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Physics of Long-Range Interacting Systems$

A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo

Print publication date: 2014

Print ISBN-13: 9780199581931

Published to Oxford Scholarship Online: October 2014

DOI: 10.1093/acprof:oso/9780199581931.001.0001

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(p.390) Appendix E Autocorrelation of the Fluctuations of the One-Particle Density

(p.390) Appendix E Autocorrelation of the Fluctuations of the One-Particle Density

Physics of Long-Range Interacting Systems
Oxford University Press

Using the definition of the Fourier transform and formula (8.13), we get


The expression of the discrete density function (8.3) leads then to


where we used the following definition of the correlation function h2:


(p.391) Substituting expression (E.5) in Eq. (E.2), we find that


In the passage from (E.8) to (E.9), we have used the fact that h2 depends only on the difference θθ. Besides, it decays rapidly to 0 in a range (θθ)1/N, so that μ(k,p,p) is of order 1.