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The present argumentation also holds for more realistic models with vector displacements
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Note that the quantity D (t) of Ref. is defined by D (t) = ∫0t dτZ (τ). The Laplace transform of this quantity is equal to D (s) /s, from which follows D (t→∞ ) = D0
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Note that the quantity D (t) of Ref. is defined by D (t) = ∫ 0 t d τ Z (τ). The Laplace transform of this quantity is equal to D (s) / s, from which follows D (t → ∞) = D 0.
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) ), see Ref
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)), see Ref..
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This argumentation, of course, holds only for independent fluctuations Δ D Vc. This excludes the vicinity of a phase transition or a percolation transition. In such a case the correlation length must be at least smaller than the wavelength (in the sound case) for a given frequency or the mean-square displacement (in the diffusion case) for a given time. In the case of a Lorentz model, which exhibits a percolation transition, a crossover from the Rayleigh-type long-time tail to a critical tail has been observed (Ref.)
-
This argumentation, of course, holds only for independent fluctuations Δ D V c. This excludes the vicinity of a phase transition or a percolation transition. In such a case the correlation length must be at least smaller than the wavelength (in the sound case) for a given frequency or the mean-square displacement (in the diffusion case) for a given time. In the case of a Lorentz model, which exhibits a percolation transition, a crossover from the Rayleigh-type long-time tail to a critical tail has been observed (Ref.).
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31
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77957592102
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We use periodic boundary conditions for the plane waves normalized in the total volume V
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We use periodic boundary conditions for the plane waves normalized in the total volume V.
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77957595925
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/2 breaks down for cases in which K (k) becomes critical for k→0, i.e., near phase transitions or for long-range interactions t (r)
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/ 2 breaks down for cases in which K (k) becomes critical for k → 0, i.e., near phase transitions or for long-range interactions t (r).
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35
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77957580692
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/2 is required for a correct low-frequency behavior of the density of states in the scalar phonon model
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/ 2 is required for a correct low-frequency behavior of the density of states in the scalar phonon model.
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36
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77957564480
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Without some nontrivial balancing mechanism, which, however, would have to be proven
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Without some nontrivial balancing mechanism, which, however, would have to be proven.
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