Geodesic
Two-dimensional limit theorem for the particle number and energy in
Izvestiya. Mathematics , Tome 4 (1970) no. 5, pp. 1183-1202

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It is shown that in the grand canonical ensemble for the case of a lattice system the fluctuations in energy and particle number have an asymptotic (for large volume) normal distribution. A similar statement is valid for fluctuations in energy and particle number for a sufficiently large region of the lattice even in the case of a limiting Gibbs distribution. Both theorems are proved provided the activity $z$ is small enough or the temperature $T$ is sufficiently high. 
@article{IM2_1970_4_5_a10,
     author = {R. A. Minlos and A. M. Khalfina},
     title = {Two-dimensional limit theorem for the particle number and energy in},
     journal = {Izvestiya. Mathematics },
     pages = {1183--1202},
     publisher = {mathdoc},
     volume = {4},
     number = {5},
     year = {1970},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1970_4_5_a10/}
}
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R. A. Minlos; A. M. Khalfina. Two-dimensional limit theorem for the particle number and energy in. Izvestiya. Mathematics , Tome 4 (1970) no. 5, pp. 1183-1202. http://geodesic.mathdoc.fr/item/IM2_1970_4_5_a10/