Understanding nonequilibrium transport phenomena in bosonic systems is highly challenging. Magnons, as bosons, exhibit different transport behavior from fermionic electron spins. This study focuses on the key factors influencing the nonequilibrium transport of magnons in steady states within magnetic insulators by taking Y
3Fe
5O
12(YIG) for example. By incorporating the Bose-Einstein distribution function with a non-zero chemical potential
$ {\mu }_{m} $
into the Boltzmann transport equation, analytical expressions for transport parameters in power of
$ \alpha $
(
$ =-{\mu }_{{\mathrm{m}}}/({k}_{{\mathrm{B}}}T) $
) are obtained under the condition
α<1. It is the biggest different from previous researches that our theory establishes a nonlinear relationship between the chemical potential and the nonequilibrium particle density
$ \delta {n}_{{\mathrm{m}}}\propto -{\alpha }^{1/2}\propto $
$ -{(-{\mu }_{{\mathrm{m}}})}^{1/2} $
for magnons under
α
$\ll 1 $
. For a large chemical potential, higher-order terms of
αmust be taken into account. Owing to this nonlinear relationship, the magnon diffusion equation markedly differs from that governing electron spin,which evolves into more complex nonlinear differential equation. We specifically focus on the ferrimagnetic insulator YIG by making a comparison of the spatial distribution of the nonequilibrium magnon density
$ \delta {n}_{m} $
and chemical potential
$ {\mu }_{m} $
between two extreme temperature gradients, namely,
$ \nabla T \sim 1\;{\mathrm{K}}/{\mathrm{m}}{\mathrm{m}} $
and
$ {10}^{4}\;{\mathrm{K}}/{\mathrm{m}}{\mathrm{m}}, $
which correspond to
$ {\mu }_{{\mathrm{m}}} $
values on the order of
$ -0.1\;{\text{μ}}{\mathrm{e}}{\mathrm{V}} $
and
$ -6.2\;{\mathrm{m}}{\mathrm{e}}{\mathrm{V}} $
, respectively, while still satisfying the prerequisite
α< 1. Given the known temperature gradient distribution, the nonequilibrium magnon density
$ \delta {n}_{{\mathrm{m}}} $
calculated based on our theory is in good agreement with the experimental result. Our theoretical and numerical findings greatly contribute to a profound understanding of the nonequilibrium magnon transport characteristics in magnetic insulators.
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