The Vlasov-Fokker-Planck Equation

XFEL&Plasma

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The Vlasov-Fokker-Planck Equation is just an application of the Boltzmann Equation (There is also a very good derivation od the equation on Wikipedia). We are to start with the Boltzmann Equation which deals all kinds of statistical systems.

The key idea of the Boltzman Equation is to determine the evolution of $\varrho [\vec r, \vec v, t]$, which is equivilent to $\varrho[\vec r, \vec p, t]$ since $\vec p=m\vec v$. we work with the later one therefore is consistent with the VFP code paper. Also on Wikipedia it assumed the total number of particals to be $N$ therefore

$$\iiint_{\vec r\ {\rm space}} \iiint_{\vec p\ {\rm space}}\varrho[\vec r, \vec p, t]{{\rm d} p_x}{{\rm d} p_y}{{\rm d} p_z}{{\rm d}x}{{\rm d}y}{{\rm d}z}=N$$ But we are using normalised convention so
$$\iiint_{\vec r\ {\rm space}} \iiint_{\vec p\ {\rm space}}\varrho[\vec r, \vec p, t]{{\rm d} p_x}{{\rm d} p_y}{{\rm d} p_z}{{\rm d}x}{{\rm d}y}{{\rm d}z}=1$$ and $\varrho [\vec r, \vec v, t]$ is interperted as probability density instead of number density.

From Boltamann Equation we have

$$\frac{\partial \varrho}{\partial t}+ \vec \nabla \varrho \cdot\frac{\vec p}{m}+\vec \nabla_{\vec p}'\varrho\cdot \vec F=\left(\frac{{\rm d} \varrho}{{\rm d}t}\right)_\rm c$$ Vlasov assumed the r.h.s. to be $0$ so that he obtian $$\frac{\partial \varrho_\bar i}{\partial t}+ \vec \nabla \varrho_\bar i \cdot\frac{\vec p}{m}+\vec \nabla_{\vec p}'\varrho_\bar i\cdot \vec F_\bar i=0$$ where $\bar i =1,2$ stands for ion or electron. And this is called Vlasov Equation.

The force that is exerted on the ions or electrons are Columb-Lorentze force ,i.e.

$$\vec F_\bar i=q_\bar i\left(\vec E+\vec v\times \vec B\right)$$ and $\vec E$, $\vec B$ can be determined by the Maxwell equaitons:
$$\vec \nabla \cdot \vec E=\frac{\rho_q}{\varepsilon_0}$$ $$\vec \nabla \times \vec E=-\frac{\partial \vec B}{\partial t}$$ $$\vec \nabla \cdot \vec E=0$$ $$\vec \nabla\times \vec B=\mu_0\left(\vec j_{\rm c}+\varepsilon_0 \frac{\partial \vec E}{\partial t}\right)$$ and the relation $$\vec j_{\rm c}[\vec r]=\sum_{\bar i=1}^{2}N_{\bar i}q_\bar i\int_0^\infty\vec v_{\bar i}\varrho [\vec r, \vec p, t]{{\rm d} p_x}{{\rm d} p_y}{{\rm d} p_z}$$

The Vlasov Equation perfectly 'determine' the elovution of the plasma and by principle we can 'calculate' it, if there is no collision.

To count into the collision term we need to evaluate the r.h.s., where the 'Fokker Planck' invovles. We need the Fokker-Planck Equation