# Linearzing technique for the Vlasov equation

I am currently reading a paper on a quantum algorithm to solve the linear Vlasov equation with collisions (https://doi.org/10.1103/PhysRevA.107.062412, A. Ameri, E. Ye, P. Cappellaro, H. Krovi, N. Loureiro) and am trying to understand how to linearize the Vlasov equation about the a Maxwellian background with a perturbation term.

I have tried to look for resources explaining the linearization of PDEs via adding a perturbation term and cancelling out quadratic terms online but was unable to find any and was wondering if someone could perhaps point me to some.

• The Vlasov equation with collisions is the Boltzmann equation, is it not? In any case, just define $f = f_{o} + \delta f$ and keep only terms to first order in $\delta f$ and first order in derivatives of $\delta f$. That's usually the simplest approach. You assume that $f_{o}$ is uniform enough that $\partial_{x} f_{o}$ = 0 and $\partial_{v} f_{o}$ = 0. There are fancier ways of going about this but you should start with the simplest approach first before adding complications. Commented Oct 13, 2023 at 12:33

The simple way to do it is to consider the (1-D) electrostatic case without collisions (i.e., Vlasov's equation). Assume perturbations around a stationary equilibrium with zero electric potential ($$\phi_{eq}=0$$). This constraints the equilibrium distribution to depend on the velocity: $$f_{eq}(v)$$. Now, the electric potential and the total distribution can be respectively written as $$\phi(x,t)=\delta \phi(x,t)$$ and $$f(x, v, t)=f_{eq}(v)+\delta f(x, v, t)$$, with $$\left \langle \delta f \right \rangle=0$$.
The (1-D) Vlasov equation is $$\frac{\partial f}{\partial t}+v\frac{\partial f}{\partial x}-\frac{q}{m} \frac{\partial \phi}{\partial x} \frac{\partial f}{\partial v}=0$$ Based on the above and assuming small perturbations, i.e., $$\delta f\ll f_{eq}$$, the non-linear (third) term can be linearized so that at leading order we have: $$\frac{\partial \delta f}{\partial t}+v\frac{\partial \delta f}{\partial x}-\frac{q}{m} \frac{\partial \phi}{\partial x} \frac{\partial f_{eq}}{\partial v}=0$$