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The maximum Lyapunov exponent is aproximately proportional to (Formula presented) 1420
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since (Formula presented) 14 for Brownian motion. The extension of dynamic entropy to the scale dependent quantity (Formula presented) resolves this unsatisfactory situation
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The existence of a finite, positive (Formula presented) is often taken as a defining characteristic of “chaos” in a dynamical system, but this definition excludes Brownian motion, the prototypical model of chaotic motion in molecular physics [N. Wiener, Am. J. Math. 60, 897 (1938)] since (Formula presented) 14 for Brownian motion. The extension of dynamic entropy to the scale dependent quantity (Formula presented) resolves this unsatisfactory situation.
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The Lennard-Jones interaction parameters (Formula presented) and (Formula presented) are given by (Formula presented) (Formula presented) (Formula presented) (Formula presented) (Formula presented) and (Formula presented) Lengths are defined in units of (Formula presented) temperature (Formula presented) in units of (Formula presented) and time (Formula presented) in units of (Formula presented)
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The Lennard-Jones interaction parameters (Formula presented) and (Formula presented) are given by (Formula presented) (Formula presented) (Formula presented) (Formula presented) (Formula presented) and (Formula presented) Lengths are defined in units of (Formula presented) temperature (Formula presented) in units of (Formula presented) and time (Formula presented) in units of (Formula presented)
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59
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Recall that for this same system simulated along a different path, the same value of (Formula presented) was found from an analysis of the minority (Formula presented) particles only 39
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Recall that for this same system simulated along a different path, the same value of (Formula presented) was found from an analysis of the minority (Formula presented) particles only 39.
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0001505258
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exhibit a breakdown of the inverse scaling between (Formula presented) and the particle rotational relaxation time (which should scale as (Formula presented) in mode-coupling theory)
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It is important to note that (Formula presented) scales in inverse proportion to (Formula presented) in mode-coupling theory, and that Eq. (3.11) suggests an important shortcoming of this model. Recent data for (Formula presented) dumbbell particles interacting with the same LJ parameters of the present calculation [S. Kammerer, W. Kob, and R. Shilling, Phys. Rev. E 56, 5450 (1998)] exhibit a breakdown of the inverse scaling between (Formula presented) and the particle rotational relaxation time (which should scale as (Formula presented) in mode-coupling theory).
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0000719817
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Recent work has shown that the fraction of unstable shoulder modes for the present LJ model tends to vanish as (Formula presented) [C. Donati et al. (unpublished)]
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