Deterministic chaos and the foundations of the kinetic theory of gases

Dorfman, J. R.

doi:10.1016/s0370-1573(98)00009-x

Cited by 13 publications

(7 citation statements)

References 58 publications

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“…(15), (17), (21) and (22). By considering the rate of restituting collisions that produce the right (k,ê, V ; t), we find that ω(k,ê, V ; t) satisfies the equation…”

Section: Generalized Boltzmann Equationmentioning

confidence: 98%

“…For the Lorentz gas at low density in two and three dimensions, Van Beijeren, Dorfman and co-workers have set up a kinetic theory in which all Lyapunov exponents could be calculated and relations with transport coefficients could be verified [9][10][11][12][13][14][15][16][17][18]. The hard disk and the hard sphere gas, which were next in line for kinetic investigation, proved harder to deal with.…”

Section: Introductionmentioning

confidence: 99%

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Zon

Beijeren

2002

Journal of Statistical Physics

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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.

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“…(15), (17), (21) and (22). By considering the rate of restituting collisions that produce the right (k,ê, V ; t), we find that ω(k,ê, V ; t) satisfies the equation…”

Section: Generalized Boltzmann Equationmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Untitled

Zon

Beijeren

2002

Journal of Statistical Physics

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“…Furthermore, it has been suggested that a close connection exists between molecular chaos and dynamical chaos, according to which stochastic-like behaviour is possible even for deterministic mechanical systems (e.g. Gaspard 1998, p. 225;Dorfman 1998). In fact, even low-dimensional deterministic dynamical systems are known to give rise to diffusive transport (usually named deterministic diffusion (Gaspard 1998, p. 293)).…”

Section: Chaotic Motion and Time-reversibilitymentioning

confidence: 99%

Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions

et al. 2002

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The dynamics of macroscopically homogeneous sheared suspensions of neutrally buoyant, non-Brownian spheres is investigated in the limit of vanishingly small Reynolds numbers using Stokesian dynamics. We show that the complex dynamics of sheared suspensions can be characterized as a chaotic motion in phase space and determine the dependence of the largest Lyapunov exponent on the volume fraction ϕ. We also offer evidence that the chaotic motion is responsible for the loss of memory in the evolution of the system and demonstrate this loss of correlation in phase space. The loss of memory at the microscopic level of individual particles is also shown in terms of the autocorrelation functions for the two transverse velocity components. Moreover, a negative correlation in the transverse particle velocities is seen to exist at the lower concentrations, an effect which we explain on the basis of the dynamics of two isolated spheres undergoing simple shear. In addition, we calculate the probability distribution function of the transverse velocity fluctuations and observe, with increasing ϕ, a transition from exponential to Gaussian distributions.The simulations include a non-hydrodynamic repulsive interaction between the spheres which qualitatively models the effects of surface roughness and other irreversible effects, such as residual Brownian displacements, that become particularly important whenever pairs of spheres are nearly touching. We investigate, for very dilute suspensions, the effects of such a non-hydrodynamic interparticle force on the scaling of the particle tracer diffusion coefficients Dy and Dz, respectively, along and normal to the plane of shear, and show that, when this force is very short-ranged, both are proportional to ϕ2 as ϕ → 0. In contrast, when the range of the non-hydrodynamic interaction is increased, we observe a crossover in the dependence of Dy on ϕ, from ϕ2 to ϕ as ϕ → 0. We also estimate that a similar crossover exists for Dz but at a value of ϕ one order of magnitude lower than that which we were able to reach in our simulations.

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“…Dynamic entropy extends the equilibrium definition of entropy from statistical mechanics to the time domain. The dynamical entropy provides an estimate of the rate of growth of "information" (per unit time) required to describe the evolution of a dynamical system [14,15,21] and it is also a measure of the "complexity" of a dynamical system [22]. The dynamic entropy characteristically decreases as a system orders and its exploration of its phase space becomes more restricted [23,24].…”

Section: Introductionmentioning

confidence: 99%

Dynamic entropy as a measure of caging and persistent particle motion in supercooled liquids

1999

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The length-scale dependence of the dynamic entropy is studied in a molecular dynamics simulation of a binary Lennard-Jones liquid above the mode-coupling critical temperature T(c). A number of methods exist for estimating the entropy of dynamical systems, and we utilize an approximation based on calculating the mean first-passage time (MFPT) for particle displacement because of its tractability and its accessibility in real and simulation measurements. The MFPT dynamic entropy S(epsilon) is defined as equal to the inverse of the average first-passage time for a particle to exit a sphere of radius epsilon. This measure of the degree of chaotic motion allows us to identify characteristic time and space scales and to quantify the increasingly correlated particle motion and intermittency occurring in supercooled liquids. In particular, we identify a "cage" size defining the scale at which the particles are transiently localized, and we observe persistent particle motion at intermediate length scales beyond the scale where caging occurs. Furthermore, we find that the dynamic entropy at the scale of one interparticle spacing extrapolates to zero as the mode-coupling temperature T(c) is approached.

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Deterministic chaos and the foundations of the kinetic theory of gases

Cited by 13 publications

References 58 publications

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Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions

Dynamic entropy as a measure of caging and persistent particle motion in supercooled liquids

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