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Analysis of the Bloch equations for the nuclear magnetization model. (English. Russian original) Zbl 1288.82059

Proc. Steklov Inst. Math. 281, Suppl. 1, S64-S81 (2013); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 18, No. 2, 123-140 (2012).
Summary: We consider a system of three ordinary first-order differential equations known in the theory of nuclear magnetism as the Bloch equations. The system contains four dimensionless parameters as coefficients. Equilibrium states and the dependence of their stability on these parameters are investigated. The possibility of the appearance of two stable equilibrium states is discovered. The equations are integrable in the absence of dissipation. For the problem with small dissipation far from equilibrium, approximate solutions are constructed by the method of averaging.

MSC:

82D40 Statistical mechanics of magnetic materials
78A25 Electromagnetic theory (general)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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