We report the nonlocal spin Seebeck effect (nlSSE) in a lateral configuration of Pt/Y$_3$Fe$_5$O$_{12}$(YIG)/Pt systems as a function of the magnetic field $B$ (up to 10 T) at various temperatures $T$ (3 K < $T$ < 300 K). The nlSSE voltage decreases with increasing $B$ in a linear regime with respect to the input power (the applied charge-current squared $I^2$). The reduction of the nlSSE becomes substantial when the Zeeman energy exceeds thermal energy at low temperatures, which can be interpreted as freeze-out of magnons relevant for the nlSSE. Furthermore, we found the non-linear power dependence of the nlSSE with increasing $I$ at low temperatures ($T$ < 20 K), at which the $B$-induced signal reduction becomes less visible. Our experimental results suggest that in the non-linear regime high-energy magnons are over populated than those expected from the thermal energy. We also estimate the magnon spin diffusion length as functions of $B$ and $T$.
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