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The Lognormal Distribution and Quantum Monte Carlo Data

Published online by Cambridge University Press:  03 June 2015

Mervlyn Moodley
Mervlyn Moodley*
Affiliation:
School of Chemistry and Physics, Quantum Research Group, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban, 4000, South Africa
*
*Corresponding author.Email:moodleym2@ukzn.ac.za
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Abstract

Quantum Monte Carlo data are often afflicted with distributions that resemble lognormal probability distributions and consequently their statistical analysis cannot be based on simple Gaussian assumptions. To this extent a method is introduced to estimate these distributions and thus give better estimates to errors associated with them. This method entails reconstructing the probability distribution of a set of data, with given mean and variance, that has been assumed to be lognormal prior to undergoing a blocking or renormalization transformation. In doing so, we perform a numerical evaluation of the renormalized sum of lognormal random variables. This technique is applied to a simple quantum model utilizing the single-thread Monte Carlo algorithm to estimate the ground state energy or dominant eigenvalue of a Hamiltonian matrix.

Keywords

02.70.Ss02.50.Cw02.50.Ng02.50.-rLognormal distributionrenormalization transformationquantum Monte Carlo
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 5 , May 2014 , pp. 1352 - 1367
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Noughabi, H. A. and Arghami, N. R., Monte Carlo comparison of seven normality tests, J. Stat. Comput. Sim., 81 (2011), 965–972.Google Scholar
[2]Hetherington, J. H., Observation on the statistical iteration of matrices, Phys. Rev. A, 30 (1984), 2713–2719.CrossRefGoogle Scholar
[3]Romeo, M., Da, V. Costa and Bardou, F., Broad distribution effects in sums of lognormal random variables, Euro. Phys. J. B, 32 (2003), 513–525.Google Scholar
[4]Limpert, E., Stahel, W. A., and Abbt, M., Log-normal distributions across the sciences: keys and clues, Bioscience, 51(5) (2001), 341–352.Google Scholar
[5]Dufresne, D., The log-normal approximation in financial and other computations, Adv. Appl. Prob., 36 (2004), 747–773.Google Scholar
[6]Fenton, L., The sum of lognormal probability distributions in scatter transmission systems, IRE Trans. Commun. Sys., 8 (1960), 57–67.Google Scholar
[7]Beaulieu, N. C., Abu-Dayya, A. A., and McLane, P. J., Estimating the distribution of a sum of independent lognormal random variables, IEEE Trans. Commun., 43 (1995), 2869–2873.CrossRefGoogle Scholar
[8]Dufresne, D., Sums of lognormals, in Proceedings of the 43rd Actuarial Research Conference, University of Regina, Regina, Canada, August 2008.Google Scholar
[9]Schleher, D., Generalized Gram-Charlier series with application to the sum of log-normal variates, IEEE Trans. Inform. Theory, 23 (1977), 275–280.CrossRefGoogle Scholar
[10]Schwartz, S. C. and Yeh, Y. S., On the distribution function and moments of power sums with log-normal components, Bell Systems Tech. J., 61(7) (1982), 1441–1462.Google Scholar
[11]Mehta, Neelesh B.et al., Approximating a sum of random variables with a lognormal, IEEE Trans. Wirel. Commun., 6 (2007), 2690–2699.CrossRefGoogle Scholar
[12]Beaulieu, N. C. and Xie, Q., On optimal lognormal approximation to lognormal sum distributions, IEEE Trans. Veh. Technol., 53 (2004), 479–489.Google Scholar
[13]Beaulieu, N. C. and Rajwani, F., Highly accurate simple closed-form approximations to lognormal sum distributions and densities, IEEE Commun. Lett., 8 (2004), 709–711.Google Scholar
[14]Senaratne, D. and Tellambura, C., Numerical computation of the lognormal sum distribution, in Proceedings of the 28th IEEE conference onGlobal telecommunications (GLOBECOM’09), Mehmet Ulema (Ed.), IEEE Press, Piscataway, NJ, USA, (2009), 3966–3971.Google Scholar
[15]Honerkamp, J., Statistical Physics, Springer-Verlag, Berlin, 1998.Google Scholar
[16]Jona-Lasinio, G., The renormalization group: a probabilistic view, Nuovo Cimento B, 26 (1975), 99–119.Google Scholar
[17]Jona-Lasinio, G., Renormalization group and probability theory, Phys. Reports, 352 (2001), 439–458.Google Scholar
[18]Press, W. H, Flannery, B. P., Teukolsky, S. and Vetterling, W. T., Numerical Recipes, Cambridge University Press, Cambridge, 1992.Google Scholar
[19]Nightingale, M. P. and Umrigar, C. J., Monte Carlo Eigenvalue Methods in Quantum Mechanics and Statistical Mechanics, Advances in Chemical Physics, Vol. 105, Monte Carlo Methods in Chemistry, edited by Ferguson, David M., Ilja Siepmann, J., and Truhlar, Donald G., series editors Prigogine, I. and Rice, Stuart A., John Wiley & Sons, New York, 1999.Google Scholar
[20]Nightingale, M. P., Basics, Quantum Monte Carlo and Statistical Mechanics, in Quantum Monte Carlo Methods in Physics and Chemistry, edited by Nightingale, M. P. and Umrigar, C. J., NATO Science Series, Series C; Mathematical and Physical Sciences-Vol. 525, Kluwer, 1999.Google Scholar