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In the previous paper we have shown analytically that the drift function of the 𝑑-dimensional Langevin equation is the Langevin function with a properly chosen scale factor when the evolution of the drift function is a martingale associated with the histories generated by the very Langevin equation. Moreover, we numerically demonstrated that those generated histories from common initial data, which become asymptotically ballistic, show their orientations to obey the classical canonical spin statistics under the external field corresponding to the initial data. In the present paper we show that the canonical density of a 𝑑-dimensional spin evolves as a martingale associated with the process generated by the above mentioned Langevin equation. Finally, as a by-product, we give an analytical explanation of the above numerical findings. These results elucidate a physical link between the martingale and canonical spin statistics.
PHYSICAL REVIEW E
By: Ken Sekimoto.
Phys. Rev. E 111, 044102 – Published 1 April, 2025
DOI: 10.1103/PhysRevE.111.044102