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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

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

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

δfk,p,0δfk,p,0=02πdθ2π02πdθ2πei(kθ+kθ)δfθ,p,0δfθ,p,0=02πdθ2π02πdθ2πei(kθ+kθ)
(E.2)
×Nfdθ,p,0fdθ,p,0f0pf0p.

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

(E.3)
fdθ,p,0fdθ,p,0=1N2j=1Nδθθjδppjδθθδpp+ijδθθjδppjδθθiδppi
(E.4)
=1N2[Nfdθ,p,0δθθδpp+N(N1)f2(0,θ,p,θ,p)]
(E.5)
=1Nf0pδθθδpp+f0pf0p+h2(θ,p,θ,p,0),

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

(E.6)
f2(θ,p,θ,p,0)=δθθjδppjδθθiδppi
(E.7)
=NN1f0pf0p+h2(θ,p,θ,p).

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

(E.8)
δfk,p,0δfk,p,0=02πdθ2πf0p2πei(k+k)θδpp+02πdθ2π02πdθ2πNei(kθ+kθ)h2(θ,p,θ,p)
(E.9)
=f0p2πδk,kδpp+12πδk,kμ(k,p,p)
(E.10)
=δk,k2πf0(p)δ(pp)+μ(k,p,p).

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.