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The van der Waals (VDW) equation of state predicts the existence of a first-order liquid-gas phase transition and contains a critical point. The VDW equation with Fermi statistics is applied to a description of the nuclear matter. The nucleon number fluctuations near the critical point of nuclear matter are studied. The scaled variance, skewness, and kurtosis diverge at the critical point. It is found that the crossover region of the phase diagram is characterized by the large values of the scaled variance, the almost zero skewness, and the significantly negative kurtosis. The rich structures of the skewness and kurtosis are observed in the phase diagram in the wide region around the critical point, namely, they both may attain large positive or negative values.

The binomial acceptance correction procedure is studied for particle number distributions detected in high energy reactions in finite regions of the momentum space. We present acceptance correction formulas for scaled variance, skewness, and kurtosis. Our considerations include various specific types of particles -positively or negatively charged, baryons and antibaryons -as well as conserved charges, namely, the net baryon number and electric charge. A simple model with effects of exact charge conservation, namely the Bessel distribution, is studied in some detail where effects of multiparticle correlations are present. The accuracy of the binomial filter is studied with UrQMD model simulations of inelastic proton-proton reactions. Binomial acceptance correction procedure works well when used inside a small region of phase space as well as for certain efficiency corrections, in particular for constructing net proton fluctuation from net baryon ones. Its performance is less accurate when applied to obtain UrQMD fluctuations in a finite rapidity window from fluctuations in the full 4π space.