Geodesic
Nonindependent splittings and Gibbs states
Matematičeskie zametki, Tome 64 (1998) no. 4, pp. 598-605

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We discuss a nonindependent (beam) splitting for which the related thinning leaves the class of equilibrium states for a one mode electromagnetic field invariant. The thinning affects only the parameters of the state, showing a nonlinear loss of energy. After the splitting, the energy values of both split parts are independent. This independence is a characteristic property of the geometric distribution, the distribution of energy values in the equilibrium state. Also, we observe that the class of states where the full states of the split parts are independent is formed by the so-called phase states. 
@article{MZM_1998_64_4_a12,
     author = {K. Fichtner and W. Freudenberg and V. Liebscher},
     title = {Nonindependent splittings and {Gibbs} states},
     journal = {Matemati\v{c}eskie zametki},
     pages = {598--605},
     publisher = {mathdoc},
     volume = {64},
     number = {4},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_4_a12/}
}
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K. Fichtner; W. Freudenberg; V. Liebscher. Nonindependent splittings and Gibbs states. Matematičeskie zametki, Tome 64 (1998) no. 4, pp. 598-605. http://geodesic.mathdoc.fr/item/MZM_1998_64_4_a12/