Lecture XXX. Approximation Solutions to Boltzmann Equation: Relaxation Time Approximation. Readings: Brennan Chapter 6.2 & Notes. Prepared By: Hua Fan
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1 Prepared y: Hua Fan Lecture XXX Apprxiatin Slutins t ltann Equatin: Relaxatin ie Apprxiatin Readings: rennan Chapter 6. & Ntes Gergia Insitute echnlgy ECE 645-Hua Fan
2 Apprxiatin Slutins the ltann Equatin (Sectin 6.) General Slutin General ltann equatin (E) is F ext { ( x,, t)[ ( x, ', t)] S(, ') + + x d' t h ( x, ', t)[ ( x,, t) S( l', )} () which is a integral-dierential equatin. In rder t ind the slutin, drastic apprxiatins hae t be used, which are ten inalid r the particular situatin. herere, a technique based n the relaxatin tie apprxiatin is use t sle r the ltann equatin. It shuld be nted that this apprxiatin des nt always wr, but because it greatly reduces the cplexity the ltann equatin and a clsed r slutin can be btained, it will be inestigated here. Fr cases that the apprxiatin can nt be used, re sphisticated ethds such as drit-diusin and the Mnte Carl ethd (Sectin 6.3) shuld be used. Gergia Insitute echnlgy ECE 645-Hua Fan
3 Apprxiatin Slutins the ltann Equatin (Sectin 6.) Issues (hw t sle it?) he prble with the ltann equatin is the integral scattering ter. As a apprxiatin, let s replace it with a cnstant relaxatin ter, which reduces the riginal integral-dierential equatin int nly a dierential equatin. In additin, the partial deriatie the distributin unctin with respect t tie due t cllisins is assued t be inersely prprtinal t a lietie that characteries the ean ree tie between cllisins. Hence, i is the equilibriu distributin unctin and is the nnequilibriu distributin unctin r which the ltann equatin is sled, the let hand side the ltann equatin changes t t cllisins Equatin () shws that the syste will relax t equilibriu ater tie, which is the relaxatin tie and represents the aerage tie it taes r the syste t relax r the nnequilibriu state t the equilibriu state thrugh cllisins. () Gergia Insitute echnlgy ECE 645-Hua Fan 3
4 Apprxiatin Slutins the ltann Equatin (Sectin 6.) Further Inestigatin Apply electric ield F with x 0 (n diusin gradient), at t<0 ltann equatin beces t cllisins then F is reed, s r t>0 ltann equatin reduces t ( x,, t) + t (3) cllisins Using () and adding int the deriatie (since 0 ), t ( ) t t t the abe dierential equatin is sled thrugh y ( ) dy y dt t y t y e (4) (5) (6) Gergia Insitute echnlgy ECE 645-Hua Fan 4
5 Apprxiatin Slutins the ltann Equatin (Sectin 6.) Further Inestigatin (Cnt d) Ater substituting (6) int (5), (5) ealuates t ( t) + [ (0) ] e Fr the abe results, it s clear that as t, relaxes t, hweer, the syste desn t need that uch tie, it will achiee equilibriu when t. t (7) Again, hw des the syste relax? (Clesn student s answer: by sitting n the cuch?) he syste cpletely relaxes thrugh scattering(cllisin) eents. I scattering desn t tae place, the rate change(8) the nnequilibriu state beces er, which eans that there is n pssible slutin r (7) such that will relax t. t 0 (8) Gergia Insitute echnlgy ECE 645-Hua Fan 5
6 Apprxiatin Slutins the ltann Equatin (Sectin 6.) Unir Electric Field Unir electric ield is applied t a steady state syste such that there s n spatial gradient, hence (9) x 0 0 t bere apprxiatins the E is t F + h with the abe assuptins, the E beces F q ext F F ext ext t + x cllisins t cllisins (0) () () Gergia Insitute echnlgy ECE 645-Hua Fan 6
7 ECE 645-Hua Fan 7 Gergia Insitute echnlgy Apprxiatin Slutins the ltann Equatin (Sectin 6.) Unir Electric Field (Cnt d) Assue ield is nly in directin, and is nt ar r, which can be equated t the Maxwellian. (3) (5) b e ~ (4) (6) e e e csθ ) (
8 Apprxiatin Slutins the ltann Equatin (Sectin 6.) Unir Electric Field (Cnt d) Equatin (6) cpletely describes the distributin unctin the nnequilibriu state, and it is clear that the nnequilibriu state is just shited ersin the equilibriu state (Figure 6.. in b), whereas the equilibriu distributin unctin is centered abut 0 (ag. 0). equilibriu nnequilibriu 0 S why is this slight shit s iprtant? he current density was deterined r the equilibriu case earlier (current density is er at equilibriu), but with the shit in the distributin the current density r the nnequilibriu case beces nn-er. Gergia Insitute echnlgy ECE 645-Hua Fan 8
9 Apprxiatin Slutins the ltann Equatin (Sectin 6.3) Current Density π 0 Current density is deined as j the aerage elcity r the nnequilibriu case is nt er, s current density is nt er. 3 ( ) D( ) d 3 3 Vl 3 D( ) d dxdyd (8) ( ) D( ) d 3 h Gergia Insitute echnlgy nq csθ sinθdθ 0 π π π 0 0 π π csθ dφ ( ) θ d θdθ cs sin π csθ dφ ( ) dsinθdθ 0 π π ( ) ( ) 4 cs θ sinθ dθd sinθddθ (7) (9) (0) ECE 645-Hua Fan 9
10 Apprxiatin Slutins the ltann Equatin (Sectin 6.3) Current Density (Cnt d) π 0 π cs θ sinθdθ /3 sinθdθ ( ) ( ) d ( ) d 3 ( ) () ( ) 3 () j Gergia Insitute echnlgy nq nq nq σ nqu F σf (3) (4) ECE 645-Hua Fan 0
11 Apprxiatin Slutins the ltann Equatin (Sectin 6.3) Current Density (Cnt d) u F q (5) Fr equatin (3), the current density r a nnquilibriu state is nt er and the subsequent paraeters electrn bility u (5) and cnductiity σ (4) are als deried r the nnequilibriu case. Gergia Insitute echnlgy ECE 645-Hua Fan
12 Apprxiatin Slutins the ltann Equatin (Sectin 6.) Exaple Cnsider a unir istrpic substance at cnstant teperature in the presence a cnstant applied electric ield F. I steady state cnditins are achieed, the nnequilibriu distributin unctin can be written as Deterine the paraeter. Slutin: Reeber that i an unir electric ield is applied un steady state cnditins, ( csθ ) λ Gergia Insitute echnlgy ( csθ ) λ + ( csθ ) ( λ) ( + λ) Can yu ind the current density??? ECE 645-Hua Fan
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