox. In the sixties a new revolutionary field of the science of the 20th century was born: Chaos. Sometimes compared to the scientific revolutions implied by quantum mechanics and special relativity, the phenomenon of chaos has been discovered in most of the natural sciences such as physics, chemistry, meteorology, and geophysics. Dynamical systems theory describes systems with few degrees of freedom which, although ruled by deterministic equations, exhibit a stochastic behavior. This remarkable feature led to the idea that the tools of statistical mechanics could be applied to such systems, and as result to the hypothesis of microscopic chaos as the key element of the emergence of irreversibility from the reversible equations of mechanics. The purpose of this paper is to sketch the historical development of non-equilibrium statistical mechanics by using viscosity as the leading concept, from Newton – the founder of classical mechanics in its modern form on the one 2 hand, and of the viscosity law on the other hand – to the recent advances providing a dynamical interpretation of the irreversibility of macroscopic processes without any stochastic assumptions – such as the Stoßzahlansatz largely used in the frame of the kinetic theory of gases. The paper is organized as follows: we start in section 2 with the presentation of the discovery of the concept of irreversibility as a fundamental property in nature. Section 3 is dedicated to the development of hydrodynamics as the domain of physics being particularly concerned by the process of viscosity. Further, in section 4, we focus on the kinetic theory that has a long history and has played an important role in the establishment of statistical mechanics, that is to say, in the understanding of the relationship between the microscopic dynamics and macroscopic behaviors. In section 5, we cover the brief history of the developmentof the theory of chaos and of dynamical systems. Finally, section 6 is dedicated to the recent theories of non-equilibrium statistical mechanics based on the hypothesis of microscopic chaos. Conclusions are drawn in section 7. 2. Irreversibility 2.1. Mechanics. Energy conservation and reversibility The history of modern science holds its richness in the variety of conceptions developed successively throughout the years.1 In the 17th century, it was the doctrine of the clockmaker God or mechanical philosophy which prevailed above the others.2 The developmentof these ideas beganwith the work of Boyle (1627-1691)and others. Accordingto his doctrine, the world wouldwork like a perfectclock: once created and energized, it can run foreverin a deterministic way and without any need for a divine intervention. To ensure that this “clockwork universe” never runs down, Descartes (1596-1650) more or less intuitively introduced the statement that the total amount of quantity as the scalar momentum mv (quantity of matter3 multiplied by its scalar speed).4 However, as Descartes himself later observed, experiments did not confirm his enunciation of the conservation law of motion. Huygens (1629-1695) corrected it in 1668 by modifying it into the vector form. Huygens also worked on the problem of collisions.5 This revised law allowed him to claim that the vector sum of the product of the mass by the vector velocity mv remained unchanged after a collision even if it was inelastic and with dissipation of energy after the collision itself. Introducing this modification, it then appeared that the world could stop after a certain amount of time. This was totally contradictory with the clockworld concept. The only way to avoid this possibility was to postulate that the matter be composed of elastic particles. If macroscopic objects lose some motion after a collision, it is only in appearance because the motion is transferred to the objects’ invisible particles. Therefore, the idea emerged that heat was related to a rapid motion of the invisible parts inside the macroscopic bodies. In the 17th century, the study of the motion of bodies and its change after collisions was of great importance. In addition to this study, another quantity mv2 was introduced for the first time. Remarkably, its conservation law in elastic collisions was proved by the

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