Abstract
Abstract
We consider the jellium model of N particles on a line confined in an external harmonic potential and with a pairwise one-dimensional Coulomb repulsion of strength α > 0. Using a Coulomb gas method, we study the statistics of
s
=
(
1
/
N
)
∑
i
=
1
N
f
(
x
i
)
where f(x), in principle, is an arbitrary smooth function. While the mean of s is easy to compute, the variance is nontrivial due to the long-range Coulomb interactions. In this paper we demonstrate that the fluctuations around this mean are Gaussian with a variance
Var
(
s
)
≈
b
/
N
3
for large N. In this paper, we provide an exact compact formula for the constant
b
=
1
/
(
4
α
)
∫
−
2
α
2
α
[
f
′
(
x
)
]
2
d
x
. In addition, we also calculate the full large deviation function characterizing the tails of the full distribution
for several different examples of f(x). Our analytical predictions are confirmed by numerical simulations.
Subject
General Physics and Astronomy,Mathematical Physics,Modeling and Simulation,Statistics and Probability,Statistical and Nonlinear Physics
Cited by
5 articles.
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