## 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

Source:
Physics of Long-Range Interacting Systems
Publisher:
Oxford University Press

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

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(E.2)
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The expression of the discrete density function (8.3) leads then to

(E.3)
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(E.4)
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(E.5)
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where we used the following definition of the correlation function h2:

(E.6)
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(E.7)
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(p.391) Substituting expression (E.5) in Eq. (E.2), we find that

(E.8)
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(E.9)
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(E.10)
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In the passage from (E.8) to (E.9), we have used the fact that h2 depends only on the difference $θ−θ′$. Besides, it decays rapidly to 0 in a range $(θ−θ′)∼1/N$, so that $μ(k,p,p′)$ is of order 1.