# Analytic solution of Boltzmann equation

This question is related to nonlinear Hall effect proposed in this paper. The Boltzmann equation in the electric field under relaxation time approximation is:

$$-e E_a \partial_a f+\partial_t f=\dfrac{f_0-f}{\tau} \tag{1}$$

in which $$\partial_a=\dfrac{\partial}{\partial_{k_a}}$$ and $$f_0$$ is the equilibrium distribution in the absence of external fields. Note that we have adopted the Einstein convention for vector analysis. To solve Eq. $$(1)$$ we take the following two assumpations:

• The electric field is given as: $$E_a(t)=Re\{\mathcal{E}_ae^{i\omega t}\} \qquad \mathcal{E}_a \in C \tag{2}$$

• The distribution is expanded up to second order: $$f=Re\{f_0+f_1+f_2\} \tag{3}$$ in which $$f_n\propto |\mathcal{E}|^n \tag{4}$$

Under these conditions, the authors obtain the following analytic solutions: $$f_1=f_1^\omega e^{i\omega t} \quad f_1^\omega=\dfrac{e\tau\mathcal{E}_a\partial_a f_0}{1+i\omega\tau} \tag{5}$$ $$f_2=f_2^0+f_2^{2\omega}e^{2i\omega t} \quad f_2^0=\dfrac{(e\tau)^2\mathcal{E}_a^* \mathcal{E}_b\partial_{ab}f_0}{2(1+i\omega \tau)} \quad f_2^{2\omega}=\dfrac{(e\tau)^2\mathcal{E}_a\mathcal{E}_b\partial_{ab}f_0}{2(1+i\omega\tau)(1+2i\omega\tau)} \tag{6}$$

My question is how to obtain Eqs. $$(5)$$ and $$(6)$$?

• Finally, I figure out this question on my own. Put Eq.$$(2)$$ and $$(3)$$ into Eq.$$(1)$$: $$$$-e \Re\{\mathcal{E}_a e^{i\omega t}\} (\partial_a f_0+\partial_a f_1)+(\partial_tf_1+\partial_tf_2)=\dfrac{-f_1-f_2}{\tau} \tag{7}$$$$ in which $$\Re\{\mathcal{E}_a e^{i\omega t}\}f_2\propto |\mathcal{E}|^3$$ is droped and $$\partial_t f_0=0$$. Furthermore, we can investigate Eq$$(7)$$ by the order of $$|\mathcal{E}|$$.

• For $$|\mathcal{E}|^1$$ term: $$$$-e \mathcal{E}_a e^{i\omega t} \partial_a f_0+\partial_t f_1=-\dfrac{f_1}{\tau} \tag{8}$$$$ in which $$\partial_t f_0$$ is real.

• For $$|\mathcal{E}|^2$$ term:

$$$$-e \Re\{ \mathcal{E}_a e^{i\omega t} \}\partial_af_1+\partial_t f_2=-\dfrac{f_2}{\tau} \tag{9}$$$$ in which $$\Re\{ \mathcal{E}_a e^{i\omega t} \}$$ is real.

• Eqs.$$(8)$$ and $$(9)$$ are linear inhomogenious differential equation of order one, which can be solved easily. $$$$\dfrac{dx(t)}{dt}+p(t)x(t)=q(t) \Rightarrow x(t)=e^{-\int p(t) dt}[c+\int q(t) e^{\int p(t) dt}dt]\tag{10}$$$$
• +10. Great work figuring it out and taking the time to type out an answer in case someone in the future might find it helpful! Aug 10 '21 at 4:41
• @NikeDattani Thanks.
– Jack
Aug 10 '21 at 5:09