Lévy Distribution of Single Molecule Line Shape Cumulants in Glasses

Abstract

We investigate the distribution of single molecule line shape cumulants, κ1, κ2, · · ·, in low temperature glasses based on the sudden jump, standard tunneling model. We find that the cumulants are described by Lévy stable laws, thus generalized central limit theorem is applicable for this problem.Pacs: 05.40.Fb, 61.43.FS, 78.66.Jg Recent experimental advances [1] have made it possible to measure the spectral line shape of a single molecule (SM) embedded in a condensed phase. Because each molecule is in a unique static and dynamic environment, the line shapes of chemically identical SMs vary from molecule to molecule [2]. In this way, the dynamic properties of the host are encoded in the distribution of single molecule spectral line shapes [1][2][3][4][5][6][7][8][9]. We examine the statistical properties of the line shapes and show how these are related to the underlying microscopic dynamical events occurring in the condensed phase.We use the Geva-Skinner [5] model for the SM line shape in a low temperature glass based on the sudden jump picture of Kubo and Anderson [10,11]. In this model, a random distribution of low-density (and noninteracting) dynamical defects [e.g., spins or two level systems (TLS)] interacts with the molecule via long range interaction (e.g., dipolar). We show that Lévy statistics fully characterizes the properties of the SM spectral line both in the fast and slow modulation limits, while far from these limits Lévy statistics describes the mean and variance of the line shape. We then compare our analytical results, derived in the slow modulation limit, with results obtained from numerical simulation. The good agreement indicates that the slow modulation limit is correct for the parameter set relevant to experiment.Lévy stable distributions serve as a natural generalization of the normal Gaussian distribution. The importance of the Gaussian in statistical physics stems from the central limit theorem. Lévy stable laws are used when analyzing sums of the type x i , with {x i } being independent identically distributed random variables characterized by a diverging variance. In this case the ordinary Gaussian central limit theorem must be replaced with the generalized central limit theorem. With this generalization, Lévy stable probability densities, L γ,η (x), replace the Gaussian of the standard central limit theorem. Khintchine and Lévy found that stable characteristic functions,L γ,η (k), are of the form [12] ln L γ,η (k) = iµk − z γ |k| γ 1 − iη k |k| tan πγ 2for 0 < γ ≤ 2 (for the case η = 0, γ = 1 see [12]). Four parameters are needed for a full description of a stable law. The constant γ is called the characteristic exponent, the parameter µ is a location parameter which is unimportant in the present case, z γ > 0 is a scale parameter and −1 ≤ η ≤ 1 is the index of symmetry. When An important issue is the slow and fast modulation limits [6,11]. Briefly, the fast (slow) modulation limit is valid if important contributions to the line shape are from TLSs which satisfy ν ≪ K (...

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