The collision frequencies of charged particles in the complex plasmas with the non-Maxwellian velocity distributions

Abstract

We study the collision frequencies of charged particles in the complex plasmas with the two-parameter and three-parameter non-Maxwellian velocity distributions. The average collision frequencies of electron–ion, electron–electron and ion–ion for the distributions are derived, respectively. We show that the average collision frequencies in the complex plasmas depend strongly on the parameters in the non-Maxwellian distributions and thus are significantly different from those cases in the plasmas with a Maxwell velocity distribution. The significant effects of the non-Maxwell parameters on the average collision frequencies are numerically analyzed. The results have important impacts on the transport coefficients and transport properties of charged particles in the highly or fully ionized complex plasmas.

Appendices

Appendix 1

Equation (15) is calculated as follows.

$$\begin{gathered} F_{{ei}} \; = \; - \frac{{3N_{e} N_{i} Z_{i}^{2} e^{4} \ln \Lambda }}{{16\pi ^{2} \varepsilon _{0}^{2} m_{e} }}\frac{{E_{{r,\sigma }} }}{{X_{{r,\sigma }}^{{3/2}} {\text{ }}v_{{Te}}^{3} }} \hfill \\ \;\;\;\;\;\;\;\int {d{\mathbf{v}}\frac{{\mathbf{v}}}{{v^{3} }}} \left\{ {1 + \frac{{2\sigma (1 + r){\mathbf{v}} \cdot {\mathbf{u}}}}{{(\sigma - 1)X_{{r,\sigma }} v_{{Te}}^{2} }}\left( {\frac{{{\mathbf{v}}^{2} }}{{X_{{r,\sigma }} v_{{Te}}^{2} }}} \right)^{r} \left[ {1 + \frac{1}{{\sigma - 1}}\left( {\frac{{{\mathbf{v}}^{2} }}{{X_{{r,\sigma }} v_{{Te}}^{2} }}} \right)^{{1 + r}} } \right]^{{ - 1}} } \right\}\left[ {1 + \frac{1}{{\sigma - 1}}\left( {\frac{{{\mathbf{v}}^{2} }}{{X_{{r,\sigma }} v_{{Te}}^{2} }}} \right)^{{1 + r}} } \right]^{{ - \sigma }} \hfill \\ \;\;\;\;\; = - \frac{{3N_{e} N_{i} Z_{i}^{2} e^{4} \ln \Lambda }}{{8\pi ^{2} \varepsilon _{0}^{2} k_{B} T_{e} }}\frac{{E_{{r,\sigma }} }}{{X_{{r,\sigma }}^{{5/2}} {\text{ }}v_{{Te}}^{3} }}\frac{{\sigma (1 + r)}}{{\sigma - 1}}\int {d{\mathbf{v}}\frac{{{\mathbf{vv}} \cdot {\mathbf{u}}}}{{v^{3} }}} \left( {\frac{{{\mathbf{v}}^{2} }}{{X_{{r,\sigma }} v_{{Te}}^{2} }}} \right)^{r} \left[ {1 + \frac{1}{{\sigma - 1}}\left( {\frac{{{\mathbf{v}}^{2} }}{{X_{{r,\sigma }} v_{{Te}}^{2} }}} \right)^{{1 + r}} } \right]^{{ - \sigma - 1}} . \hfill \\ \end{gathered}$$

(42)

Due to ${\mathbf{u}} = (0,0,u_{z} )$, we obtain that

$$\begin{gathered} F_{{ei}} = - \frac{{3N_{e} N_{i} Z_{i}^{2} e^{4} u_{z} \ln \Lambda }}{{8\pi ^{2} \varepsilon _{0}^{2} k_{B} T_{e} }}\frac{{E_{{r,\sigma }} }}{{X_{{r,\sigma }}^{{5/2}} {\text{ }}v_{{Te}}^{3} }}\frac{{\sigma (1 + r)}}{{\sigma - 1}}{\text{ }} \hfill \\ \;\;\;\;\;\;\;\int_{{ - v_{{\max }} }}^{{v_{{\max }} }} {d{\mathbf{v}}\frac{{v_{z}^{2} }}{{v^{3} }}} \left( {\frac{{{\mathbf{v}}^{2} }}{{X_{{r,\sigma }} v_{{Te}}^{2} }}} \right)^{r} \left[ {1 + \frac{1}{{\sigma - 1}}\left( {\frac{{{\mathbf{v}}^{2} }}{{X_{{r,\sigma }} v_{{Te}}^{2} }}} \right)^{{1 + r}} } \right]^{{ - \sigma - 1}} \hfill \\ = - \frac{{N_{e} N_{i} Z_{i}^{2} e^{4} u_{z} \ln \Lambda }}{{2\pi \varepsilon _{0}^{2} k_{B} T_{e} }}\frac{{\sigma (1 + r)E_{{r,q}} }}{{(\sigma - 1)X_{{r,q}}^{{5/2}} v_{{Te}}^{3} }}\int_{0}^{{v_{{\max }} }} {dv} \left( {\frac{{v^{2} }}{{X_{{r,\sigma }} v_{{Te}}^{2} }}} \right)^{r} v\left[ {1 + \frac{1}{{\sigma - 1}}\left( {\frac{{v^{2} }}{{X_{{r,\sigma }} v_{{Te}}^{2} }}} \right)^{{1 + r}} } \right]^{{ - \sigma - 1}} . \hfill \\ \end{gathered}$$

(43)

If $v_{\max } = \infty$, we derive that

$$F_{ei} = - \frac{{N_{e} N_{i} Z_{i}^{2} e^{4} u_{z} m_{e}^{1/2} \ln \Lambda }}{{4\pi \varepsilon_{0}^{2} (k_{B} T_{e} )^{3/2} }}\frac{{E_{r,\sigma } }}{{X_{r,\sigma }^{3/2} }}.$$

(44)

Appendix 2

Equation (30) is calculated as follows.

$$\begin{aligned} F_{{ei}} = & - \frac{{3N_{e} N_{i} Z_{i}^{2} e^{4} \ln \Lambda }}{{16\pi ^{2} \varepsilon _{0}^{2} m_{e} }}\frac{{\rho _{{\alpha ,r,\sigma }} }}{{X_{{r,\sigma }}^{{3/2}} {\text{ }}v_{{Te}}^{3} }}\int {d{\mathbf{v}}\frac{{\mathbf{v}}}{{v^{3} }}} \left\{ {1 + {\mathbf{u}}B^{{ - 1}} \left( {\frac{{{\mathbf{v}}_{e}^{2} }}{{X_{{r,\sigma }} v_{{Te}}^{2} }}} \right)^{r} \frac{{2\sigma (1 + r)}}{{X_{{r,\sigma }} (\sigma - 1)}} - {\mathbf{u}}A^{{ - 1}} \frac{{4\alpha {\mathbf{v}}_{e}^{2} }}{{v_{{Te}}^{4} }}} \right\}AB^{{ - \sigma }} \\ = & - \frac{{3N_{e} N_{i} Z_{i}^{2} e^{4} \ln \Lambda }}{{16\pi ^{2} \varepsilon _{0}^{2} k_{B} T_{e} }}\frac{{\rho _{{\alpha ,r,\sigma }} }}{{X_{{r,\sigma }}^{{3/2}} {\text{ }}v_{{Te}}^{3} }}\int {d{\mathbf{v}}\frac{{{\mathbf{vv}} \cdot {\mathbf{u}}}}{{v^{3} }}} \left[ {AB^{{ - \sigma - 1}} \left( {\frac{{{\mathbf{v}}^{2} }}{{X_{{r,\sigma }} v_{{Te}}^{2} }}} \right)^{r} \frac{{2\sigma (1 + r)}}{{X_{{r,\sigma }} (\sigma - 1)}} - \frac{{4\alpha {\mathbf{v}}_{e}^{2} }}{{v_{{Te}}^{2} }}B} \right] \\ \end{aligned}$$

(45)

Due to ${\mathbf{u}} = (0,0,u_{z} )$, we obtain that

$$\begin{gathered} F_{{ei}} = - \frac{{3N_{e} N_{i} Z_{i}^{2} e^{4} u_{z} \ln \Lambda }}{{16\pi ^{2} \varepsilon _{0}^{2} k_{B} T_{e} }}\frac{{\rho _{{\alpha ,r,\sigma }} }}{{X_{{r,\sigma }}^{{3/2}} {\text{ }}v_{{Te}}^{3} }}\int_{{ - v_{{\max }} }}^{{v_{{\max }} }} {\frac{{v_{z}^{2} }}{{v^{3} }}} \hfill \\ \;\;\;\;\;\;\;\;\left[ {AB^{{ - \sigma - 1}} \left( {\frac{{{\mathbf{v}}^{2} }}{{X_{{r,\sigma }} v_{{Te}}^{2} }}} \right)^{r} \frac{{2\sigma (1 + r)}}{{X_{{r,\sigma }} (\sigma - 1)}} - \frac{{4\alpha {\mathbf{v}}^{2} }}{{v_{{Te}}^{2} }}B} \right]d{\mathbf{v}} \hfill \\ \;\;\;\; = - \frac{{N_{e} N_{i} Z_{i}^{2} e^{4} u_{z} \ln \Lambda }}{{4\pi \varepsilon _{0}^{2} k_{B} T_{e} }}\frac{{\rho _{{\alpha ,r,\sigma }} }}{{X_{{r,\sigma }}^{{3/2}} {\text{ }}v_{{Te}}^{3} }}\int_{0}^{{v_{{\max }} }} v \hfill \\ \;\;\;\;\;\;\;\;\left[ {AB^{{ - \sigma - 1}} \left( {\frac{{v^{2} }}{{X_{{r,\sigma }} v_{{Te}}^{2} }}} \right)^{r} \frac{{2\sigma (1 + r)}}{{(\sigma - 1)X_{{r,\sigma }} }} - \frac{{4\alpha v^{2} }}{{v_{{Te}}^{2} }}B} \right]dv \hfill \\ \end{gathered}$$

(46)

If $v_{\max } = \infty$, we have that

$$F_{ei} = - \frac{{N_{e} N_{i} Z_{i}^{2} e^{4} u_{z} m_{e}^{1/2} \ln \Lambda }}{{4\pi \varepsilon_{0}^{2} (k_{B} T_{e} )^{3/2} }}\frac{{\rho_{\alpha ,r,\sigma } }}{{X_{r,\sigma }^{3/2} }},$$

(47)