**Abstract**

The aim of the study is mathematical modeling of the process of nucleation of crystallites in liquid iron. To solve the problem, the authors used the stationarity principle and put forward a hypothesis that short-range order regions are formed in the volume of a liquid (metal melt) due to temperature fluctuations and the temperature of the short-range region is during the crystallization time the fluctuation of temperature near zero on the Kelvin scale. Studies were carried out for the substance in the liquid state, in particular for pig iron in the paramagnetic state at the melting temperature. In constructing a mathematical model of the process, analytical expressions are obtained for determining the average number of atoms in the short-range order region for crystallized cast iron. To obtain a regularity, the method of statistical physics, probability theory and mathematical statistics were used. The results of the computational experiment showed that the short-range order regions contain a sufficiently large number of atoms in order for the interaction with the pulsed magnetic field to lead to an increasing the number of crystal nuclei and a decrease in their dimensions. The results obtained are applicable for making decisions on the conduct of technological measures aimed at obtaining a homogeneous structure of metals, in particular when designing installations for the action of a pulsed magnetic field on melts, particularly for ferrous and non-ferrous metallurgy.

**Keywords**

Mathematical model of fluctuations, crystallization of metallic melts, pulsed magnetic field, short-range order, the energy of thermal motion.

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