Front propagation techniques to calculate the largest Lyapunov exponent of dilute hard disk gases

A kinetic approach is adopted to describe the exponential growth of a small deviation of the initial phase space point, measured by the largest Lyapunov exponent, for a dilute system of hard disks, both in equilibrium and in a uniform shear flow. We derive a generalized Boltzmann equation for an extended one-particle distribution that includes deviations from the reference phase space point. The equation is valid for very low densities n, and requires an unusual expansion in powers of 1/|ln n|. It reproduces and extends results from the earlier, more heuristic clock model and may be interpreted as describing a front propagating into an unstable state. The asymptotic speed of propagation of the front is proportional to the largest Lyapunov exponent of the system. Its value may be found by applying the standard front speed selection mechanism for pulled fronts to the case at hand. For the equilibrium case, an explicit expression for the largest Lyapunov exponent is given and for sheared systems we give explicit expressions that may be evaluated numerically to obtain the shear rate dependence of the largest Lyapunov exponent.Comment: 26 pages REVTeX, 1 eps figure. Added remarks, a reference and corrected some typo

An Elementary Proof of Lyapunov Exponent Pairing for Hard-Sphere Systems at Constant Kinetic Energy

The conjugate pairing of Lyapunov exponents for a field-driven system with smooth inter-particle interaction at constant total kinetic energy was first proved by Dettmann and Morriss [Phys. Rev. E {\bf 53}, R5545 (1996)] using simple methods of geometry. Their proof was extended to systems interacting via hard-core inter-particle potentials by Wojtkowski and Liverani [Comm. Math. Phys. {\bf 194}, 47 (1998)], using more sophisticated methods. Another, and somewhat more direct version of the proof for hard-sphere systems has been provided by Ruelle [J. Stat. Phys. {\bf 95}, 393 (1999)]. However, these approaches for hard-sphere systems are somewhat difficult to follow. In this paper, a proof of the pairing of Lyapunov exponents for hard-sphere systems at constant kinetic energy is presented, based on a very simple explicit geometric construction, similar to that of Ruelle. Generalizations of this construction to higher dimensions and arbitrary shapes of scatterers or particles are trivial. This construction also works for hard-sphere systems in an external field with a Nos\'e-Hoover thermostat. However, there are situations of physical interest, where these proofs of conjugate pairing rule for systems interacting via hard-core inter-particle potentials break down.Comment: 16 pages, 4 figures, to appear in J. Stat. Phy

Non-equilibrium Thermodynamics and Fluctuations

In the last ten years, a number of ``Conventional Fluctuation Theorems'' have been derived for systems with deterministic or stochastic dynamics, in a transient or in a non-equilibrium stationary state. These theorems gave explicit expressions for the ratio of the probability to find the system with a certain value of entropy (or heat) production to that of finding the opposite value. A similar theorem for the fluctuations of the work done on a system has recently been demonstrated experimentally for a simple system in a transient state, consisting of a Brownian particle in water, confined by a moving harmonic potential. In this paper we show that because of the interaction between the stochastic motion of the particle in water and its deterministic motion in the potential, very different new heat theorems are found than in the conventional case. One of the consequences of these new heat Fluctuation Theorems is that the ratio of the probability for the Brownian particle to absorb heat from rather than supply heat to the water is much larger than in the Conventional Fluctuation Theorem. This could be of relevance for micro/nano-technology.Comment: 10 pages, 6 figures. Some corrections in the text were made. Submitted to Physica

Kinetic Theory Estimates for the Kolmogorov-Sinai Entropy and the Largest Lyapunov Exponents for Dilute, Hard-Ball Gases and for Dilute, Random Lorentz Gases

The kinetic theory of gases provides methods for calculating Lyapunov exponents and other quantities, such as Kolmogorov-Sinai entropies, that characterize the chaotic behavior of hard-ball gases. Here we illustrate the use of these methods for calculating the Kolmogorov-Sinai entropy, and the largest positive Lyapunov exponent, for dilute hard-ball gases in equilibrium. The calculation of the largest Lyapunov exponent makes interesting connections with the theory of propagation of hydrodynamic fronts. Calculations are also presented for the Lyapunov spectrum of dilute, random Lorentz gases in two and three dimensions, which are considerably simpler than the corresponding calculations for hard-ball gases. The article concludes with a brief discussion of some interesting open problems.Comment: 41 pages (REVTEX); 7 figs., 4 of which are included in LaTeX source. (Fig.7 doesn't print well on some printers) This revised paper will appear in "Hard Ball Systems and the Lorentz Gas", D. Szasz ed., Encyclopaedia of Mathematical Sciences, Springe

Effective pair potentials for spherical nanoparticles

An effective description for spherical nanoparticles in a fluid of point particles is presented. The points inside the nanoparticles and the point particles are assumed to interact via spherically symmetric additive pair potentials, while the distribution of points inside the nanoparticles is taken to be spherically symmetric and smooth. The resulting effective pair interactions between a nanoparticle and a point particle, as well as between two nanoparticles, are then given by spherically symmetric potentials. If overlap between particles is allowed, the effective potential generally has non-analytic points, but for each effective potential the expressions for different overlapping cases can be written in terms of one analytic auxiliary potential. Effective potentials for hollow nanoparticles (appropriate e.g. for buckyballs) are also considered, and shown to be related to those for solid nanoparticles. Finally, explicit expressions are given for the effective potentials derived from basic pair potentials of power law and exponential form, as well as from the commonly used London-Van der Waals, Morse, Buckingham, and Lennard-Jones potential. The applicability of the latter is demonstrated by comparison with an atomic description of nanoparticles with an internal face centered cubic structure.Comment: 27 pages, 12 figures. Unified description of overlapping and nonoverlapping particles added, as well as a comparison with an idealized atomic descriptio

Theorem on the Distribution of Short Time Single Particle Displacements

The distribution of the initial very short-time displacements of a single particle is considered for a class of classical systems with Gaussian initial velocity distributions and arbitrary initial particle positions. A very brief sketch is given of a rather intricate and lengthy proof that for this class of systems the nth order cumulants behave as t^{2n} for all n>2, rather than as t^{n}. We also briefly discuss some physical consequences for liquids.Comment: Short 8 page pedagogical review of cond-mat/0505734 for Proc. of "News, Expectations and Trends in Statistical Physics", Crete 200