% math symbols

\newcommand{\Fr}{F_{\scriptscriptstyle r}}
\newcommand{\Fc}{F_{\scriptscriptstyle c}}
\newcommand{\Ft}{F_{\scriptscriptstyle t}}

\newcommand{\Er}{E_{\scriptscriptstyle r}}
\newcommand{\er}{e_{\scriptscriptstyle r}}
\newcommand{\eg}{e_{\scriptscriptstyle g}}
\newcommand{\et}{e_{\scriptscriptstyle t}}
\newcommand{\ei}{e_{\scriptscriptstyle ion}}

\newcommand{\Pe}{P_{\scriptscriptstyle e}}
\newcommand{\Pg}{P_{\scriptscriptstyle g}}
\newcommand{\pr}{P_{\scriptscriptstyle r}}
\newcommand{\Pv}{P_{\scriptscriptstyle v}}
\newcommand{\Pt}{P_{\scriptscriptstyle t}}
\newcommand{\PQ}{P_{\scriptscriptstyle Q}}

\newcommand{\St}{S_{\scriptscriptstyle t}}
\newcommand{\Kt}{K_{\scriptscriptstyle t}}
\newcommand{\Mt}{M_{\scriptscriptstyle t}}
\newcommand{\Dt}{{D_t}}
\newcommand{\nt}{\nu_{\scriptscriptstyle t}}
\newcommand{\Ct}{C_{\scriptscriptstyle t}}
\newcommand{\Cr}{C_{\scriptscriptstyle r}}
\newcommand{\Teff}{T_{\scriptscriptstyle eff}}
\newcommand{\fG}{f_{\scriptscriptstyle G}}

\newcommand{\Me}{M_{\scriptscriptstyle ext}}
\newcommand{\sr}{\sigma_{\scriptscriptstyle \rho}}
\newcommand{\se}{\sigma_{\scriptscriptstyle e}}
\newcommand{\tr}{\tau_{\scriptscriptstyle r}}
\newcommand{\ellQ}{\ell_{\scriptscriptstyle Q}}
\newcommand{\nuQ}{\nu_{\scriptscriptstyle Q}}
\newcommand{\mq}{\mu_{\scriptscriptstyle Q}}
\newcommand{\CQ}{C_{\scriptscriptstyle Q}}
\newcommand{\lQ}{l_{\scriptscriptstyle Q}}
\newcommand{\cs}{c_{\scriptscriptstyle s}}
\newcommand{\cv}{c_{\scriptscriptstyle V}}
\newcommand{\cp}{c_{\scriptscriptstyle P}}
\newcommand{\ar}{a_{\scriptscriptstyle r}}
\newcommand{\ur}{u_{\scriptscriptstyle rel}}
\newcommand{\fe}{f_{\scriptscriptstyle E}}
\newcommand{\gE}{g_{\scriptscriptstyle E}}
\newcommand{\kb}{k_{\scriptscriptstyle B}}
\newcommand{\kP}{\kappa_{\scriptscriptstyle P}}
\newcommand{\kE}{\kappa_{\scriptscriptstyle E}}
\newcommand{\kF}{\chi_{\scriptscriptstyle F}}
\newcommand{\kR}{\chi_{\scriptscriptstyle Ross}}
\newcommand{\en}{\varepsilon_{\scriptscriptstyle N}}

\newcommand{\om}{\varpi}

\newcommand{\p}{\partial}
\newcommand{\overp}{\over\partial}

\newcommand{\eavh}{\hat{\mbox{\itbf e}}_{\alpha}}
\newcommand{\evh}{\hat{\mbox{\itbf e}}}

\newcommand{\eavhs}{\hat{\mbox{\itbfs e}}_{\alpha}}
\newcommand{\evhs}{\hat{\mbox{\itbfs e}}}

\newcommand{\faxt}{f_{\alpha}(\mbox{\itbf x}, \, t)}
\newcommand{\faxetl}{f_{\alpha}(\mbox{\itbf x} + 
            \hat{\mbox{\itbf e}}_{\alpha}, \, t + 1)}

\newcommand{\frgt}{f(\mbox{\itbf r},\, \Gamma,\, t)}

\newcommand{\Hkv}{H(\mbox{\itbf k})}

\newcommand{\kv}{\mbox{\itbf k}}
\newcommand{\kvh}{\hat{\mbox{\itbf k}}}
\newcommand{\kvs}{\mbox{\itbfs k}}
\newcommand{\kvhs}{\hat{\mbox{\itbfs k}}}

\newcommand{\Naxt}{N_{\alpha}(\mbox{\itbf x}, \, t) }
\newcommand{\Naxetl}{N_{\alpha}(\mbox{\itbf x} + 
            \hat{\mbox{\itbf e}}_{\alpha}, \, t + 1) }

\newcommand{\phiaxt}{\phi_{\alpha}(\mbox{\itbf x}, \, t)}
\newcommand{\phibxt}{\phi_{\beta}(\mbox{\itbf x}, \, t)}
\newcommand{\phiaxetl}{\phi_{\alpha}(\mbox{\itbf x} + 
            \hat{\mbox{\itbf e}}_{\alpha}, \, t + 1)}
\newcommand{\phiakt}{\phi_{\alpha}(\mbox{\itbf k}, \, t)}
\newcommand{\phibkt}{\phi_{\beta}(\mbox{\itbf k}, \, t)}
\newcommand{\phiaktl}{\phi_{\alpha}(\mbox{\itbf k}, \, t + 1)}
\newcommand{\phikt}{| \phi (\mbox{\itbf k}, \, t) \rangle}
\newcommand{\phiktl}{| \phi(\mbox{\itbf k}, \, t + 1) \rangle}
\newcommand{\phitkt}{\phi_{t}(\mbox{\itbf k}, \, t)}
\newcommand{\phitktl}{\phi_{t}(\mbox{\itbf k}, \, t + 1)}
\newcommand{\phipmkt}{\phi_{\pm}(\mbox{\itbf k}, \, t)}
\newcommand{\phipmktl}{\phi_{\pm}(\mbox{\itbf k}, \, t + 1)}

\newcommand{\pv}{\mbox{\itbf p}}
\newcommand{\pvp}{\mbox{\itbf p}^{\prime}}

\newcommand{\rhort}{\rho (\mbox{\itbf r},\, t)}

\newcommand{\rv}{\mbox{\itbf r}}

\newcommand{\sv}{\mbox{\itbf s}}
\newcommand{\svp}{\mbox{\itbf s}'}

\newcommand{\td}{\tilde{d}}

\newcommand{\temprt}{T(\mbox{\itbf r},\, t)}

\newcommand{\vvec}{\mbox{\itbf v}}
\newcommand{\vvecrt}{\mbox{\itbf v}(\mbox{\itbf r},\, t)}

\newcommand{\xiv}{\mbox{\boldmath $\xi$}}

\newcommand{\xv}{\mbox{\itbf x}}
\newcommand{\xvh}{\hat{\mbox{\itbf x}}}
\newcommand{\yvh}{\hat{\mbox{\itbf y}}}

% making plots for collision tables.

\newcommand{\stagmesh}{     
\bpt(00,80)
 \thinlines
 \put(060,15){\line( 0,2){60}}
 \put(105,41){X}
 \put(150,15){\line( 0,2){45}}
 \put(195,41){X}
 \put(240,15){\line( 0,2){60}}
 \put(295,41){X}
 
 \put(054,08){k-1}
 \put(101,30){k-1}
 \put(144,08){ k }
 \put(139,65){r,m,u}
 \put(192,30){ k }
 \put(192,55){e,P}
 \put(234,08){k+1}
 \put(291,30){k+1}
 \put(351,41){k}

 \thicklines
 \put(45,45){\vector( 1,0){300}}
 \put(45,45){\line  (-1,0){030}}
\ept}

\newcommand{\sodinit}{     
\bpt(00,135)
 \thinlines
 \put(030,15){\line( 1,0){300}}
 
 \put(090,090){P=1.0}
 \put(090,080){d=1.0}
 \put(270,050){P=0.1}
 \put(270,040){d=0.125}

 \put(024,03){0.0}
 \put(176,03){0.5}
 \put(324,03){1.0}

 \thicklines
 \put(030,015){\line( 0,2){100}}
 \put(180,025){\line( 0,2){090}}
 \put(330,015){\line( 0,2){010}}

 \put(030,115){\line( 1,0){150}}
 \put(180,025){\line( 1,0){150}}
\ept}

\newcommand{\woodinit}{     
\bpt(00,145)
 \thinlines
 \put(030,15){\line( 1,0){300}}
 
 \put(025,120){P=1000}
 \put(175,030){P=0}
 \put(175,020){d=1}
 \put(298,030){P=100}

 \put(024,03){0.0}
 \put(056,03){0.1}
 \put(296,03){0.9}
 \put(324,03){1.0}

 \thicklines
 \put(030,015){\line( 0,2){100}}
 \put(060,015){\line( 0,2){100}}
 \put(300,015){\line( 0,2){010}}
 \put(330,015){\line( 0,2){010}}

 \put(030,115){\line( 1,0){030}}
 \put(060,015){\line( 1,0){240}}
 \put(300,025){\line( 1,0){030}}
\ept}


\newcommand{\hexad}{     
\bpt(90,90)
 \thinlines
 \put(15,45){\line(1,0){60}}
 \put(30,15){\line(1,2){30}}
 \put(60,15){\line(-1,2){30}}
 
 \put(76,41){1}
 \put(61,71){2}
 \put(23,71){3}
 \put(8,41){4}
 \put(23,10){5}
 \put(61,10){6}

 \thicklines
 \put(45,45){\vector(1,0){25}}     % velocity 1 
 \put(45,45){\vector(-1,0){25}}    % velocity 4 
\ept}

\newcommand{\hexbe}{
\bpt(90,90)
 \thinlines
 \put(15,45){\line(1,0){60}}
 \put(30,15){\line(1,2){30}}
 \put(60,15){\line(-1,2){30}}
 
 \put(76,41){1}
 \put(61,71){2}
 \put(23,71){3}
 \put(8,41){4}
 \put(23,10){5}
 \put(61,10){6}

 \thicklines
 \put(45,45){\vector(1,2){12.5}}   % velocity 2 
 \put(45,45){\vector(-1,-2){12.5}} % velocity 5 
\ept}

\newcommand{\hexcf}{
\bpt(90,90)
 \thinlines
 \put(15,45){\line(1,0){60}}
 \put(30,15){\line(1,2){30}}
 \put(60,15){\line(-1,2){30}}
 
 \put(76,41){1}
 \put(61,71){2}
 \put(23,71){3}
 \put(8,41){4}
 \put(23,10){5}
 \put(61,10){6}

 \thicklines
 \put(45,45){\vector(-1,2){12.5}}  % velocity 3 
 \put(45,45){\vector(1,-2){12.5}}  % velocity 6 
\ept}

\newcommand{\hexace}{
\bpt(90,90)
 \thinlines
 \put(15,45){\line(1,0){60}}
 \put(30,15){\line(1,2){30}}
 \put(60,15){\line(-1,2){30}}
 
 \put(76,41){1}
 \put(61,71){2}
 \put(23,71){3}
 \put(8,41){4}
 \put(23,10){5}
 \put(61,10){6}

 \thicklines
 \put(45,45){\vector(1,0){25}}     % velocity 1 
 \put(45,45){\vector(-1,2){12.5}}  % velocity 3 
 \put(45,45){\vector(-1,-2){12.5}} % velocity 5 
\ept}

\newcommand{\hexbdf}{
\bpt(90,90)
 \thinlines
 \put(15,45){\line(1,0){60}}
 \put(30,15){\line(1,2){30}}
 \put(60,15){\line(-1,2){30}}
 
 \put(76,41){1}
 \put(61,71){2}
 \put(23,71){3}
 \put(8,41){4}
 \put(23,10){5}
 \put(61,10){6}

 \thicklines
 \put(45,45){\vector(1,2){12.5}}   % velocity 2 
 \put(45,45){\vector(-1,0){25}}    % velocity 4 
 \put(45,45){\vector(1,-2){12.5}}  % velocity 6 
\ept}

\newcommand{\hexadb}{
\bpt(90,90)
 \thinlines
 \put(15,45){\line(1,0){60}}
 \put(30,15){\line(1,2){30}}
 \put(60,15){\line(-1,2){30}}
 
 \put(76,41){1}
 \put(61,71){2}
 \put(23,71){3}
 \put(8,41){4}
 \put(23,10){5}
 \put(61,10){6}

 \thicklines
 \put(45,45){\vector(1,0){25}}     % velocity 1 
 \put(45,45){\vector(1,2){12.5}}   % velocity 2 
 \put(45,45){\vector(-1,0){25}}    % velocity 4 
\ept}

\newcommand{\hexcfb}{
\bpt(90,90)
 \thinlines
 \put(15,45){\line(1,0){60}}
 \put(30,15){\line(1,2){30}}
 \put(60,15){\line(-1,2){30}}
 
 \put(76,41){1}
 \put(61,71){2}
 \put(23,71){3}
 \put(8,41){4}
 \put(23,10){5}
 \put(61,10){6}

 \thicklines
 \put(45,45){\vector(1,2){12.5}}   % velocity 2 
 \put(45,45){\vector(-1,2){12.5}}  % velocity 3 
 \put(45,45){\vector(1,-2){12.5}}  % velocity 6 
\ept}

\newcommand{\hexadbe}{
\bpt(90,90)
 \thinlines
 \put(15,45){\line(1,0){60}}
 \put(30,15){\line(1,2){30}}
 \put(60,15){\line(-1,2){30}}
 
 \put(76,41){1}
 \put(61,71){2}
 \put(23,71){3}
 \put(8,41){4}
 \put(23,10){5}
 \put(61,10){6}

 \thicklines
 \put(45,45){\vector(1,0){25}}     % velocity 1 
 \put(45,45){\vector(1,2){12.5}}   % velocity 2 
 \put(45,45){\vector(-1,0){25}}    % velocity 4 
 \put(45,45){\vector(-1,-2){12.5}} % velocity 5 
\ept}

\newcommand{\hexbecf}{
\bpt(90,90)
 \thinlines
 \put(15,45){\line(1,0){60}}
 \put(30,15){\line(1,2){30}}
 \put(60,15){\line(-1,2){30}}
 
 \put(76,41){1}
 \put(61,71){2}
 \put(23,71){3}
 \put(8,41){4}
 \put(23,10){5}
 \put(61,10){6}

 \thicklines
 \put(45,45){\vector(1,2){12.5}}   % velocity 2 
 \put(45,45){\vector(-1,2){12.5}}  % velocity 3 
 \put(45,45){\vector(-1,-2){12.5}} % velocity 5 
 \put(45,45){\vector(1,-2){12.5}}  % velocity 6 
\ept}

\newcommand{\hexadcf}{
\bpt(90,90)
 \thinlines
 \put(15,45){\line(1,0){60}}
 \put(30,15){\line(1,2){30}}
 \put(60,15){\line(-1,2){30}}
 
 \put(76,41){1}
 \put(61,71){2}
 \put(23,71){3}
 \put(8,41){4}
 \put(23,10){5}
 \put(61,10){6}

 \thicklines
 \put(45,45){\vector(1,0){25}}     % velocity 1 
 \put(45,45){\vector(-1,2){12.5}}  % velocity 3 
 \put(45,45){\vector(-1,0){25}}    % velocity 4 
 \put(45,45){\vector(1,-2){12.5}}  % velocity 6 
\ept}

\newcommand{\hexsixb}{
\bpt(260,540)
 \put(0,0){\line(1,0){260}}
 \put(0,180){\line(1,0){260}}
 \put(0,360){\line(1,0){260}}
 \put(0,540){\line(1,0){260}}

 \put(0,0){\line(0,1){540}}
 \put(130,0){\line(0,1){540}}
 \put(260,0){\line(0,1){540}}

 \put(20,405){\hexad}
 \put(150,450){\hexbe}
 \put(150,360){\hexcf}

 \put(20,270){\hexace}
 \put(150,270){\hexbdf}

 \put(20,180){\hexadb}
 \put(150,180){\hexcfb}

 \put(20,45){\hexadbe}
 \put(150,90){\hexbecf}
 \put(150,0){\hexadcf}
\ept}

\newcommand{\sqac}{
\bpt(80,80)
 \thinlines
 \put(0,40){\line(1,0){80}}
 \put(40,0){\line(0,1){80}}
 \thicklines
 \put(40,40){\vector(1,0){30}}
 \put(40,40){\vector(-1,0){30}}
\ept}

\newcommand{\sqbd}{
\bpt(80,80)
 \thinlines
 \put(0,40){\line(1,0){80}}
 \put(40,0){\line(0,1){80}}
 \thicklines
 \put(40,40){\vector(0,1){30}}
 \put(40,40){\vector(0,-1){30}}
\ept}

\newcommand{\dblarw}{
\bpt(60,20)
 \thinlines
 \put(0,10){\line(1,1){10}}
 \put(0,10){\line(1,-1){10}}
 \put(10,20){\line(0,-1){6}}
 \put(10,0){\line(0,1){6}}
 \put(10,14){\line(1,0){40}}
 \put(10,6){\line(1,0){40}}
 \put(50,6){\line(0,-1){6}}
 \put(50,14){\line(0,1){6}}
 \put(50,0){\line(1,1){10}}
 \put(50,20){\line(1,-1){10}}
\ept}

.