David M. Barnett, Ph.D.

Barnett et al. Reply [to comment by Torcini et. al. on Lyapunov Exponent of a Many Body System and its Transport Coefficients],

Torcini et al. raise some interesting issues. Their main point is that the diffusion coefficient of a dilute gas diverges with decreasing density, while the Lyapunov exponent tends to zero; surely they cannot be related by a 1/3 power rule.

However, Torcini et al. are being a little hasty. Elementary dimensional analysis shows that the proportionality constant in the relationship would have to be density dependent. Equivalently, the Lyapunov exponent and diffusion constant must be rescaled by system parameters into dimensionless quantities before being compared by the 1/3 power rule. This they have not done...
[Torcini et al. full text PDF - 26kB]

Relation between Lyapunov Exponent and Dielectric Response Function in Dilute One Component Plasmas

An analytical model is developed for the N-body largest Lyapunov exponent in the dilute plasma, and it is shown that the Lyapunov exponent relates to the dielectric response function. The relation provides a bridge between microscopic mechanical and macroscopic statistical quantities and it is expected to also be applicable for a weakly nonequilibrium system. In thermal equilibrium, the model shows that the Lyapunov exponent of dilute one component plasmas is of the same order as the plasma frequency and independent of the coulomb coupling constant. These results agree fairly well with three dimensiional particle simulations.

Particle simulation of Lyapunov exponents in one-component strongly-coupled plasmas

The Lyapunov exponents and instantaneous expansion rates in a phase space of Coulomb many-body systems are measured with the use of a three-dimensional particle code SCOPE [K. Nishihara, Kakuyugo Kenkyu 66, 253 (1991)]. The code calculates particle dynamics determined by Coulomb forces among individual particles. The Lyapunov exponents normalized by plasma frequency are found to be proportional to Γ^-2/5 in the range of 1 ≤ Γ ≤ 160, where Γ is the Coulomb coupling constant of the ion one-component plasma. There is a large jump of the Lyapunov exponent near Γ ~ 170, which corresponds to the phase transition from the liquid to the solid state in the one-component plasma. In the solid stae, the normalized Lyapunov exponents are proportional to Γ^-6/5 for 170 < Γ < 300. The observed dependence is discussed in analogy to a rigid-body particle system and a weakly nonlinear lattice system for liquid and solid states, respectively. Diffusion coefficients are found to be proportional to the third power Lyapunov exponent in the liquid state, that is for 1 ≤ Γ ≤ 160. These results imply that the Lyapunov exponent is in close relation to the transport processes. The instantaneous expansion rate starts from a small value and increases rapidly to a large peak value before declining slowly torwads an asymptotic value. This stage is called the Lyapunov transient stage. Products of the transient time and the Lyapunov exponent are found to be 1.5-2. Information of the initial state is lost after the transient time. The chaotic behavior of the instantaneous expansion rate is also shown.

Fluctuations and the many-body Lyapunov exponent

An ab initio theoretical method is derived for calculating the maximal Lyapunov exponent of an N-body system obeying Hamilton's equations. The theory is developed in detail for a dilute gas. It shows the Lyapunov exponent to be a function of the time integral of the correlation function for fluctuations in the second derivative of the inter-particle potential (approximately a power 1/3 law). We apply the theory to a one component plasma and derive the dependence of the Lyapunov exponent on plasma parameter.

Lyapunov Exponent of a Many Body System and its Transport Coefficients

An ab initio theoretical expression for the N-body Lyapunov exponent of a dilute gas is derived. It shows the Lyapunov exponent to be a function of the time integral of the correlation function for the second derivative of the interparticle potential (approximately a power 1/3 law). This establishes a link between the Lyapunov exponent and the transport coefficients. We compare the theory with the numerical simulations of a one component plasma.

Lyapunov Exponents of Many Body Systems

For many body systems, we show that the separation rate between two adjacent trajectories in phase space tends to an asymptotic level with relatively small fluctuations about that level. We define an instantaneous phase space expansion rate to be the separation rate in this asymptotic regime. The long time average of the instantaneous expansion rate is the largest Lyapunov exponent familiar in the literature. Gross changes in the instantaneous expansion rate are shown to be a microscopic correlative of macroscopic changes in the system. Instantaneous expansion rates may be definable in circumstances when statistical quantities such as temperature and pressure are hard to calculate or have no meaning.

This work advances the computational technique and theory for Lyapunov expansion rates. We also show that the size of the transient separation rate depends on the choice of metric for defining distance on phase space.

An ab initio theoretical expression for the Lyapunov exponent of a dilute gas is derived. It shows the Lyapunov exponent to be a function of the time integral of the two time auto-correlation function for the second derivative of the interparticle potential (approximately a power 1/3 law). Such auto-correlations are related to the system's response functions via the fluctuation-dissipation theorem, establishing a link between the Lyapunov exponent and the transport coefficients. We apply the theory to a one component plasma and compare it with numerical simulations. The Lyapunov exponent is proportional to the diffusion coefficient to the power 1/3, as predicted. The theoretical plasma parameter dependence also corresponds well.