Skip to main content
Physics LibreTexts

8.5: Diffusion and the Lorentz model

  • Page ID
    18594
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand\bes{\begin{equation}\begin{split}}\)
    \( \newcommand\ltwid{\propto}\)
    \( \newcommand\ees{\end{split}\end{equation}}\)
    \( \newcommand\mib{\mathbf}\)
    \( \newcommand\Sa{\textsf a}\)
    \( \newcommand\Sb{\textsf b}\)
    \( \newcommand\Sc{\textsf c}\)
    \( \newcommand\Sd{\textsf d}\)
    \( \newcommand\Se{\textsf e}\)
    \( \newcommand\Sf{\textsf f}\)
    \( \newcommand\Sg{\textsf g}\)
    \( \newcommand\Sh{\textsf h}\)
    \( \newcommand\Si{\textsf i}\)
    \( \newcommand\Sj{\textsf j}\)
    \( \newcommand\Sk{\textsf k}\)
    \( \newcommand\Sl{\textsf l}\)
    \( \newcommand\Sm{\textsf m}\)
    \( \newcommand\Sn{\textsf n}\)
    \( \newcommand\So{\textsf o}\)
    \( \newcommand\Sp{\textsf p}\)
    \( \newcommand\Sq{\textsf q}\)
    \( \newcommand\Sr{\textsf r}\)
    \( \newcommand\Ss{\textsf s}\)
    \( \newcommand\St{\textsf t}\)
    \( \newcommand\Su{\textsf u}\)
    \( \newcommand\Sv{\textsf v}\)
    \( \newcommand\Sw{\textsf w}\)
    \( \newcommand\Sx{\textsf x}\)
    \( \newcommand\Sy{\textsf y}\)
    \( \newcommand\Sz{\textsf z}\)
    \( \newcommand\SA{\textsf A}\)
    \( \newcommand\SB{\textsf B}\)
    \( \newcommand\SC{\textsf C}\)
    \( \newcommand\SD{\textsf D}\)
    \( \newcommand\SE{\textsf E}\)
    \( \newcommand\SF{\textsf F}\)
    \( \newcommand\SG{\textsf G}\)
    \( \newcommand\SH{\textsf H}\)
    \( \newcommand\SI{\textsf I}\)
    \( \newcommand\SJ{\textsf J}\)
    \( \newcommand\SK{\textsf K}\)
    \( \newcommand\SL{\textsf L}\)
    \( \newcommand\SM{\textsf M}\)
    \( \newcommand\SN{\textsf N}\)
    \( \newcommand\SO{\textsf O}\)
    \( \newcommand\SP{\textsf P}\)
    \( \newcommand\SQ{\textsf Q}\)
    \( \newcommand\SR{\textsf R}\)
    \( \newcommand\SS{\textsf S}\)
    \( \newcommand\ST{\textsf T}\)
    \( \newcommand\SU{\textsf U}\)
    \( \newcommand\SV{\textsf V}\)
    \( \newcommand\SW{\textsf W}\)
    \( \newcommand\SX{\textsf X}\)
    \( \newcommand\SY{\textsf Y}\)
    \( \newcommand\SZ{\textsf Z}\)
    \( \newcommand\Ha{\hat a}\)
    \( \newcommand\Hb{\hat b}\)
    \( \newcommand\Hc{\hat c}\)
    \( \newcommand\Hd{\hat d}\)
    \( \newcommand\He{\hat e}\)
    \( \newcommand\Hf{\hat f}\)
    \( \newcommand\Hg{\hat g}\)
    \( \newcommand\Hh{\hat h}\)
    \( \newcommand\Hi{\hat \imath}\)
    \( \newcommand\Hj{\hat \jmath}\)
    \( \newcommand\Hk{\hat k}\)
    \( \newcommand\Hl{\hat l}\)
    \( \newcommand\Hm{\hat m}\)
    \( \newcommand\Hn{\hat n}\)
    \( \newcommand\Ho{\hat o}\)
    \( \newcommand\Hp{\hat p}\)
    \( \newcommand\Hq{\hat q}\)
    \( \newcommand\Hr{\hat r}\)
    \( \newcommand\Hs{\hat s}\)
    \( \newcommand\Ht{\hat t}\)
    \( \newcommand\Hu{\hat u}\)
    \( \newcommand\Hv{\hat v}\)
    \( \newcommand\Hw{\hat w}\)
    \( \newcommand\Hx{\hat x}\)
    \( \newcommand\Hy{\hat y}\)
    \( \newcommand\Hz{\hat z}\)
    \( \newcommand\HA{\hat A}\)
    \( \newcommand\HB{\hat B}\)
    \( \newcommand\HC{\hat C}\)
    \( \newcommand\HD{\hat D}\)
    \( \newcommand\HE{\hat E}\)
    \( \newcommand\HF{\hat F}\)
    \( \newcommand\HG{\hat G}\)
    \( \newcommand\HH{\hat H}\)
    \( \newcommand\HI{\hat I}\)
    \( \newcommand\HJ{\hat J}\)
    \( \newcommand\HK{\hat K}\)
    \( \newcommand\HL{\hat L}\)
    \( \newcommand\HM{\hat M}\)
    \( \newcommand\HN{\hat N}\)
    \( \newcommand\HO{\hat O}\)
    \( \newcommand\HP{\hat P}\)
    \( \newcommand\HQ{\hat Q}\)
    \( \newcommand\HR{\hat R}\)
    \( \newcommand\HS{\hat S}\)
    \( \newcommand\HT{\hat T}\)
    \( \newcommand\HU{\hat U}\)
    \( \newcommand\HV{\hat V}\)
    \( \newcommand\HW{\hat W}\)
    \( \newcommand\HX{\hat X}\)
    \( \newcommand\HY{\hat Y}\)
    \( \newcommand\HZ{\hat Z}\)
    \( \newcommand\Halpha{\hat\alpha}\)
    \( \newcommand\Hbeta{\hat\beta}\)
    \( \newcommand\Hgamma{\hat\gamma}\)
    \( \newcommand\Hdelta{\hat\delta}\)
    \( \newcommand\Hepsilon{\hat\epsilon}\)
    \( \newcommand\Hvarepsilon{\hat\varepsilon}\)
    \( \newcommand\Hzeta{\hat\zeta}\)
    \( \newcommand\Heta{\hat\eta}\)
    \( \newcommand\Htheta{\hat\theta}\)
    \( \newcommand\Hvartheta{\hat\vartheta}\)
    \( \newcommand\Hiota{\hat\iota}\)
    \( \newcommand\Hkappa{\hat\kappa}\)
    \( \newcommand\Hlambda{\hat\lambda}\)
    \( \newcommand\Hmu{\hat\mu}\)
    \( \newcommand\Hnu{\hat\nu}\)
    \( \newcommand\Hxi{\hat\xi}\)
    \( \newcommand\Hom{\hat\omicron}\)
    \( \newcommand\Hpi{\hat\pi}\)
    \( \newcommand\Hvarpi{\hat\varpi}\)
    \( \newcommand\Hrho{\hat\rho}\)
    \( \newcommand\Hvarrho{\hat\varrho}\)
    \( \newcommand\Hsigma{\hat\sigma}\)
    \( \newcommand\Hvarsigma{\hat\varsigma}\)
    \( \newcommand\Htau{\var\tau}\)
    \( \newcommand\Hupsilon{\hat\upsilon}\)
    \( \newcommand\Hphi{\hat\phi}\)
    \( \newcommand\Hvarphi{\hat\varphi}\)
    \( \newcommand\Hchi{\hat\chi}\)
    \( \newcommand\Hxhi{\hat\xhi}\)
    \( \newcommand\Hpsi{\hat\psi}\)
    \( \newcommand\Homega{\hat\omega}\)
    \( \newcommand\HGamma{\hat\Gamma}\)
    \( \newcommand\HDelta{\hat\Delta}\)
    \( \newcommand\HTheta{\hat\Theta}\)
    \( \newcommand\HLambda{\hat\Lambda}\)
    \( \newcommand\HXi{\hat\Xi}\)
    \( \newcommand\HPi{\hat\Pi}\)
    \( \newcommand\HSigma{\hat\Sigma}\)
    \( \newcommand\HUps{\hat\Upsilon}\)
    \( \newcommand\HPhi{\hat\Phi}\)
    \( \newcommand\HPsi{\hat\Psi}\)
    \( \newcommand\HOmega{\hat\Omega}\)
    \( \newcommand\xhat{\hat\Bx}\)
    \( \newcommand\yhat{\hat\By}\)
    \( \newcommand\zhat{\hat\Bz}\)
    \( \newcommand\ehat{\hat\Be}\)
    \( \newcommand\khat{\hat\Bk}\)
    \( \newcommand\nhat{\hat\Bn}\)
    \( \newcommand\rhat{\hat\Br}\)
    \( \newcommand\phihat{\hat\Bphi}\)
    \( \newcommand\thetahat{\hat\Btheta}\)
    \( \newcommand\MA{\mathbb A}\)
    \( \newcommand\MB{\mathbb B}\)
    \( \newcommand\MC{\mathbb C}\)
    \( \newcommand\MD{\mathbb D}\)
    \( \newcommand\ME{\mathbb E}\)
    \( \newcommand\MF{\mathbb F}\)
    \( \newcommand\MG{\mathbb G}\)
    \( \newcommand\MH{\mathbb H}\)
    \( \newcommand\MI{\mathbb I}\)
    \( \newcommand\MJ{\mathbb J}\)
    \( \newcommand\MK{\mathbb K}\)
    \( \newcommand\ML{\mathbb L}\)
    \( \newcommand\MM{\mathbb M}\)
    \( \newcommand\MN{\mathbb N}\)
    \( \newcommand\MO{\mathbb O}\)
    \( \newcommand\MP{\mathbb P}\)
    \( \newcommand\MQ{\mathbb Q}\)
    \( \newcommand\MR{\mathbb R}\)
    \( \newcommand\MS{\mathbb S}\)
    \( \newcommand\MT{\mathbb T}\)
    \( \newcommand\MU{\mathbb U}\)
    \( \newcommand\MV{\mathbb V}\)
    \( \newcommand\MW{\mathbb W}\)
    \( \newcommand\MX{\mathbb X}\)
    \( \newcommand\MY{\mathbb Y}\)
    \( \newcommand\MZ{\mathbb Z}\)
    \( \newcommand\CA{\mathcal A}\)
    \( \newcommand\CB{\mathcal B}\)
    \( \newcommand\CC{\mathcal C}\)
    \( \newcommand\CD{\mathcal D}\)
    \( \newcommand\CE{\mathcal E}\)
    \( \newcommand\CF{\mathcal F}\)
    \( \newcommand\CG{\mathcal G}\)
    \( \newcommand\CH{\mathcal H}\)
    \( \newcommand\CI{\mathcal I}\)
    \( \newcommand\CJ{\mathcal J}\)
    \( \newcommand\CK{\mathcal K}\)
    \( \newcommand\CL{\mathcal L}\)
    \( \newcommand\CM{\mathcal M}\)
    \( \newcommand\CN{\mathcal N}\)
    \( \newcommand\CO{\mathcal O}\)
    \( \newcommand\CP{\mathcal P}\)
    \( \newcommand\CQ{\mathcal Q}\)
    \( \newcommand\CR{\mathcal R}\)
    \( \newcommand\CS{\mathcal S}\)
    \( \newcommand\CT{\mathcal T}\)
    \( \newcommand\CU{\mathcal U}\)
    \( \newcommand\CV{\mathcal V}\)
    \( \newcommand\CW{\mathcal W}\)
    \( \newcommand\CX{\mathcal X}\)
    \( \newcommand\CY{\mathcal Y}\)
    \( \newcommand\CZ{\mathcal Z}\)
    \( \newcommand\Fa{\mathfrak a}\)
    \( \newcommand\Fb{\mathfrak b}\)
    \( \newcommand\Fc{\mathfrak c}\)
    \( \newcommand\Fd{\mathfrak d}\)
    \( \newcommand\Fe{\mathfrak e}\)
    \( \newcommand\Ff{\mathfrak f}\)
    \( \newcommand\Fg{\mathfrak g}\)
    \( \newcommand\Fh{\mathfrak h}\)
    \( \newcommand\Fi{\mathfrak i}\)
    \( \newcommand\Fj{\mathfrak j}\)
    \( \newcommand\Fk{\mathfrak k}\)
    \( \newcommand\Fl{\mathfrak l}\)
    \( \newcommand\Fm{\mathfrak m}\)
    \( \newcommand\Fn{\mathfrak n}\)
    \( \newcommand\Fo{\mathfrak o}\)
    \( \newcommand\Fp{\mathfrak p}\)
    \( \newcommand\Fq{\mathfrak q}\)
    \( \newcommand\Fr{\mathfrak r}\)
    \( \newcommand\Fs{\mathfrak s}\)
    \( \newcommand\Ft{\mathfrak t}\)
    \( \newcommand\Fu{\mathfrak u}\)
    \( \newcommand\Fv{\mathfrak v}\)
    \( \newcommand\Fw{\mathfrak w}\)
    \( \newcommand\Fx{\mathfrak x}\)
    \( \newcommand\Fy{\mathfrak y}\)
    \( \newcommand\Fz{\mathfrak z}\)
    \( \newcommand\FA{\mathfrak A}\)
    \( \newcommand\FB{\mathfrak B}\)
    \( \newcommand\FC{\mathfrak C}\)
    \( \newcommand\FD{\mathfrak D}\)
    \( \newcommand\FE{\mathfrak E}\)
    \( \newcommand\FF{\mathfrak F}\)
    \( \newcommand\FG{\mathfrak G}\)
    \( \newcommand\FH{\mathfrak H}\)
    \( \newcommand\FI{\mathfrak I}\)
    \( \newcommand\FJ{\mathfrak J}\)
    \( \newcommand\FK{\mathfrak K}\)
    \( \newcommand\FL{\mathfrak L}\)
    \( \newcommand\FM{\mathfrak M}\)
    \( \newcommand\FN{\mathfrak N}\)
    \( \newcommand\FO{\mathfrak O}\)
    \( \newcommand\FP{\mathfrak P}\)
    \( \newcommand\FQ{\mathfrak Q}\)
    \( \newcommand\FR{\mathfrak R}\)
    \( \newcommand\FS{\mathfrak S}\)
    \( \newcommand\FT{\mathfrak T}\)
    \( \newcommand\FU{\mathfrak U}\)
    \( \newcommand\FV{\mathfrak V}\)
    \( \newcommand\FW{\mathfrak W}\)
    \( \newcommand\FX{\mathfrak X}\)
    \( \newcommand\FY{\mathfrak Y}\)
    \( \newcommand\FZ{\mathfrak Z}\)
    \( \newcommand\Da{\dot a}\)
    \( \newcommand\Db{\dot b}\)
    \( \newcommand\Dc{\dot c}\)
    \( \newcommand\Dd{\dot d}\)
    \( \newcommand\De{\dot e}\)
    \( \newcommand\Df{\dot f}\)
    \( \newcommand\Dg{\dot g}\)
    \( \newcommand\Dh{\dot h}\)
    \( \newcommand\Di{\dot \imath}\)
    \( \newcommand\Dj{\dot \jmath}\)
    \( \newcommand\Dk{\dot k}\)
    \( \newcommand\Dl{\dot l}\)
    \( \newcommand\Dm{\dot m}\)
    \( \newcommand\Dn{\dot n}\)
    \( \newcommand\Do{\dot o}\)
    \( \newcommand\Dp{\dot p}\)
    \( \newcommand\Dq{\dot q}\)
    \( \newcommand\Dr{\dot r}\)
    \( \newcommand\Ds{\dot s}\)
    \( \newcommand\Dt{\dot t}\)
    \( \newcommand\Du{\dot u}\)
    \( \newcommand\Dv{\dot v}\)
    \( \newcommand\Dw{\dot w}\)
    \( \newcommand\Dx{\dot x}\)
    \( \newcommand\Dy{\dot y}\)
    \( \newcommand\Dz{\dot z}\)
    \( \newcommand\DA{\dot A}\)
    \( \newcommand\DB{\dot B}\)
    \( \newcommand\DC{\dot C}\)
    \( \newcommand\DD{\dot D}\)
    \( \newcommand\DE{\dot E}\)
    \( \newcommand\DF{\dot F}\)
    \( \newcommand\DG{\dot G}\)
    \( \newcommand\DH{\dot H}\)
    \( \newcommand\DI{\dot I}\)
    \( \newcommand\DJ{\dot J}\)
    \( \newcommand\DK{\dot K}\)
    \( \newcommand\DL{\dot L}\)
    \( \newcommand\DM{\dot M}\)
    \( \newcommand\DN{\dot N}\)
    \( \newcommand\DO{\dot O}\)
    \( \newcommand\DP{\dot P}\)
    \( \newcommand\DQ{\dot Q}\)
    \( \newcommand\DR{\dot R}\)
    \( \newcommand\DS{\dot S}\)
    \( \newcommand\DT{\dot T}\)
    \( \newcommand\DU{\dot U}\)
    \( \newcommand\DV{\dot V}\)
    \( \newcommand\DW{\dot W}\)
    \( \newcommand\DX{\dot X}\)
    \( \newcommand\DY{\dot Y}\)
    \( \newcommand\DZ{\dot Z}\)
    \( \newcommand\Dalpha

    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[1], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Dbeta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[2], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Dgamma
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[3], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Ddelta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[4], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Depsilon
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[5], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Dvarepsilon
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[6], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Dzeta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[7], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Deta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[8], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Dtheta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[9], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Dvartheta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[10], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Diota
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[11], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Dkappa
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[12], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Dlambda
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[13], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Dmu{\dot\mu}\)
    \( \newcommand\Dnu{\dot\nu}\)
    \( \newcommand\Dxi{\dot\xi}\)
    \( \newcommand\Dom{\dot\omicron}\)
    \( \newcommand\Dpi{\dot\pi}\)
    \( \newcommand\Dvarpi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[14], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Drho{\dot\rho}\)
    \( \newcommand\Dvarrho{\dot\varrho}\)
    \( \newcommand\Dsigma{\dot\sigma}\)
    \( \newcommand\Dvarsigma{\dot\varsigma}\)
    \( \newcommand\Dtau{\var\tau}\)
    \( \newcommand\Dupsilon{\dot\upsilon}\)
    \( \newcommand\Dphi{\dot\phi}\)
    \( \newcommand\Dvarphi{\dot\varphi}\)
    \( \newcommand\Dchi{\dot\chi}\)
    \( \newcommand\Dpsi{\dot\psi}\)
    \( \newcommand\Domega{\dot\omega}\)
    \( \newcommand\DGamma
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[15], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\DDelta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\DTheta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[17], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\DLambda{\dot\Lambda}\)
    \( \newcommand\DXi{\dot\Xi}\)
    \( \newcommand\DPi{\dot\Pi}\)
    \( \newcommand\DSigma{\dot\Sigma}\)
    \( \newcommand\DUps{\dot\Upsilon}\)
    \( \newcommand\DPhi{\dot\Phi}\)
    \( \newcommand\DPsi{\dot\Psi}\)
    \( \newcommand\DOmega{\dot\Omega}\)
    \( \newcommand\Va{\vec a}\)
    \( \newcommand\Vb{\vec b}\)
    \( \newcommand\Vc{\vec c}\)
    \( \newcommand\Vd{\vec d}\)
    \( \newcommand\Ve{\vec e}\)
    \( \newcommand\Vf{\vec f}\)
    \( \newcommand\Vg{\vec g}\)
    \( \newcommand\Vh{\vec h}\)
    \( \newcommand\Vi{\vec \imath}\)
    \( \newcommand\Vj{\vec \jmath}\)
    \( \newcommand\Vk{\vec k}\)
    \( \newcommand\Vl{\vec l}\)
    \( \newcommand\Vm{\vec m}\)
    \( \newcommand\Vn{\vec n}\)
    \( \newcommand\Vo{\vec o}\)
    \( \newcommand\Vp{\vec p}\)
    \( \newcommand\Vq{\vec q}\)
    \( \newcommand\Vr{\vec r}\)
    \( \newcommand\Vs{\vec s}\)
    \( \newcommand\Vt{\vec t}\)
    \( \newcommand\Vu{\vec u}\)
    \( \newcommand\Vv{\vec v}\)
    \( \newcommand\Vw{\vec w}\)
    \( \newcommand\Vx{\vec x}\)
    \( \newcommand\Vy{\vec y}\)
    \( \newcommand\Vz{\vec z}\)
    \( \newcommand\VA{\vec A}\)
    \( \newcommand\VB{\vec B}\)
    \( \newcommand\VC{\vec C}\)
    \( \newcommand\VD{\vec D}\)
    \( \newcommand\VE{\vec E}\)
    \( \newcommand\VF{\vec F}\)
    \( \newcommand\VG{\vec G}\)
    \( \newcommand\VH{\vec H}\)
    \( \newcommand\VI{\vec I}\)
    \( \newcommand\VJ{\vec J}\)
    \( \newcommand\VK{\vec K}\)
    \( \newcommand\VL{\vec L}\)
    \( \newcommand\VM{\vec M}\)
    \( \newcommand\VN{\vec N}\)
    \( \newcommand\VO{\vec O}\)
    \( \newcommand\VP{\vec P}\)
    \( \newcommand\VQ{\vec Q}\)
    \( \newcommand\VR{\vec R}\)
    \( \newcommand\VS{\vec S}\)
    \( \newcommand\VT{\vec T}\)
    \( \newcommand\VU{\vec U}\)
    \( \newcommand\VV{\vec V}\)
    \( \newcommand\VW{\vec W}\)
    \( \newcommand\VX{\vec X}\)
    \( \newcommand\VY{\vec Y}\)
    \( \newcommand\VZ{\vec Z}\)
    \( \newcommand\Valpha{\vec\alpha}\)
    \( \newcommand\Vbeta{\vec\beta}\)
    \( \newcommand\Vgamma{\vec\gamma}\)
    \( \newcommand\Vdelta{\vec\delta}\)
    \( \newcommand\Vepsilon{\vec\epsilon}\)
    \( \newcommand\Vvarepsilon{\vec\varepsilon}\)
    \( \newcommand\Vzeta{\vec\zeta}\)
    \( \newcommand\Veta{\vec\eta}\)
    \( \newcommand\Vtheta{\vec\theta}\)
    \( \newcommand\Vvartheta{\vec\vartheta}\)
    \( \newcommand\Viota{\vec\iota}\)
    \( \newcommand\Vkappa{\vec\kappa}\)
    \( \newcommand\Vlambda{\vec\lambda}\)
    \( \newcommand\Vmu
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[18], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vnu
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[19], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vxi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vom
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vpi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vvarpi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vrho
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vvarrho
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vsigma
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vvarsigma
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vtau
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vupsilon
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[29], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vphi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[30], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vvarphi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[31], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vchi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[32], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vpsi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[33], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vomega
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[34], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\VGamma
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[35], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\VDelta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[36], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\VTheta{\vec\Theta}\)
    \( \newcommand\VLambda{\vec\Lambda}\)
    \( \newcommand\VXi{\vec\Xi}\)
    \( \newcommand\VPi{\vec\Pi}\)
    \( \newcommand\VSigma{\vec\Sigma}\)
    \( \newcommand\VUps{\vec\Upsilon}\)
    \( \newcommand\VPhi{\vec\Phi}\)
    \( \newcommand\VPsi{\vec\Psi}\)
    \( \newcommand\VOmega{\vec\Omega}\)
    \( \newcommand\BA{\mib A}\)
    \( \newcommand\BB{\mib B}\)
    \( \newcommand\BC{\mib C}\)
    \( \newcommand\BD{\mib D}\)
    \( \newcommand\BE{\mib E}\)
    \( \newcommand\BF{\mib F}\)
    \( \newcommand\BG{\mib G}\)
    \( \newcommand\BH{\mib H}\)
    \( \newcommand\BI{\mib I}}\)
    \( \newcommand\BJ{\mib J}\)
    \( \newcommand\BK{\mib K}\)
    \( \newcommand\BL{\mib L}\)
    \( \newcommand\BM{\mib M}\)
    \( \newcommand\BN{\mib N}\)
    \( \newcommand\BO{\mib O}\)
    \( \newcommand\BP{\mib P}\)
    \( \newcommand\BQ{\mib Q}\)
    \( \newcommand\BR{\mib R}\)
    \( \newcommand\BS{\mib S}\)
    \( \newcommand\BT{\mib T}\)
    \( \newcommand\BU{\mib U}\)
    \( \newcommand\BV{\mib V}\)
    \( \newcommand\BW{\mib W}\)
    \( \newcommand\BX{\mib X}\)
    \( \newcommand\BY{\mib Y}\)
    \( \newcommand\BZ{\mib Z}\)
    \( \newcommand\Ba{\mib a}\)
    \( \newcommand\Bb{\mib b}\)
    \( \newcommand\Bc{\mib c}\)
    \( \newcommand\Bd{\mib d}\)
    \( \newcommand\Be{\mib e}\)
    \( \newcommand\Bf{\mib f}\)
    \( \newcommand\Bg{\mib g}\)
    \( \newcommand\Bh{\mib h}\)
    \( \newcommand\Bi{\mib i}\)
    \( \newcommand\Bj{\mib j}\)
    \( \newcommand\Bk{\mib k}\)
    \( \newcommand\Bl{\mib l}\)
    \( \newcommand\Bm{\mib m}\)
    \( \newcommand\Bn{\mib n}\)
    \( \newcommand\Bo{\mib o}\)
    \( \newcommand\Bp{\mib p}\)
    \( \newcommand\Bq{\mib q}\)
    \( \newcommand\Br{\mib r}\)
    \( \newcommand\Bs{\mib s}\)
    \( \newcommand\Bt{\mib t}\)
    \( \newcommand\Bu{\mib u}\)
    \( \newcommand\Bv{\mib v}\)
    \( \newcommand\Bw{\mib w}\)
    \( \newcommand\Bx{\mib x}\)
    \( \newcommand\By{\mib y}\)
    \( \newcommand\Bz{\mib z}\)\)
    \( \newcommand\vrh{\varrho}\)
    \( \newcommand\vsig{\varsigma}\)
    \( \newcommand\ups{\upsilon}\)
    \( \newcommand\eps{\epsilon}\)
    \( \newcommand\ve{\varepsilon}\)
    \( \newcommand\vth{\vartheta}\)
    \( \newcommand\vphi{\varphi}\)
    \( \newcommand\xhi{\chi}\)
    \( \newcommand\Ups{\Upsilon}\)
    \( \newcommand\Balpha{\mib\alpha}\)
    \( \newcommand\Bbeta{\mib\beta}\)
    \( \newcommand\Bgamma{\mib\gamma}\)
    \( \newcommand\Bdelta{\mib\delta}\)
    \( \newcommand\Beps{\mib\epsilon}\)
    \( \newcommand\Bve{\mib\varepsilon}\)
    \( \newcommand\Bzeta{\mib\zeta}\)
    \( \newcommand\Beta{\mib\eta}\)
    \( \newcommand\Btheta{\mib\theta}\)
    \( \newcommand\Bvth{\mib\vartheta}\)
    \( \newcommand\Biota{\mib\iota}\)
    \( \newcommand\Bkappa{\mib\kappa}\)
    \( \newcommand\Blambda{\mib\lambda}\)
    \( \newcommand\Bmu{\mib\mu}\)
    \( \newcommand\Bnu{\mib\nu}\)
    \( \newcommand\Bxi{\mib\xi}\)
    \( \newcommand\Bom{\mib\omicron}\)
    \( \newcommand\Bpi{\mib\pi}\)
    \( \newcommand\Bvarpi{\mib\varpi}\)
    \( \newcommand\Brho{\mib\rho}\)
    \( \newcommand\Bvrh{\mib\varrho}\)
    \( \newcommand\Bsigma{\mib\sigma}\)
    \( \newcommand\Bvsig{\mib\varsigma}\)
    \( \newcommand\Btau{\mib\tau}\)
    \( \newcommand\Bups{\mib\upsilon}\)
    \( \newcommand\Bphi{\mib\phi}\)
    \( \newcommand\Bvphi{\mib\vphi}\)
    \( \newcommand\Bchi{\mib\chi}\)
    \( \newcommand\Bpsi{\mib\psi}\)
    \( \newcommand\Bomega{\mib\omega}\)
    \( \newcommand\BGamma{\mib\Gamma}\)
    \( \newcommand\BDelta{\mib\Delta}\)
    \( \newcommand\BTheta{\mib\Theta}\)
    \( \newcommand\BLambda{\mib\Lambda}\)
    \( \newcommand\BXi{\mib\Xi}\)
    \( \newcommand\BPi{\mib\Pi}\)
    \( \newcommand\BSigma{\mib\Sigma}\)
    \( \newcommand\BUps{\mib\Upsilon}\)
    \( \newcommand\BPhi{\mib\Phi}\)
    \( \newcommand\BPsi{\mib\Psi}\)
    \( \newcommand\BOmega{\mib\Omega}\)
    \( \newcommand\Bxhi{\raise.35ex\hbox{$\Bchi$}}\)
    \( \newcommand\RGamma{ \Gamma}\)
    \( \newcommand\RDelta{ \Delta}\)
    \( \newcommand\RTheta{ \Theta}\)
    \( \newcommand\RLambda{ \Lambda}\)
    \( \newcommand\RXi{ \Xi}\)
    \( \newcommand\RPi{ \Pi}\)
    \( \newcommand\RSigma{ \Sigma}\)
    \( \newcommand\RUps{ \Upsilon}\)
    \( \newcommand\RPhi{ \Phi}\)
    \( \newcommand\RPsi{ \Psi}\)
    \( \newcommand\ROmega{ \Omega}\)
    \( \newcommand\RA{ A}\)
    \( \newcommand\RB{ B}\)
    \( \newcommand\RC{ C}\)
    \( \newcommand\RD{ D}\)
    \( \newcommand\RE{ E}\)
    \( \newcommand\RF{ F}\)
    \( \newcommand\RG{ G}\)
    \( \newcommand\RH{ H}\)
    \( \newcommand\RI{ I}\)
    \( \newcommand\RJ{ J}\)
    \( \newcommand\RK{ K}\)
    \( \newcommand\RL{ L}\)
    \( \newcommand { M}\)
    \( \newcommand\RN{ N}\)
    \( \newcommand\RO{ O}\)
    \( \newcommand\RP{ P}\)
    \( \newcommand\RQ{ Q}\)
    \( \newcommand\RR{ R}\)
    \( \newcommand\RS{ S}\)
    \( \newcommand\RT{ T}\)
    \( \newcommand\RU{ U}\)
    \( \newcommand\RV{ V}\)
    \( \newcommand\RW{ W}\)
    \( \newcommand\RX{ X}\)
    \( \newcommand\RY{ Y}\)
    \( \newcommand\RZ{ Z}\)
    \( \newcommand\Ra{ a}\)
    \( \newcommand\Rb{ b}\)
    \( \newcommand\Rc{ c}\)
    \( \newcommand\Rd{ d}\)
    \( \newcommand\Re{ e}\)
    \( \newcommand\Rf{ f}\)
    \( \newcommand\Rg{ g}\)
    \( \newcommand\Rh{ h}\)
    \( \newcommand\Ri{ i}\)
    \( \newcommand\Rj{ j}\)
    \( \newcommand\Rk{ k}\)
    \( \newcommand\Rl{ l}\)
    \( \newcommand { m}\)
    \( \newcommand\Rn{ n}\)
    \( \newcommand\Ro{ o}\)
    \( \newcommand\Rp{ p}\)
    \( \newcommand\Rq{ q}\)
    \( \newcommand\Rr{ r}\)
    \( \newcommand\Rs{ s}\)
    \( \newcommand\Rt{ t}\)
    \( \newcommand\Ru{ u}\)
    \( \newcommand\Rv{ v}\)
    \( \newcommand\Rw{ w}\)
    \( \newcommand\Rx{ x}\)
    \( \newcommand\Ry{ y}\)
    \( \newcommand\Rz{ z}\)
    \( \newcommand\BBA{\boldsymbol\RA}\)
    \( \newcommand\BBB{\boldsymbol\RB}\)
    \( \newcommand\BBC{\boldsymbol\RC}\)
    \( \newcommand\BBD{\boldsymbol\RD}\)
    \( \newcommand\BBE{\boldsymbol\RE}\)
    \( \newcommand\BBF{\boldsymbol\RF}\)
    \( \newcommand\BBG{\boldsymbol\RG}\)
    \( \newcommand\BBH{\boldsymbol\RH}\)
    \( \newcommand\BBI{\boldsymbol\RI}\)
    \( \newcommand\BBJ{\boldsymbol\RJ}\)
    \( \newcommand\BBK{\boldsymbol\RK}\)
    \( \newcommand\BBL{\boldsymbol\RL}\)
    \( \newcommand\BBM{\boldsymbol }\)
    \( \newcommand\BBN{\boldsymbol\RN}\)
    \( \newcommand\BBO{\boldsymbol\RO}\)
    \( \newcommand\BBP{\boldsymbol\RP}\)
    \( \newcommand\BBQ{\boldsymbol\RQ}\)
    \( \newcommand\BBR{\boldsymbol\RR}\)
    \( \newcommand\BBS{\boldsymbol\RS}\)
    \( \newcommand\BBT{\boldsymbol\RT}\)
    \( \newcommand\BBU{\boldsymbol\RU}\)
    \( \newcommand\BBV{\boldsymbol\RV}\)
    \( \newcommand\BBW{\boldsymbol\RW}\)
    \( \newcommand\BBX{\boldsymbol\RX}\)
    \( \newcommand\BBY{\boldsymbol\RY}\)
    \( \newcommand\BBZ{\boldsymbol\RZ}\)
    \( \newcommand\BBa{\boldsymbol\Ra}\)
    \( \newcommand\BBb{\boldsymbol\Rb}\)
    \( \newcommand\BBc{\boldsymbol\Rc}\)
    \( \newcommand\BBd{\boldsymbol\Rd}\)
    \( \newcommand\BBe{\boldsymbol\Re}\)
    \( \newcommand\BBf{\boldsymbol\Rf}\)
    \( \newcommand\BBg{\boldsymbol\Rg}\)
    \( \newcommand\BBh{\boldsymbol\Rh}\}\)
    \( \newcommand\BBi{\boldsymbol\Ri}\)
    \( \newcommand\BBj{\boldsymbol\Rj}\)
    \( \newcommand\BBk{\boldsymbol\Rk}\)
    \( \newcommand\BBl{boldsymbol\Rl}\)
    \( \newcommand\BBm{\boldsymbol }\)
    \( \newcommand\BBn{\boldsymbol\Rn}\)
    \( \newcommand\BBo{\boldsymbol\Ro}\)
    \( \newcommand\BBp{\boldsymbol\Rp}\)
    \( \newcommand\BBq{\boldsymbol\Rq}\)
    \( \newcommand\BBr{\boldsymbol\Rr}\)
    \( \newcommand\BBs{\boldsymbol\Rs}\)
    \( \newcommand\BBt{\boldsymbol\Rt}\)
    \( \newcommand\BBu{\boldsymbol\Ru}\)
    \( \newcommand\BBv{\boldsymbol\Rv}\)
    \( \newcommand\BBw{\boldsymbol\Rw}\)
    \( \newcommand\BBx{\boldsymbol\Rx}\)
    \( \newcommand\BBy{\boldsymbol\Ry}\)
    \( \newcommand\BBz{\boldsymbol\Rz}\)
    \( \newcommand\tcb{\textcolor{blue}\)
    \( \newcommand\tcr{\textcolor{red}\)
    \( \newcommand\bnabla{\boldsymbol{\nabla}}\)
    \( \newcommand\Bell{\boldsymbol\ell}\)
    \( \newcommand\dbar{\,{\mathchar'26\mkern-12mu d}} \)
    \( \newcommand\ns{^\vphantom{*}}\)
    \( \newcommand\uar{\uparrow}\)
    \( \newcommand\dar{\downarrow}\)
    \( \newcommand\impi{\int\limits_{-\infty}^{\infty}\!\!}\)
    \( \newcommand\izpi{\int\limits_{0}^{\infty}\!\!}\)
    \( \newcommand\etc{\it etc.\/}\)
    \( \newcommand\etal{\it et al.\/}\)
    \( \newcommand\opcit{\it op. cit.\/}\)
    \( \newcommand\ie{\it i.e.\/}\)
    \( \newcommand\Ie{\it I.e.\/}\)
    \( \newcommand\viz{\it viz.\/}\)
    \( \newcommand\eg{\it e.g.\/}\)
    \( \newcommand\Eg{\it E.g.\/}\)
    \( \newcommand\dbar{\,{\mathchar'26\mkern-12mu d}} \)
    \( \def\sss#1{\scriptscriptstyle #1}\)
    \( \def\ss#1{\scriptstyle #1}\)
    \( \def\ssr#1{\scriptstyle #1}\)
    \( \def\ssf#1{\scriptstyle #1}\)
    \( \newcommand\NA{N_{\ssr{\!A}}}\)
    \( \newcommand\lala{\langle\!\langle}\)
    \( \newcommand\rara{\rangle\!\rangle}\)
    \( \newcommand\blan{\big\langle}\)
    \( \newcommand\bran{\big\rangle}\)
    \( \newcommand\Blan{\Big\langle}\)
    \( \newcommand\Bran{\Big\rangle}\)
    \( \newcommand\intl{\int\limits}\)
    \( \newcommand\half{\frac{1}{2}}\)
    \( \newcommand\third{\frac{1}{3}}\)
    \( \newcommand\fourth{\frac{1}{4}}\)
    \( \newcommand\eighth{\frac{1}{8}}\)
    \( \newcommand\uar{\uparrow}\)
    \( \newcommand\dar{\downarrow}\)
    \( \newcommand\undertext#1{$\underline{\hbox{#1}}$}\)
    \( \newcommand\Tra{\mathop{\textsf{Tr}}\,}\)
    \( \newcommand\det{\mathop{\textsf{det}}\,}\)
    \( \def\tket#1{|  #1 \rangle}\)
    \( \def\tbra#1{\langle #1|}\)
    \( \def\tbraket#1#2{\langle #1  |   #2 \rangle}\)
    \( \def\texpect#1#2#3{\langle #1 |   #2  |  #3 \rangle}\)
    \( \def\sket#1{|  \, #1 \,  \rangle}\)
    \( \def\sbra#1{\langle \,  #1 \, |}\)
    \( \def\sbraket#1#2{\langle \, #1  \, |  \, #2 \,  \rangle}\)
    \( \def\sexpect#1#2#3{\langle \, #1 \, | \,  #2  \, | \, #3 \, \rangle}\)
    \(\def\ket#1{\big| \, #1\, \big\rangle}\)
    \( \def\bra#1{\big\langle \, #1 \, \big|}\)
    \( \def\braket#1#2{\big\langle \, #1\, \big| \,#2 \,\big\rangle}\)
    \( \def\expect#1#2#3{\big\langle\, #1\, \big|\, #2\, \big| \,#3\, \big\rangle}\)
    \( \newcommand\pz{\partial}\)
    \( \newcommand\pzb{\bar{\partial}}\)
    \( \newcommand\svph{\vphantom{\int}}\)
    \( \newcommand\vph{\vphantom{\sum_i}}\)
    \( \newcommand\bvph{\vphantom{\sum_N^N}}\)
    \( \newcommand\nd{^{\vphantom{\dagger}}}\)
    \( \newcommand\ns{^{\vphantom{*}}}\)
    \( \newcommand\yd{^\dagger}\)
    \( \newcommand\zb{\bar z}\)
    \( \newcommand\zdot{\dot z}\)
    \( \newcommand\zbdot{\dot{\bar z}}\)
    \( \newcommand\kB{k_{\sss{B}}}\)
    \( \newcommand\kT{k_{\sss{B}}T}\)
    \( \newcommand\gtau{g_\tau}\)
    \( \newcommand\Htil{\tilde H}\)
    \( \newcommand\pairo{(\phi\nd_0,J\nd_0)}\)
    \( \newcommand\pairm{(\phi\nd_0,J)}\)
    \( \newcommand\pairob{(\Bphi\nd_0,\BJ\nd_0)}\)
    \( \newcommand\pairmb{(\Bphi\nd_0,\BJ)}\)
    \( \newcommand\pair{(\phi,J)}\)
    \( \newcommand\Hz{H\nd_0}\)
    \( \newcommand\Ho{H\nd_1}\)
    \( \newcommand\Htz{\Htil\nd_0}\)
    \( \newcommand\Hto{\Htil\nd_1}\)
    \( \newcommand\oc{\omega_\Rc}\)

    \(\newcommand \gtwid{\approx}\)

    \( \newcommand\index{\textsf{ind}}\)
    \( \newcommand\csch{\,{ csch\,}}\)
    \( \newcommand\ctnh{\,{ ctnh\,}}\)
    \( \newcommand\ctn{\,{ ctn\,}}\)
    \( \newcommand\sgn{\,{ sgn\,}}\)
    \( \def\tmapright#1{\xrightarrow \limits^{#1}}\)
    \( \def\bmapright#1{\xrightarrow\limits_{#1}}\)
    \( \newcommand\hfb{\hfill\break}\)
    \( \newcommand\Rep{\textsf{Re}\,}\)
    \( \newcommand\Imp{\textsf{Im}\,}\)
    \( \newcommand\ncdot{\!\cdot\!}\)
    \( \def\tmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits^{#1}}\ }\)
    \( \def\bmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits_{#1}}\ }\)
    \( \newcommand\bsqcap{\mbox{\boldmath{$\sqcap$}}}\)

    \( \def\pabc#1#2#3{\left({\pz #1\over\pz #2}\right)\ns_{\!\!#3}}\)
    \( \def\spabc#1#2#3{\big({\pz #1\over\pz #2}\big)\ns_{\!#3}}\)
    \( \def\qabc#1#2#3{\pz^2\! #1\over\pz #2\,\pz #3}\)
    \( \def\rabc#1#2#3#4{(\pz #1,\pz #2)\over (\pz #3,\pz #4)}\)
    \( \newcommand\subA{\ns_\ssr{A}}\)
    \( \newcommand\subB{\ns_\ssr{B}}\)
    \( \newcommand\subC{\ns_\ssr{C}}\)
    \( \newcommand\subD{\ns_\ssr{D}}\)
    \( \newcommand\subAB{\ns_\ssr{AB}}\)
    \( \newcommand\subBC{\ns_\ssr{BC}}\)
    \( \newcommand\subCD{\ns_\ssr{CD}}\)
    \( \newcommand\subDA{\ns_\ssr{DA}}\)
    \( \def\lmapright#1{\ \ \smash{\mathop{\hbox to 55pt{\rightarrowfill}}\limits^{#1}}\ \ }\)
    \( \def\enth#1{\RDelta {\textsf H}^0_\Rf[{ #1}]}\)
    \( \newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{$\longrightharpoonup\joinrel$}\crcr  \lower1pt\hbox{$\longleftharpoondown\joinrel$}}}}}}\)
    \( \newcommand\longrightharpoonup{\relbar\joinrel\rightharpoonup}\)
    \( \newcommand\longleftharpoondown{\leftharpoondown\joinrel\relbar}\)
    \( \newcommand\cds{\,\bullet\,}\)
    \( \newcommand\ccs{\,\circ\,}\)
    \( \newcommand\nsub{_{\vphantom{\dagger}}}\)
    \( \newcommand\rhohat{\hat\rho}\)
    \( \newcommand\vrhhat{\hat\vrh}\)
    \( \newcommand\impi{\int\limits_{-\infty}^\infty\!\!\!}\)
    \( \newcommand\brangle{\big\rangle}\)
    \( \newcommand\blangle{\big\langle}\)
    \( \newcommand\vet{\tilde\ve}\)
    \( \newcommand\zbar{\bar z}\)
    \( \newcommand\ftil{\tilde f}\)
    \( \newcommand\XBE{\RXi\ns_\ssr{BE}}\)
    \( \newcommand\XFD{\RXi\ns_\ssr{FD}}\)
    \( \newcommand\OBE{\Omega\ns_\ssr{BE}}\)
    \( \newcommand\OFD{\Omega\ns_\ssr{FD}}\)
    \( \newcommand\veF{\ve\ns_\RF}\)
    \( \newcommand\kF{k\ns_\RF}\)
    \( \newcommand\kFu{k\ns_{\RF\uar}}\)
    \( \newcommand\SZ{\textsf Z}}\) \( \newcommand\kFd{k\ns_{\RF\dar}\)
    \( \newcommand\muB{\mu\ns_\ssr{B}}\)
    \( \newcommand\mutB{\tilde\mu}\ns_\ssr{B}\)
    \( \newcommand\xoN{\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N}\)
    \( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
    \( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
    \( \newcommand\xhihOZ
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/span[1], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\jhz{\HJ(0)}\)
    \( \newcommand\nda{\nd_\alpha}\)
    \( \newcommand\ndap{\nd_{\alpha'}}\)
    \( \newcommand\labar
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/span[2], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.05:_Diffusion_and_the_Lorentz_model), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\msa{m\ns_\ssr{A}}\)
    \( \newcommand\msb{m\ns_\ssr{B}}\)
    \( \newcommand\mss{m\ns_\Rs}\)
    \( \newcommand\HBx{\hat\Bx}\)
    \( \newcommand\HBy{\hat\By}\)
    \( \newcommand\HBz{\hat\Bz}\)
    \( \newcommand\thm{\theta\ns_m}\)
    \( \newcommand\thp{\theta\ns_\phi}\)
    \( \newcommand\mtil{\widetilde m}\)
    \( \newcommand\phitil{\widetilde\phi}\)
    \( \newcommand\delf{\delta\! f}\)
    \( \newcommand\coll{\bigg({\pz f\over\pz t}\bigg)\nd_{\! coll}}\)
    \( \newcommand\stre{\bigg({\pz f\over\pz t}\bigg)\nd_{\! str}}\)
    \( \newcommand\idrp{\int\!\!{d^3\!r\,d^3\!p\over h^3}\>}\)
    \( \newcommand\vbar{\bar v}\)
    \( \newcommand\BCE{\mbox{\boldmath{$\CE$}}\!}\)
    \( \newcommand\BCR{\mbox{\boldmath{$\CR$}}\!}\)
    \( \newcommand\gla{g\nd_{\RLambda\nd}}\)
    \( \newcommand\TA{T\ns_\ssr{A}}\)
    \( \newcommand\TB{T\ns_\ssr{B}}\)
    \( \newcommand\ncdot{\!\cdot\!}\)
    \( \newcommand\NS{N\ns_{\textsf S}}\)

    Failure of the relaxation time approximation

    As we remarked above, the relaxation time approximation fails to conserve any of the collisional invariants. It is therefore unsuitable for describing hydrodynamic phenomena such as diffusion. To see this, let \(f(\Br,\Bv,t)\) be the distribution function, here written in terms of position, velocity, and time rather than position, momentum, and time as befor7. In the absence of external forces, the Boltzmann equation in the relaxation time approximation is \[{\pz f\over\pz t} + \Bv\cdot {\pz f\over\pz\Br} = - {f-f^0\over\tau}\ .\] The density of particles in velocity space is given by \[{\tilde n}(\Bv,t)=\int\!\!d^3\!r\>f(\Br,\Bv,t)\ .\] In equilibrium, this is the Maxwell distribution times the total number of particles: \({\tilde n}\ns_0(\Bv)=N P\ns_\ssr{M}(\Bv)\). The number of particles as a function of time, \(N(t)=\int\!d^3\!v\,{\tilde n}(\Bv,t)\), should be a constant.

    Integrating the Boltzmann equation one has \[{\pz {\tilde n}\over\pz t} = - {{\tilde n}-{\tilde n}\ns_0\over\tau}\ .\] Thus, with \(\delta {\tilde n}(\Bv,t)={\tilde n}(\Bv,t)-{\tilde n}\ns_0(\Bv)\), we have \[\delta {\tilde n}(\Bv,t)=\delta {\tilde n}(\Bv,0)\,e^{-t/\tau}\ .\] Thus, \({\tilde n}(\Bv,t)\) decays exponentially to zero with time constant \(\tau\), from which it follows that the total particle number exponentially relaxes to \(N\ns_0\). This is physically incorrect; local density perturbations can’t just vanish. Rather, they diffuse.

    Modified Boltzmann equation and its solution

    To remedy this unphysical aspect, consider the modified Boltzmann equation, \[{\pz f\over\pz t} + \Bv\cdot {\pz f\over\pz\Br} = {1\over\tau}\bigg[- f + \int\!{d{\hat\Bv}\over 4\pi}\> f \bigg] \equiv {1\over\tau}\big(\,\MP\,-1\big) f\ , \label{Lormod}\] where \(\,\MP\,\) is a projector onto a space of isotropic functions of \(\Bv\): \(\,\MP\, F = \int\!{d{\hat\Bv}\over 4\pi}\, F(\Bv)\) for any function \(F(\Bv)\). Note that \(\,\MP\, F\) is a function of the speed \(v=|\Bv|\). For this modified equation, known as the Lorentz model, one finds \(\pz\ns_t{\tilde n}=0\).

    The model in Equation [Lormod] is known as the Lorentz model8. To solve it, we consider the Laplace transform, \[\Hf(\Bk,\Bv,s)=\izpi dt \> e^{-st} \!\int\!\!d^3\!r\>e^{-i\Bk\cdot\Br}\> f(\Br,\Bv,t)\ .\] Taking the Laplace transform of Equation [Lormod], we find \[\big(s+i\Bv\cdot\Bk + \tau^{-1}\big)\,\Hf(\Bk,\Bv,s) = \tau^{-1}\,\MP\, \Hf(\Bk,\Bv,s) + f(\Bk,\Bv,t=0)\ .\] We now solve for \(\,\MP\,\Hf(\Bk,\Bv,s)\): \[\Hf(\Bk,\Bv,s)={\tau^{-1}\over s + i\Bv\cdot\Bk + \tau^{-1}}\,\,\MP\,\Hf(\Bk,\Bv,s) + {f(\Bk,\Bv,t=0)\over s + i\Bv\cdot\Bk + \tau^{-1}}\ ,\] which entails \[\,\MP\,\Hf(\Bk,\Bv,s) = \left[ \int\!\!{d{\hat\Bv}\over 4\pi}\>{\tau^{-1}\over s + i\Bv\cdot\Bk + \tau^{-1}} \right] \,\MP\,\Hf(\Bk,\Bv,\Bs) + \int\!\!{d{\hat\Bv}\over 4\pi}\> {f(\Bk,\Bv,t=0)\over s + i\Bv\cdot\Bk + \tau^{-1}}\ .\] Now we have \[\begin{split} \int\!\!{d{\hat\Bv}\over 4\pi}\>{\tau^{-1}\over s + i\Bv\cdot\Bk + \tau^{-1}} &= \int\limits_{-1}^1\!\!dx\> {\tau^{-1}\over s + ivkx + \tau^{-1}} \\ &={1\over vk}\tan^{-1}\!\bigg({vk\tau\over 1 + \tau s}\bigg)\ . \end{split}\] Thus, \[\,\MP\, f(\Bk,\Bv,s)=\Bigg[ 1 - {1\over vk\tau}\tan^{-1}\!\bigg({vk\tau\over 1 + \tau s}\bigg) \Bigg]^{-1} \!\! \int\!\!{d{\hat\Bv}\over 4\pi}\> {f(\Bk,\Bv,t=0)\over s + i\Bv\cdot\Bk + \tau^{-1}}\ .\] We now have the solution to Lorentz’s modified Boltzmann equation: \[\begin{split} \Hf(\Bk,\Bv,s)&={\tau^{-1}\over s + i\Bv\cdot\Bk + \tau^{-1}} \Bigg[ 1 - {1\over vk\tau}\tan^{-1}\!\bigg({vk\tau\over 1 + \tau s}\bigg) \Bigg]^{-1} \!\! \int\!\!{d{\hat\Bv}\over 4\pi}\> {f(\Bk,\Bv,t=0)\over s + i\Bv\cdot\Bk + \tau^{-1}} \\ &\hskip1.0in + {f(\Bk,\Bv,t=0)\over s + i\Bv\cdot\Bk + \tau^{-1}} \ . \end{split}\]

    Let us assume an initial distribution which is perfectly localized in both \(\Br\) and \(\Bv\): \[f(\Br,\Bv,t=0)=\delta(\Bv-\Bv\ns_0)\ .\] For these initial conditions, we find \[\int\!\!{d{\hat\Bv}\over 4\pi}\> {f(\Bk,\Bv,t=0)\over s + i\Bv\cdot\Bk + \tau^{-1}} = {1\over s + i\Bv\ns_0\cdot\Bk + \tau^{-1}}\cdot{\delta(v-v\ns_0)\over 4\pi v_0^2}\ .\] We further have that \[1 - {1\over vk\tau}\tan^{-1}\!\bigg({vk\tau\over 1 + \tau s}\bigg) = s\tau + \third k^2 v^2 \tau^2 + \ldots\ ,\] and therefore \[\begin{split} \Hf(\Bk,\Bv,s)&={\tau^{-1}\over s + i\Bv\cdot\Bk + \tau^{-1}} \cdot {\tau^{-1}\over s + i\Bv\ns_0\cdot\Bk + \tau^{-1}}\cdot{1\over s + {1\over 3} v_0^2\, k^2 \, \tau + \ldots} \cdot{\delta(v-v\ns_0)\over 4\pi v_0^2}\\ &\hskip 1.0in + {\delta(\Bv-\Bv\ns_0)\over s+i\Bv\ns_0\cdot\Bk+\tau^{-1}}\ . \end{split}\] We are interested in the long time limit \(t\gg\tau\) for \(f(\Br,\Bv,t)\). This is dominated by \(s\sim t^{-1}\), and we assume that \(\tau^{-1}\) is dominant over \(s\) and \(i\Bv\cdot\Bk\). We then have \[\Hf(\Bk,\Bv,s)\approx {1\over s + {1\over 3} v_0^2\, k^2 \, \tau} \cdot{\delta(v-v\ns_0)\over 4\pi v_0^2}\ .\] Performing the inverse Laplace and Fourier transforms, we obtain \[f(\Br,\Bv,t)=(4\pi Dt)^{-3/2} \, e^{-r^2/4Dt}\cdot {\delta(v-v\ns_0)\over 4\pi v_0^2}\ ,\] where the diffusion constant is \[D=\third v_0^2\,\tau\ .\] The units are \([D]=L^2/T\). Integrating over velocities, we have the density \[n(\Br,t)=\int\!\!d^3\!v\>f(\Br,\Bv,t) = (4\pi Dt)^{-3/2}\,e^{-r^2/4Dt}\ .\] Note that \[\int\!\!d^3\!r\>n(\Br,t) = 1\] for all time. Total particle number is conserved!


    This page titled 8.5: Diffusion and the Lorentz model is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

    • Was this article helpful?