8.11: Appendix II- Distributions and Functionals

$\newcommand\bes{\begin{equation}\begin{split}}$
$\newcommand\ltwid{\propto}$
$\newcommand\ees{\end{split}\end{equation}}$
$\newcommand\mib{\mathbf}$
$\newcommand\Sa{\textsf a}$
$\newcommand\Sb{\textsf b}$
$\newcommand\Sc{\textsf c}$
$\newcommand\Sd{\textsf d}$
$\newcommand\Se{\textsf e}$
$\newcommand\Sf{\textsf f}$
$\newcommand\Sg{\textsf g}$
$\newcommand\Sh{\textsf h}$
$\newcommand\Si{\textsf i}$
$\newcommand\Sj{\textsf j}$
$\newcommand\Sk{\textsf k}$
$\newcommand\Sl{\textsf l}$
$\newcommand\Sm{\textsf m}$
$\newcommand\Sn{\textsf n}$
$\newcommand\So{\textsf o}$
$\newcommand\Sp{\textsf p}$
$\newcommand\Sq{\textsf q}$
$\newcommand\Sr{\textsf r}$
$\newcommand\Ss{\textsf s}$
$\newcommand\St{\textsf t}$
$\newcommand\Su{\textsf u}$
$\newcommand\Sv{\textsf v}$
$\newcommand\Sw{\textsf w}$
$\newcommand\Sx{\textsf x}$
$\newcommand\Sy{\textsf y}$
$\newcommand\Sz{\textsf z}$
$\newcommand\SA{\textsf A}$
$\newcommand\SB{\textsf B}$
$\newcommand\SC{\textsf C}$
$\newcommand\SD{\textsf D}$
$\newcommand\SE{\textsf E}$
$\newcommand\SF{\textsf F}$
$\newcommand\SG{\textsf G}$
$\newcommand\SH{\textsf H}$
$\newcommand\SI{\textsf I}$
$\newcommand\SJ{\textsf J}$
$\newcommand\SK{\textsf K}$
$\newcommand\SL{\textsf L}$
$\newcommand\SM{\textsf M}$
$\newcommand\SN{\textsf N}$
$\newcommand\SO{\textsf O}$
$\newcommand\SP{\textsf P}$
$\newcommand\SQ{\textsf Q}$
$\newcommand\SR{\textsf R}$
$\newcommand\SS{\textsf S}$
$\newcommand\ST{\textsf T}$
$\newcommand\SU{\textsf U}$
$\newcommand\SV{\textsf V}$
$\newcommand\SW{\textsf W}$
$\newcommand\SX{\textsf X}$
$\newcommand\SY{\textsf Y}$
$\newcommand\SZ{\textsf Z}$
$\newcommand\Ha{\hat a}$
$\newcommand\Hb{\hat b}$
$\newcommand\Hc{\hat c}$
$\newcommand\Hd{\hat d}$
$\newcommand\He{\hat e}$
$\newcommand\Hf{\hat f}$
$\newcommand\Hg{\hat g}$
$\newcommand\Hh{\hat h}$
$\newcommand\Hi{\hat \imath}$
$\newcommand\Hj{\hat \jmath}$
$\newcommand\Hk{\hat k}$
$\newcommand\Hl{\hat l}$
$\newcommand\Hm{\hat m}$
$\newcommand\Hn{\hat n}$
$\newcommand\Ho{\hat o}$
$\newcommand\Hp{\hat p}$
$\newcommand\Hq{\hat q}$
$\newcommand\Hr{\hat r}$
$\newcommand\Hs{\hat s}$
$\newcommand\Ht{\hat t}$
$\newcommand\Hu{\hat u}$
$\newcommand\Hv{\hat v}$
$\newcommand\Hw{\hat w}$
$\newcommand\Hx{\hat x}$
$\newcommand\Hy{\hat y}$
$\newcommand\Hz{\hat z}$
$\newcommand\HA{\hat A}$
$\newcommand\HB{\hat B}$
$\newcommand\HC{\hat C}$
$\newcommand\HD{\hat D}$
$\newcommand\HE{\hat E}$
$\newcommand\HF{\hat F}$
$\newcommand\HG{\hat G}$
$\newcommand\HH{\hat H}$
$\newcommand\HI{\hat I}$
$\newcommand\HJ{\hat J}$
$\newcommand\HK{\hat K}$
$\newcommand\HL{\hat L}$
$\newcommand\HM{\hat M}$
$\newcommand\HN{\hat N}$
$\newcommand\HO{\hat O}$
$\newcommand\HP{\hat P}$
$\newcommand\HQ{\hat Q}$
$\newcommand\HR{\hat R}$
$\newcommand\HS{\hat S}$
$\newcommand\HT{\hat T}$
$\newcommand\HU{\hat U}$
$\newcommand\HV{\hat V}$
$\newcommand\HW{\hat W}$
$\newcommand\HX{\hat X}$
$\newcommand\HY{\hat Y}$
$\newcommand\HZ{\hat Z}$
$\newcommand\Halpha{\hat\alpha}$
$\newcommand\Hbeta{\hat\beta}$
$\newcommand\Hgamma{\hat\gamma}$
$\newcommand\Hdelta{\hat\delta}$
$\newcommand\Hepsilon{\hat\epsilon}$
$\newcommand\Hvarepsilon{\hat\varepsilon}$
$\newcommand\Hzeta{\hat\zeta}$
$\newcommand\Heta{\hat\eta}$
$\newcommand\Htheta{\hat\theta}$
$\newcommand\Hvartheta{\hat\vartheta}$
$\newcommand\Hiota{\hat\iota}$
$\newcommand\Hkappa{\hat\kappa}$
$\newcommand\Hlambda{\hat\lambda}$
$\newcommand\Hmu{\hat\mu}$
$\newcommand\Hnu{\hat\nu}$
$\newcommand\Hxi{\hat\xi}$
$\newcommand\Hom{\hat\omicron}$
$\newcommand\Hpi{\hat\pi}$
$\newcommand\Hvarpi{\hat\varpi}$
$\newcommand\Hrho{\hat\rho}$
$\newcommand\Hvarrho{\hat\varrho}$
$\newcommand\Hsigma{\hat\sigma}$
$\newcommand\Hvarsigma{\hat\varsigma}$
$\newcommand\Htau{\var\tau}$
$\newcommand\Hupsilon{\hat\upsilon}$
$\newcommand\Hphi{\hat\phi}$
$\newcommand\Hvarphi{\hat\varphi}$
$\newcommand\Hchi{\hat\chi}$
$\newcommand\Hxhi{\hat\xhi}$
$\newcommand\Hpsi{\hat\psi}$
$\newcommand\Homega{\hat\omega}$
$\newcommand\HGamma{\hat\Gamma}$
$\newcommand\HDelta{\hat\Delta}$
$\newcommand\HTheta{\hat\Theta}$
$\newcommand\HLambda{\hat\Lambda}$
$\newcommand\HXi{\hat\Xi}$
$\newcommand\HPi{\hat\Pi}$
$\newcommand\HSigma{\hat\Sigma}$
$\newcommand\HUps{\hat\Upsilon}$
$\newcommand\HPhi{\hat\Phi}$
$\newcommand\HPsi{\hat\Psi}$
$\newcommand\HOmega{\hat\Omega}$
$\newcommand\xhat{\hat\Bx}$
$\newcommand\yhat{\hat\By}$
$\newcommand\zhat{\hat\Bz}$
$\newcommand\ehat{\hat\Be}$
$\newcommand\khat{\hat\Bk}$
$\newcommand\nhat{\hat\Bn}$
$\newcommand\rhat{\hat\Br}$
$\newcommand\phihat{\hat\Bphi}$
$\newcommand\thetahat{\hat\Btheta}$
$\newcommand\MA{\mathbb A}$
$\newcommand\MB{\mathbb B}$
$\newcommand\MC{\mathbb C}$
$\newcommand\MD{\mathbb D}$
$\newcommand\ME{\mathbb E}$
$\newcommand\MF{\mathbb F}$
$\newcommand\MG{\mathbb G}$
$\newcommand\MH{\mathbb H}$
$\newcommand\MI{\mathbb I}$
$\newcommand\MJ{\mathbb J}$
$\newcommand\MK{\mathbb K}$
$\newcommand\ML{\mathbb L}$
$\newcommand\MM{\mathbb M}$
$\newcommand\MN{\mathbb N}$
$\newcommand\MO{\mathbb O}$
$\newcommand\MP{\mathbb P}$
$\newcommand\MQ{\mathbb Q}$
$\newcommand\MR{\mathbb R}$
$\newcommand\MS{\mathbb S}$
$\newcommand\MT{\mathbb T}$
$\newcommand\MU{\mathbb U}$
$\newcommand\MV{\mathbb V}$
$\newcommand\MW{\mathbb W}$
$\newcommand\MX{\mathbb X}$
$\newcommand\MY{\mathbb Y}$
$\newcommand\MZ{\mathbb Z}$
$\newcommand\CA{\mathcal A}$
$\newcommand\CB{\mathcal B}$
$\newcommand\CC{\mathcal C}$
$\newcommand\CD{\mathcal D}$
$\newcommand\CE{\mathcal E}$
$\newcommand\CF{\mathcal F}$
$\newcommand\CG{\mathcal G}$
$\newcommand\CH{\mathcal H}$
$\newcommand\CI{\mathcal I}$
$\newcommand\CJ{\mathcal J}$
$\newcommand\CK{\mathcal K}$
$\newcommand\CL{\mathcal L}$
$\newcommand\CM{\mathcal M}$
$\newcommand\CN{\mathcal N}$
$\newcommand\CO{\mathcal O}$
$\newcommand\CP{\mathcal P}$
$\newcommand\CQ{\mathcal Q}$
$\newcommand\CR{\mathcal R}$
$\newcommand\CS{\mathcal S}$
$\newcommand\CT{\mathcal T}$
$\newcommand\CU{\mathcal U}$
$\newcommand\CV{\mathcal V}$
$\newcommand\CW{\mathcal W}$
$\newcommand\CX{\mathcal X}$
$\newcommand\CY{\mathcal Y}$
$\newcommand\CZ{\mathcal Z}$
$\newcommand\Fa{\mathfrak a}$
$\newcommand\Fb{\mathfrak b}$
$\newcommand\Fc{\mathfrak c}$
$\newcommand\Fd{\mathfrak d}$
$\newcommand\Fe{\mathfrak e}$
$\newcommand\Ff{\mathfrak f}$
$\newcommand\Fg{\mathfrak g}$
$\newcommand\Fh{\mathfrak h}$
$\newcommand\Fi{\mathfrak i}$
$\newcommand\Fj{\mathfrak j}$
$\newcommand\Fk{\mathfrak k}$
$\newcommand\Fl{\mathfrak l}$
$\newcommand\Fm{\mathfrak m}$
$\newcommand\Fn{\mathfrak n}$
$\newcommand\Fo{\mathfrak o}$
$\newcommand\Fp{\mathfrak p}$
$\newcommand\Fq{\mathfrak q}$
$\newcommand\Fr{\mathfrak r}$
$\newcommand\Fs{\mathfrak s}$
$\newcommand\Ft{\mathfrak t}$
$\newcommand\Fu{\mathfrak u}$
$\newcommand\Fv{\mathfrak v}$
$\newcommand\Fw{\mathfrak w}$
$\newcommand\Fx{\mathfrak x}$
$\newcommand\Fy{\mathfrak y}$
$\newcommand\Fz{\mathfrak z}$
$\newcommand\FA{\mathfrak A}$
$\newcommand\FB{\mathfrak B}$
$\newcommand\FC{\mathfrak C}$
$\newcommand\FD{\mathfrak D}$
$\newcommand\FE{\mathfrak E}$
$\newcommand\FF{\mathfrak F}$
$\newcommand\FG{\mathfrak G}$
$\newcommand\FH{\mathfrak H}$
$\newcommand\FI{\mathfrak I}$
$\newcommand\FJ{\mathfrak J}$
$\newcommand\FK{\mathfrak K}$
$\newcommand\FL{\mathfrak L}$
$\newcommand\FM{\mathfrak M}$
$\newcommand\FN{\mathfrak N}$
$\newcommand\FO{\mathfrak O}$
$\newcommand\FP{\mathfrak P}$
$\newcommand\FQ{\mathfrak Q}$
$\newcommand\FR{\mathfrak R}$
$\newcommand\FS{\mathfrak S}$
$\newcommand\FT{\mathfrak T}$
$\newcommand\FU{\mathfrak U}$
$\newcommand\FV{\mathfrak V}$
$\newcommand\FW{\mathfrak W}$
$\newcommand\FX{\mathfrak X}$
$\newcommand\FY{\mathfrak Y}$
$\newcommand\FZ{\mathfrak Z}$
$\newcommand\Da{\dot a}$
$\newcommand\Db{\dot b}$
$\newcommand\Dc{\dot c}$
$\newcommand\Dd{\dot d}$
$\newcommand\De{\dot e}$
$\newcommand\Df{\dot f}$
$\newcommand\Dg{\dot g}$
$\newcommand\Dh{\dot h}$
$\newcommand\Di{\dot \imath}$
$\newcommand\Dj{\dot \jmath}$
$\newcommand\Dk{\dot k}$
$\newcommand\Dl{\dot l}$
$\newcommand\Dm{\dot m}$
$\newcommand\Dn{\dot n}$
$\newcommand\Do{\dot o}$
$\newcommand\Dp{\dot p}$
$\newcommand\Dq{\dot q}$
$\newcommand\Dr{\dot r}$
$\newcommand\Ds{\dot s}$
$\newcommand\Dt{\dot t}$
$\newcommand\Du{\dot u}$
$\newcommand\Dv{\dot v}$
$\newcommand\Dw{\dot w}$
$\newcommand\Dx{\dot x}$
$\newcommand\Dy{\dot y}$
$\newcommand\Dz{\dot z}$
$\newcommand\DA{\dot A}$
$\newcommand\DB{\dot B}$
$\newcommand\DC{\dot C}$
$\newcommand\DD{\dot D}$
$\newcommand\DE{\dot E}$
$\newcommand\DF{\dot F}$
$\newcommand\DG{\dot G}$
$\newcommand\DH{\dot H}$
$\newcommand\DI{\dot I}$
$\newcommand\DJ{\dot J}$
$\newcommand\DK{\dot K}$
$\newcommand\DL{\dot L}$
$\newcommand\DM{\dot M}$
$\newcommand\DN{\dot N}$
$\newcommand\DO{\dot O}$
$\newcommand\DP{\dot P}$
$\newcommand\DQ{\dot Q}$
$\newcommand\DR{\dot R}$
$\newcommand\DS{\dot S}$
$\newcommand\DT{\dot T}$
$\newcommand\DU{\dot U}$
$\newcommand\DV{\dot V}$
$\newcommand\DW{\dot W}$
$\newcommand\DX{\dot X}$
$\newcommand\DY{\dot Y}$
$\newcommand\DZ{\dot Z}$
\( \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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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/8.11:_Appendix_II-_Distributions_and_Functionals), /content/body/p[1]/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$}}\!}$
$\newcommand\gla{g\nd_{\RLambda\nd}}$
$\newcommand\TA{T\ns_\ssr{A}}$
$\newcommand\TB{T\ns_\ssr{B}}$
$\newcommand\ncdot{\!\cdot\!}$
$\newcommand\NS{N\ns_{\textsf S}}$

Let $x\in{\mathbb R}$ be a random variable, and $P(x)$ a probability distribution for $x$. The average of any function $\phi(x)$ is then $$\blangle\phi(x)\brangle = \int\limits_{-\infty}^\infty\!\!dx\>P(x)\,\phi(x)\Bigg/\!\!\int\limits_{-\infty}^\infty\!\!dx\>P(x)\ .$$

Let $\eta(t)$ be a random function of $t$, with $\eta(t)\in{\mathbb R}$, and let $P\big[\eta(t)\big]$ be the probability distribution functional for $\eta(t)$. Then if $\Phi\big[\eta(t)\big]$ is a functional of $\eta(t)$, the average of $\Phi$ is given by $$\int\!\!D\eta\>P\big[\eta(t)\big]\>\Phi\big[\eta(t)\big]\Bigg/\!\!\int\!\!D\eta\,P\big[\eta(t)\big]$$ The expression $\int\!D\eta\>P[\eta]\,\Phi[\eta]$ is a functional integral. A functional integral is a continuum limit of a multivariable integral. Suppose $\eta(t)$ were defined on a set of $t$ values $t\ns_n=n\tau$. A functional of $\eta(t)$ becomes a multivariable function of the values $\eta\ns_n\equiv \eta(t\ns_n)$. The metric then becomes $$D\eta\longrightarrow \prod_n d\eta\ns_n\ .$$

In fact, for our purposes we will not need to know any details about the functional measure $D\eta$; we will finesse this delicate issue¹⁸. Consider the generating functional, $$Z\big[J(t)\big]=\int\!\!D\eta\>P[\eta]\,\exp\Bigg(\int\limits_{-\infty}^\infty\!\!\!dt\>J(t)\,\eta(t)\Bigg)\ .$$ It is clear that $${1\over Z[J]}\,{\delta^n\!Z[J]\over\delta J(t\ns_1)\cdots\delta J(t\ns_n)}\Bigg|\nd_{J(t)=0}= \blangle \eta(t\ns_1)\cdots\eta(t\ns_n)\brangle\ .$$ The function $J(t)$ is an arbitrary source function. We differentiate with respect to it in order to find the $\eta$-field correlators.

[Fdiscretize] Discretization of a continuous function $\eta(t)$. Upon discretization, a functional $\Phi\big[\eta(t)\big]$ becomes an ordinary multivariable function $\Phi(\{\eta_j\})$.

Let’s compute the generating function for a class of distributions of the Gaussian form, $$\begin{aligned} P[\eta]&=\exp\!\Bigg(\!-{1\over 2\Gamma}\!\!\int\limits_{-\infty}^\infty\!\!\!dt\> \big(\tau^2\,{\dot\eta}^2 + \eta^2\big)\Bigg)\\ &=\exp\!\Bigg(\!-{1\over 2\Gamma}\!\!\int\limits_{-\infty}^\infty\!\!{d\omega\over 2\pi}\> \big( 1 + \omega^2\tau^2\big)\, \big|{\hat\eta}(\omega)\big|^2\Bigg)\ .\end{aligned}$$ Then Fourier transforming the source function $J(t)$, it is easy to see that $$Z[J]=Z[0]\cdot\exp\!\Bigg({\Gamma\over 2}\!\!\int\limits_{-\infty}^\infty\!\!{d\omega\over 2\pi}\> { \big|{\hat J}(\omega)\big|^2\over 1+\omega^2\tau^2}\Bigg)\ .$$ Note that with $\eta(t)\in {\mathbb R}$ and $J(t)\in {\mathbb R}$ we have $\eta^*(\omega)=\eta(-\omega)$ and $J^*(\omega)=J(-\omega)$. Transforming back to real time, we have $$Z[J]=Z[0]\cdot\exp\Bigg({1\over 2}\!\int\limits_{-\infty}^\infty\!\!\!dt\!\!\int\limits_{-\infty}^\infty\!\!\!dt'\> J(t)\,G(t-t')\,J(t')\Bigg)\ ,$$ where $$G(s)={\Gamma\over 2\tau}\,e^{-|s|/\tau} \qquad,\qquad {\widehat G}(\omega)={\Gamma\over 1+\omega^2\tau^2}$$ is the Green’s function, in real and Fourier space. Note that $$\int\limits_{-\infty}^\infty\!\!\!ds\>G(s)={\widehat G}(0)=\Gamma\ .$$ We can now compute $$\begin{aligned} \blangle \eta(t\ns_1)\,\eta(t\ns_2)\brangle&=G(t\ns_1-t\ns_2)\bvph\\ \blangle \eta(t\ns_1)\,\eta(t\ns_2)\,\eta(t\ns_3)\,\eta(t\ns_4)\brangle&=G(t\ns_1-t\ns_2)\,G(t\ns_3-t\ns_4)+ G(t\ns_1-t\ns_3)\,G(t\ns_2-t\ns_4)\\ &\qquad\qquad + G(t\ns_1-t\ns_4)\,G(t\ns_2-t\ns_3)\ .\nonumber\end{aligned}$$ The generalization is now easy to prove, and is known as Wick’s theorem: $$\blangle \eta(t\ns_1)\cdots\eta(t\ns_{2n})\brangle=\sum_{contractions}\!\!\! G(t\ns_{i\ns_1}-t\ns_{i\ns_2})\cdots G(t\ns_{i\ns_{2n-1}}-t\ns_{i\ns_{2n}})\ ,$$ where the sum is over all distinct contractions of the sequence $1$-$2\cdots 2n$ into products of pairs. How many terms are there? Some simple combinatorics answers this question. Choose the index $1$. There are $(2n-1)$ other time indices with which it can be contracted. Now choose another index. There are $(2n-3)$ indices with which that index can be contracted. And so on. We thus obtain $$C(n)\equiv{\hbox{\# of contractions}\atop \hbox{of 1-2-3\,$\cdots$$2n$}} = (2n-1)(2n-3)\cdots 3\cdot 1 = {(2n)!\over 2^n\,n!}\ .$$