Skip to main content
\(\require{cancel}\)
Physics LibreTexts

4.4: Ordinary Canonical Ensemble (OCE)

  • Page ID
    18565
  • \( \newcommand\bes{\begin{equation}\begin{split}}\)
    \( \newcommand\ltwid{\propto}\)
    \( \newcommand\ees{\end{split}\end{equation}}\)
    \( \newcommand\mib{\mathbf}\)
    \( \newcommand\Sa{\textsf a}\)
    \( \newcommand\Sb{\textsf b}\)
    \( \newcommand\Sc{\textsf c}\)
    \( \newcommand\Sd{\textsf d}\)
    \( \newcommand\Se{\textsf e}\)
    \( \newcommand\Sf{\textsf f}\)
    \( \newcommand\Sg{\textsf g}\)
    \( \newcommand\Sh{\textsf h}\)
    \( \newcommand\Si{\textsf i}\)
    \( \newcommand\Sj{\textsf j}\)
    \( \newcommand\Sk{\textsf k}\)
    \( \newcommand\Sl{\textsf l}\)
    \( \newcommand\Sm{\textsf m}\)
    \( \newcommand\Sn{\textsf n}\)
    \( \newcommand\So{\textsf o}\)
    \( \newcommand\Sp{\textsf p}\)
    \( \newcommand\Sq{\textsf q}\)
    \( \newcommand\Sr{\textsf r}\)
    \( \newcommand\Ss{\textsf s}\)
    \( \newcommand\St{\textsf t}\)
    \( \newcommand\Su{\textsf u}\)
    \( \newcommand\Sv{\textsf v}\)
    \( \newcommand\Sw{\textsf w}\)
    \( \newcommand\Sx{\textsf x}\)
    \( \newcommand\Sy{\textsf y}\)
    \( \newcommand\Sz{\textsf z}\)
    \( \newcommand\SA{\textsf A}\)
    \( \newcommand\SB{\textsf B}\)
    \( \newcommand\SC{\textsf C}\)
    \( \newcommand\SD{\textsf D}\)
    \( \newcommand\SE{\textsf E}\)
    \( \newcommand\SF{\textsf F}\)
    \( \newcommand\SG{\textsf G}\)
    \( \newcommand\SH{\textsf H}\)
    \( \newcommand\SI{\textsf I}\)
    \( \newcommand\SJ{\textsf J}\)
    \( \newcommand\SK{\textsf K}\)
    \( \newcommand\SL{\textsf L}\)
    \( \newcommand\SM{\textsf M}\)
    \( \newcommand\SN{\textsf N}\)
    \( \newcommand\SO{\textsf O}\)
    \( \newcommand\SP{\textsf P}\)
    \( \newcommand\SQ{\textsf Q}\)
    \( \newcommand\SR{\textsf R}\)
    \( \newcommand\SS{\textsf S}\)
    \( \newcommand\ST{\textsf T}\)
    \( \newcommand\SU{\textsf U}\)
    \( \newcommand\SV{\textsf V}\)
    \( \newcommand\SW{\textsf W}\)
    \( \newcommand\SX{\textsf X}\)
    \( \newcommand\SY{\textsf Y}\)
    \( \newcommand\SZ{\textsf Z}\)
    \( \newcommand\Ha{\hat a}\)
    \( \newcommand\Hb{\hat b}\)
    \( \newcommand\Hc{\hat c}\)
    \( \newcommand\Hd{\hat d}\)
    \( \newcommand\He{\hat e}\)
    \( \newcommand\Hf{\hat f}\)
    \( \newcommand\Hg{\hat g}\)
    \( \newcommand\Hh{\hat h}\)
    \( \newcommand\Hi{\hat \imath}\)
    \( \newcommand\Hj{\hat \jmath}\)
    \( \newcommand\Hk{\hat k}\)
    \( \newcommand\Hl{\hat l}\)
    \( \newcommand\Hm{\hat m}\)
    \( \newcommand\Hn{\hat n}\)
    \( \newcommand\Ho{\hat o}\)
    \( \newcommand\Hp{\hat p}\)
    \( \newcommand\Hq{\hat q}\)
    \( \newcommand\Hr{\hat r}\)
    \( \newcommand\Hs{\hat s}\)
    \( \newcommand\Ht{\hat t}\)
    \( \newcommand\Hu{\hat u}\)
    \( \newcommand\Hv{\hat v}\)
    \( \newcommand\Hw{\hat w}\)
    \( \newcommand\Hx{\hat x}\)
    \( \newcommand\Hy{\hat y}\)
    \( \newcommand\Hz{\hat z}\)
    \( \newcommand\HA{\hat A}\)
    \( \newcommand\HB{\hat B}\)
    \( \newcommand\HC{\hat C}\)
    \( \newcommand\HD{\hat D}\)
    \( \newcommand\HE{\hat E}\)
    \( \newcommand\HF{\hat F}\)
    \( \newcommand\HG{\hat G}\)
    \( \newcommand\HH{\hat H}\)
    \( \newcommand\HI{\hat I}\)
    \( \newcommand\HJ{\hat J}\)
    \( \newcommand\HK{\hat K}\)
    \( \newcommand\HL{\hat L}\)
    \( \newcommand\HM{\hat M}\)
    \( \newcommand\HN{\hat N}\)
    \( \newcommand\HO{\hat O}\)
    \( \newcommand\HP{\hat P}\)
    \( \newcommand\HQ{\hat Q}\)
    \( \newcommand\HR{\hat R}\)
    \( \newcommand\HS{\hat S}\)
    \( \newcommand\HT{\hat T}\)
    \( \newcommand\HU{\hat U}\)
    \( \newcommand\HV{\hat V}\)
    \( \newcommand\HW{\hat W}\)
    \( \newcommand\HX{\hat X}\)
    \( \newcommand\HY{\hat Y}\)
    \( \newcommand\HZ{\hat Z}\)
    \( \newcommand\Halpha{\hat\alpha}\)
    \( \newcommand\Hbeta{\hat\beta}\)
    \( \newcommand\Hgamma{\hat\gamma}\)
    \( \newcommand\Hdelta{\hat\delta}\)
    \( \newcommand\Hepsilon{\hat\epsilon}\)
    \( \newcommand\Hvarepsilon{\hat\varepsilon}\)
    \( \newcommand\Hzeta{\hat\zeta}\)
    \( \newcommand\Heta{\hat\eta}\)
    \( \newcommand\Htheta{\hat\theta}\)
    \( \newcommand\Hvartheta{\hat\vartheta}\)
    \( \newcommand\Hiota{\hat\iota}\)
    \( \newcommand\Hkappa{\hat\kappa}\)
    \( \newcommand\Hlambda{\hat\lambda}\)
    \( \newcommand\Hmu{\hat\mu}\)
    \( \newcommand\Hnu{\hat\nu}\)
    \( \newcommand\Hxi{\hat\xi}\)
    \( \newcommand\Hom{\hat\omicron}\)
    \( \newcommand\Hpi{\hat\pi}\)
    \( \newcommand\Hvarpi{\hat\varpi}\)
    \( \newcommand\Hrho{\hat\rho}\)
    \( \newcommand\Hvarrho{\hat\varrho}\)
    \( \newcommand\Hsigma{\hat\sigma}\)
    \( \newcommand\Hvarsigma{\hat\varsigma}\)
    \( \newcommand\Htau{\var\tau}\)
    \( \newcommand\Hupsilon{\hat\upsilon}\)
    \( \newcommand\Hphi{\hat\phi}\)
    \( \newcommand\Hvarphi{\hat\varphi}\)
    \( \newcommand\Hchi{\hat\chi}\)
    \( \newcommand\Hxhi{\hat\xhi}\)
    \( \newcommand\Hpsi{\hat\psi}\)
    \( \newcommand\Homega{\hat\omega}\)
    \( \newcommand\HGamma{\hat\Gamma}\)
    \( \newcommand\HDelta{\hat\Delta}\)
    \( \newcommand\HTheta{\hat\Theta}\)
    \( \newcommand\HLambda{\hat\Lambda}\)
    \( \newcommand\HXi{\hat\Xi}\)
    \( \newcommand\HPi{\hat\Pi}\)
    \( \newcommand\HSigma{\hat\Sigma}\)
    \( \newcommand\HUps{\hat\Upsilon}\)
    \( \newcommand\HPhi{\hat\Phi}\)
    \( \newcommand\HPsi{\hat\Psi}\)
    \( \newcommand\HOmega{\hat\Omega}\)
    \( \newcommand\xhat{\hat\Bx}\)
    \( \newcommand\yhat{\hat\By}\)
    \( \newcommand\zhat{\hat\Bz}\)
    \( \newcommand\ehat{\hat\Be}\)
    \( \newcommand\khat{\hat\Bk}\)
    \( \newcommand\nhat{\hat\Bn}\)
    \( \newcommand\rhat{\hat\Br}\)
    \( \newcommand\phihat{\hat\Bphi}\)
    \( \newcommand\thetahat{\hat\Btheta}\)
    \( \newcommand\MA{\mathbb A}\)
    \( \newcommand\MB{\mathbb B}\)
    \( \newcommand\MC{\mathbb C}\)
    \( \newcommand\MD{\mathbb D}\)
    \( \newcommand\ME{\mathbb E}\)
    \( \newcommand\MF{\mathbb F}\)
    \( \newcommand\MG{\mathbb G}\)
    \( \newcommand\MH{\mathbb H}\)
    \( \newcommand\MI{\mathbb I}\)
    \( \newcommand\MJ{\mathbb J}\)
    \( \newcommand\MK{\mathbb K}\)
    \( \newcommand\ML{\mathbb L}\)
    \( \newcommand\MM{\mathbb M}\)
    \( \newcommand\MN{\mathbb N}\)
    \( \newcommand\MO{\mathbb O}\)
    \( \newcommand\MP{\mathbb P}\)
    \( \newcommand\MQ{\mathbb Q}\)
    \( \newcommand\MR{\mathbb R}\)
    \( \newcommand\MS{\mathbb S}\)
    \( \newcommand\MT{\mathbb T}\)
    \( \newcommand\MU{\mathbb U}\)
    \( \newcommand\MV{\mathbb V}\)
    \( \newcommand\MW{\mathbb W}\)
    \( \newcommand\MX{\mathbb X}\)
    \( \newcommand\MY{\mathbb Y}\)
    \( \newcommand\MZ{\mathbb Z}\)
    \( \newcommand\CA{\mathcal A}\)
    \( \newcommand\CB{\mathcal B}\)
    \( \newcommand\CC{\mathcal C}\)
    \( \newcommand\CD{\mathcal D}\)
    \( \newcommand\CE{\mathcal E}\)
    \( \newcommand\CF{\mathcal F}\)
    \( \newcommand\CG{\mathcal G}\)
    \( \newcommand\CH{\mathcal H}\)
    \( \newcommand\CI{\mathcal I}\)
    \( \newcommand\CJ{\mathcal J}\)
    \( \newcommand\CK{\mathcal K}\)
    \( \newcommand\CL{\mathcal L}\)
    \( \newcommand\CM{\mathcal M}\)
    \( \newcommand\CN{\mathcal N}\)
    \( \newcommand\CO{\mathcal O}\)
    \( \newcommand\CP{\mathcal P}\)
    \( \newcommand\CQ{\mathcal Q}\)
    \( \newcommand\CR{\mathcal R}\)
    \( \newcommand\CS{\mathcal S}\)
    \( \newcommand\CT{\mathcal T}\)
    \( \newcommand\CU{\mathcal U}\)
    \( \newcommand\CV{\mathcal V}\)
    \( \newcommand\CW{\mathcal W}\)
    \( \newcommand\CX{\mathcal X}\)
    \( \newcommand\CY{\mathcal Y}\)
    \( \newcommand\CZ{\mathcal Z}\)
    \( \newcommand\Fa{\mathfrak a}\)
    \( \newcommand\Fb{\mathfrak b}\)
    \( \newcommand\Fc{\mathfrak c}\)
    \( \newcommand\Fd{\mathfrak d}\)
    \( \newcommand\Fe{\mathfrak e}\)
    \( \newcommand\Ff{\mathfrak f}\)
    \( \newcommand\Fg{\mathfrak g}\)
    \( \newcommand\Fh{\mathfrak h}\)
    \( \newcommand\Fi{\mathfrak i}\)
    \( \newcommand\Fj{\mathfrak j}\)
    \( \newcommand\Fk{\mathfrak k}\)
    \( \newcommand\Fl{\mathfrak l}\)
    \( \newcommand\Fm{\mathfrak m}\)
    \( \newcommand\Fn{\mathfrak n}\)
    \( \newcommand\Fo{\mathfrak o}\)
    \( \newcommand\Fp{\mathfrak p}\)
    \( \newcommand\Fq{\mathfrak q}\)
    \( \newcommand\Fr{\mathfrak r}\)
    \( \newcommand\Fs{\mathfrak s}\)
    \( \newcommand\Ft{\mathfrak t}\)
    \( \newcommand\Fu{\mathfrak u}\)
    \( \newcommand\Fv{\mathfrak v}\)
    \( \newcommand\Fw{\mathfrak w}\)
    \( \newcommand\Fx{\mathfrak x}\)
    \( \newcommand\Fy{\mathfrak y}\)
    \( \newcommand\Fz{\mathfrak z}\)
    \( \newcommand\FA{\mathfrak A}\)
    \( \newcommand\FB{\mathfrak B}\)
    \( \newcommand\FC{\mathfrak C}\)
    \( \newcommand\FD{\mathfrak D}\)
    \( \newcommand\FE{\mathfrak E}\)
    \( \newcommand\FF{\mathfrak F}\)
    \( \newcommand\FG{\mathfrak G}\)
    \( \newcommand\FH{\mathfrak H}\)
    \( \newcommand\FI{\mathfrak I}\)
    \( \newcommand\FJ{\mathfrak J}\)
    \( \newcommand\FK{\mathfrak K}\)
    \( \newcommand\FL{\mathfrak L}\)
    \( \newcommand\FM{\mathfrak M}\)
    \( \newcommand\FN{\mathfrak N}\)
    \( \newcommand\FO{\mathfrak O}\)
    \( \newcommand\FP{\mathfrak P}\)
    \( \newcommand\FQ{\mathfrak Q}\)
    \( \newcommand\FR{\mathfrak R}\)
    \( \newcommand\FS{\mathfrak S}\)
    \( \newcommand\FT{\mathfrak T}\)
    \( \newcommand\FU{\mathfrak U}\)
    \( \newcommand\FV{\mathfrak V}\)
    \( \newcommand\FW{\mathfrak W}\)
    \( \newcommand\FX{\mathfrak X}\)
    \( \newcommand\FY{\mathfrak Y}\)
    \( \newcommand\FZ{\mathfrak Z}\)
    \( \newcommand\Da{\dot a}\)
    \( \newcommand\Db{\dot b}\)
    \( \newcommand\Dc{\dot c}\)
    \( \newcommand\Dd{\dot d}\)
    \( \newcommand\De{\dot e}\)
    \( \newcommand\Df{\dot f}\)
    \( \newcommand\Dg{\dot g}\)
    \( \newcommand\Dh{\dot h}\)
    \( \newcommand\Di{\dot \imath}\)
    \( \newcommand\Dj{\dot \jmath}\)
    \( \newcommand\Dk{\dot k}\)
    \( \newcommand\Dl{\dot l}\)
    \( \newcommand\Dm{\dot m}\)
    \( \newcommand\Dn{\dot n}\)
    \( \newcommand\Do{\dot o}\)
    \( \newcommand\Dp{\dot p}\)
    \( \newcommand\Dq{\dot q}\)
    \( \newcommand\Dr{\dot r}\)
    \( \newcommand\Ds{\dot s}\)
    \( \newcommand\Dt{\dot t}\)
    \( \newcommand\Du{\dot u}\)
    \( \newcommand\Dv{\dot v}\)
    \( \newcommand\Dw{\dot w}\)
    \( \newcommand\Dx{\dot x}\)
    \( \newcommand\Dy{\dot y}\)
    \( \newcommand\Dz{\dot z}\)
    \( \newcommand\DA{\dot A}\)
    \( \newcommand\DB{\dot B}\)
    \( \newcommand\DC{\dot C}\)
    \( \newcommand\DD{\dot D}\)
    \( \newcommand\DE{\dot E}\)
    \( \newcommand\DF{\dot F}\)
    \( \newcommand\DG{\dot G}\)
    \( \newcommand\DH{\dot H}\)
    \( \newcommand\DI{\dot I}\)
    \( \newcommand\DJ{\dot J}\)
    \( \newcommand\DK{\dot K}\)
    \( \newcommand\DL{\dot L}\)
    \( \newcommand\DM{\dot M}\)
    \( \newcommand\DN{\dot N}\)
    \( \newcommand\DO{\dot O}\)
    \( \newcommand\DP{\dot P}\)
    \( \newcommand\DQ{\dot Q}\)
    \( \newcommand\DR{\dot R}\)
    \( \newcommand\DS{\dot S}\)
    \( \newcommand\DT{\dot T}\)
    \( \newcommand\DU{\dot U}\)
    \( \newcommand\DV{\dot V}\)
    \( \newcommand\DW{\dot W}\)
    \( \newcommand\DX{\dot X}\)
    \( \newcommand\DY{\dot Y}\)
    \( \newcommand\DZ{\dot Z}\)
    \( \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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /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)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\msa{m\ns_\ssr{A}}\)
    \( \newcommand\msb{m\ns_\ssr{B}}\)
    \( \newcommand\mss{m\ns_\Rs}\)
    \( \newcommand\HBx{\hat\Bx}\)
    \( \newcommand\HBy{\hat\By}\)
    \( \newcommand\HBz{\hat\Bz}\)
    \( \newcommand\thm{\theta\ns_m}\)
    \( \newcommand\thp{\theta\ns_\phi}\)
    \( \newcommand\mtil{\widetilde m}\)
    \( \newcommand\phitil{\widetilde\phi}\)
    \( \newcommand\delf{\delta\! f}\)
    \( \newcommand\coll{\bigg({\pz f\over\pz t}\bigg)\nd_{\! coll}}\)
    \( \newcommand\stre{\bigg({\pz f\over\pz t}\bigg)\nd_{\! str}}\)
    \( \newcommand\idrp{\int\!\!{d^3\!r\,d^3\!p\over h^3}\>}\)
    \( \newcommand\vbar{\bar v}\)
    \( \newcommand\BCE{\mbox{\boldmath{$\CE$}}\!}\)
    \( \newcommand\BCR{\mbox{\boldmath{$\CR$}}\!}\)
    \( \newcommand\gla{g\nd_{\RLambda\nd}}\)
    \( \newcommand\TA{T\ns_\ssr{A}}\)
    \( \newcommand\TB{T\ns_\ssr{B}}\)
    \( \newcommand\ncdot{\!\cdot\!}\)
    \( \newcommand\NS{N\ns_{\textsf S}}\)

    Canonical Distribution and Partition Function

    Consider a system \(S\) in contact with a world \(W\), and let their union \(U=W\cup S\) be called the ‘universe’. The situation is depicted in Figure [universe]. The volume \(V\ns_{\ssr{S}}\) and particle number \(N\ns_{\ssr{S}}\) of the system are held fixed, but the energy is allowed to fluctuate by exchange with the world \(W\). We are interested in the limit \(N\ns_{\ssr{S}}\to\infty\), \(N\ns_{\ssr{W}}\to\infty\), with \(N\ns_{\ssr{S}}\ll N\ns_{\ssr{W}}\), with similar relations holding for the respective volumes and energies. We now ask what is the probability that \(S\) is in a state \(\sket{n}\) with energy \(E\ns_n\). This is given by the ratio

    \[\begin{align} P\ns_n&=\lim_{\RDelta E\to 0}\ {D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n)\,\RDelta E\over D\ns_{\ssr{U}}(E\ns_{\ssr{U}})\,\RDelta E}\label{OCErat}\\ &={\hbox{ # of states accessible to $W$ given that $E\ns_{\ssr{S}}=E\ns_n$}\over \hbox{ total # of states in $U$}}\ .\bvph \end{align}\]

    Then

    \[\begin{align} \ln P\ns_n&=\ln D\ns_{\ssr{W}} (E\ns_{\ssr{U}}-E\ns_n) - \ln D\ns_{\ssr{U}} (E\ns_{\ssr{U}})\\ &=\ln D\ns_{\ssr{W}} (E\ns_{\ssr{U}})- \ln D\ns_{\ssr{U}}(E\ns_{\ssr{U}}) - E\ns_n\>{\pz\ln D\ns_{\ssr{W}}(E)\over \pz E}\bigg|\nd_{E=E\ns_{\ssr{U}}} \!\! + \ldots\\ &\equiv -\alpha-\beta E\ns_n\ . \end{align}\]

    The constant \(\beta\) is given by

    \[\beta={\pz\ln D\ns_{\ssr{W}}(E)\over\pz E}\bigg|\nd_{E=E\ns_{\ssr{U}}} = {1\over \kT}\ .\]

    Thus, we find \(P\ns_n=e^{-\alpha}\,e^{-\beta E\ns_n}\). The constant \(\alpha\) is fixed by the requirement that \(\sum_n P\ns_n=1\):

    \[P\ns_n={1\over Z}\, e^{-\beta E\ns_n}\qquad,\qquad Z(T,V,N)=\sum_n e^{-\beta E\ns_n}=\Tra e^{-\beta \HH}\ .\]

    We’ve already met \(Z(\beta)\) in Equation \ref{Zlap} – it is the Laplace transform of the density of states. It is also called the partition function of the system \(S\). Quantum mechanically, we can write the ordinary canonical density matrix as

    \[\vrhhat={e^{-\beta\HH}\over \Tra e^{-\beta\HH}}\quad,\]

    which is known as the Gibbs distribution. Note that \(\big[\vrhhat,\HH\big]=0\), hence the ordinary canonical distribution is a stationary solution to the evolution equation for the density matrix. Note that the OCE is specified by three parameters: \(T\), \(V\), and \(N\).

    The difference between \(P(E_n)\) and \(P_n\)

    Let the total energy of the Universe be fixed at \(E\ns_{\ssr{U}}\). The joint probability density \(P(E\ns_{\ssr{S}},E\ns_{\ssr{W}})\) for the system to have energy \(E\ns_\RS\) and the world to have energy \(E\ns_{\ssr{W}}\) is

    \[P(E\ns_{\ssr{S}},E\ns_{\ssr{W}})=D\ns_{\ssr{S}}(E\ns_{\ssr{S}}) \, D\ns_{\ssr{W}}(E\ns_{\ssr{W}}) \,\delta(E\ns_{\ssr{U}}-E\ns_{\ssr{S}}-E\ns_{\ssr{W}}) \big/ D\ns_{\ssr{U}}(E\ns_{\ssr{U}})\ ,\]

    where

    \[D\ns_{\ssr{U}}(E\ns_{\ssr{U}})=\impi dE\ns_{\ssr{S}}\>D\ns_{\ssr{S}}(E\ns_{\ssr{S}})\,D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_{\ssr{S}})\ ,\]

    which ensures that \(\int\!dE\ns_{\ssr{S}}\int\!dE\ns_{\ssr{W}}\,P(E\ns_{\ssr{S}},E\ns_{\ssr{W}})=1\). The probability density \(P(E\ns_{\ssr{S}})\) is defined such that \(P(E\ns_{\ssr{S}})\,dE\ns_{\ssr{S}}\) is the (differential) probability for the system to have an energy in the range \([E\ns_{\ssr{S}},E\ns_{\ssr{S}}+dE\ns_{\ssr{S}}]\). The units of \(P(E\ns_{\ssr{S}})\) are \(E^{-1}\). To obtain \(P(E\ns_{\ssr{S}})\), we simply integrate the joint probability density \(P(E\ns_{\ssr{S}},E\ns_{\ssr{W}})\) over all possible values of \(E\ns_{\ssr{W}}\), obtaining

    \[P(E\ns_{\ssr{S}})={D\ns_{\ssr{S}}(E\ns_{\ssr{S}})\,D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_{\ssr{S}})\over D\ns_{\ssr{U}}(E\ns_{\ssr{U}})}\ ,\]

    as we have in Equation \ref{OCErat}.

    Now suppose we wish to know the probability \(P\ns_n\) that the system is in a particular state \(\sket{n}\) with energy \(E\ns_n\). Clearly

    \[P\ns_n=\lim_{\RDelta E\to 0}{\hbox{ probability that $E\ns_{\ssr{S}}\in[E\ns_n,E\ns_n+\RDelta E]$}\over \hbox{ \ \# of S states with $E\ns_{\ssr{S}}\in [E\ns_n,E\ns_n+\RDelta E]$\ }} ={P(E\ns_n)\,\RDelta E\over D\ns_{\ssr{S}}(E\ns_n)\,\RDelta E} = {D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n)\over D\ns_{\ssr{U}}(E\ns_{\ssr{U}})}\ .\]

    Additional remarks

    The formula of Equation \ref{OCErat} is quite general and holds in the case where \(N\ns_{\ssr{S}}/N\ns_{\ssr{W}}=\CO(1)\), so long as we are in the thermodynamic limit, where the energy associated with the interface between S and W may be neglected. In this case, however, one is not licensed to perform the subsequent Taylor expansion, and the distribution \(P\ns_n\) is no longer of the Gibbs form. It is also valid for quantum systems6, in which case we interpret \(P\ns_n=\texpect{n}{\vrh\ns_{\ssr{S}}}{n}\) as a diagonal element of the density matrix \(\vrh\ns_{\ssr{S}}\). The density of states functions may then be replaced by

    \[\begin{split} D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n)\,\RDelta E &\to e^{S\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n\,,\,\RDelta E)} \equiv \mathop{\textsf{Tra}}_{\ssr{W}} \hskip-0.7cm\int\limits_{E\ns_{\ssr{U}}-E\ns_n}^{E\ns_{\ssr{U}}-E\ns_n+\RDelta E}\hskip-0.7cm dE\>\delta(E-\HH\ns_{\ssr{W}})\\ D\ns_{\ssr{U}}(E\ns_{\ssr{U}})\,\RDelta E &\to e^{S\ns_{\ssr{U}}(E\ns_{\ssr{U}}\,,\,\RDelta E)} \equiv \mathop{\textsf{Tra}}_{\ssr{U}} \hskip-0.4cm\int\limits_{E\ns_{\ssr{U}}}^{E\ns_{\ssr{U}}+\RDelta E}\hskip-0.4cm dE\>\delta(E-\HH\ns_{\ssr{U}})\quad. \end{split}\]

    The off-diagonal matrix elements of \(\vrh_{\ssr{S}}\) are negligible in the thermodynamic limit.

    Averages within the OCE

    To compute averages within the OCE,

    \[\big\langle\HA\big\rangle=\Tra\!\big(\vrhhat\,\HA\big) ={\sum_n\texpect{n}{\HA}{n}\>e^{-\beta E\ns_n}\over\sum_n e^{-\beta E\ns_n}}\ ,\]

    where we have conveniently taken the trace in a basis of energy eigenstates. In the classical limit, we have

    \[\vrh(\Bvphi)={1\over Z}\,e^{-\beta \HH(\Bvphi)} \quad,\quad Z=\Tra e^{-\beta \HH}=\int\!\! d\mu \> e^{-\beta \HH(\Bvphi)}\ ,\]

    with \(d\mu=\frac{1}{N!}\prod_{j=1}^N (d^d q\nd_j\,d^d p\nd_j / h^d)\) for identical particles (‘Maxwell-Boltzmann statistics’). Thus,

    \[\langle A \rangle =\Tra(\vrh A) = {\int\!\! d\mu\>A(\Bvphi)\,e^{-\beta \HH(\Bvphi)}\over \int\!\! d\mu\> e^{-\beta \HH(\Bvphi)}}\ .\]

    Entropy and Free Energy

    The Boltzmann entropy is defined by

    \[S=-\kB\Tra\!\big(\vrhhat\ln\vrhhat) = -\kB\sum_n P\ns_n\,\ln P\ns_n\ .\]

    The Boltzmann entropy and the statistical entropy \(S=\kB\ln D(E)\) are identical in the thermodynamic limit. We define the Helmholtz free energy \(F(T,V,N)\) as

    \[F(T,V,N)=-\kT\ln Z(T,V,N)\ ,\]

    hence

    \[P\ns_n=e^{\beta F}\, e^{-\beta E\ns_n} \qquad,\qquad \ln P\ns_n=\beta F-\beta E\ns_n\ .\]

    Therefore the entropy is

    \[S=-\kB\sum_n P\ns_n\, \big(\beta F-\beta E\ns_n\big)\\ =-{F\over T} + {\langle \,\HH\,\rangle\over T}\ ,\]

    which is to say \(F=E-TS\), where

    \[E=\sum_n P\ns_n \,E\ns_n = {\Tra \HH \,e^{-\beta\HH}\over \Tra e^{-\beta\HH}}\]

    is the average energy. We also see that

    \[Z=\Tra e^{-\beta\HH}=\sum_n e^{-\beta E\ns_n} \quad\Longrightarrow\quad E={\sum_n E\ns_n\,e^{-\beta E\ns_n}\over\sum_n e^{-\beta E\ns_n}}=-{\pz\over\pz\beta}\,\ln Z={\pz\over\pz\beta}\big(\beta F\big)\ .\]

    Thus, \(F(T,V,N)\) is a Legendre transform of \(E(S,V,N)\), with

    \[dF=-S\,dT - p\,dV + \mu\,dN\ ,\]

    which means

    \[S=-\pabc{F}{T}{V,N} \qquad,\qquad p=-\pabc{F}{V}{T,N} \qquad,\qquad \mu=+\pabc{F}{N}{T,V}\ .\]

    Fluctuations in the OCE

    In the OCE, the energy is not fixed. It therefore fluctuates about its average value \(E=\langle \HH\rangle\). Note that

    \[\begin{split} -{\pz E\over\pz\beta}&=\kB T^2\,{\pz E\over\pz T}={\pz^2\ln Z\over\pz\beta^2}\\ &={\Tra \HH^2\,e^{-\beta\HH}\over \Tra e^{-\beta\HH}} - \Bigg({\Tra \HH \,e^{-\beta\HH}\over \Tra e^{-\beta\HH}}\Bigg)^{\!\!2}\\ &=\blangle\HH^2\brangle - \blangle\HH\brangle^2\ . \end{split}\]

    Thus, the heat capacity is related to the fluctuations in the energy, just as we saw at the end of §4:

    \[C\ns_V=\pabc{E}{T}{V,N}={1\over \kB T^2}\, \Big(\blangle \HH^2\brangle - \blangle\HH\brangle^2\Big)\]

    For the nonrelativistic ideal gas, we found \(C\ns_V={d\over 2}\,N\kB\), hence the ratio of RMS fluctuations in the energy to the energy itself is

    \[{\sqrt{\blangle\,(\RDelta\HH)^2\,\brangle} \over\langle\HH\rangle}= {\sqrt{\kB T^2\,C\ns_V}\over {d\over 2}N\kT} = \sqrt{2\over Nd}\ ,\]

    and the ratio of the RMS fluctuations to the mean value vanishes in the thermodynamic limit.

    The full distribution function for the energy is

    \[P(\CE)=\blangle\delta(\CE-\HH)\brangle={\Tra \delta(\CE-\HH)\,e^{-\beta\HH}\over \Tra e^{-\beta\HH}}={1\over Z}\,D(\CE)\,e^{-\beta \CE}\ .\]

    Thus,

    \[P(\CE)={e^{-\beta\left[\CE-TS(\CE)\right]}\over\int\!d\CE'\,e^{-\beta\left[\CE'-TS(\CE')\right]}}\ , \label{PEOCE}\]

    where \(S(\CE)=\kB\ln D(\CE)\) is the statistical entropy. Let’s write \(\CE=E+\delta \CE\), where \(E\) extremizes the combination \(\CE-T\,S(\CE)\), the solution to \(T\,S'(E)=1\), where the energy derivative of \(S\) is performed at fixed volume \(V\) and particle number \(N\). We now expand \(S(E+\delta \CE)\) to second order in \(\delta \CE\), obtaining

    \[S(E+\delta \CE)=S(E) + {\delta \CE\over T} -{\big(\delta \CE\big)^2\over 2 T^2 \,C\ns_V}\, + \ldots\]

    Recall that \(S''(E)={\pz\over\pz E}\left({1\over T}\right) = -{1\over T^2 C\ns_V}\). Thus,

    \[\CE-T\,S(\CE)=E - T\,S(E) + {(\delta \CE)^2\over 2 T\,C\ns_V} + \CO\big((\delta \CE)^3\big)\ . \label{EminusTS}\]

    Applying this to both numerator and denominator of Equation \ref{PEOCE}, we obtain7

    \[P(\CE)=\CN\,\exp\Bigg[\!-{(\delta \CE)^2\over 2\kB T^2\,C\ns_V}\Bigg]\ ,\]

    where \(\CN=(2\pi\kB T^2 C\ns_V)^{-1/2}\) is a normalization constant which guarantees \(\int\!d\CE\,P(\CE)=1\). Once again, we see that the distribution is a Gaussian centered at \(\langle\CE\rangle = E\), and of width \((\RDelta \CE)\nd_{\ssr{RMS}}=\sqrt{\kB T^2\,C\ns_V}\). This is a consequence of the Central Limit Theorem.

    Thermodynamics revisited

    The average energy within the OCE is

    \[E=\sum_n E\ns_n P\ns_n\ ,\]

    and therefore

    \[\begin{split} dE=& \sum_n E\ns_n \,dP\ns_n + \sum_n P\ns_n\,dE\ns_n\\ &=\dbar Q-\dbar W\ , \end{split}\label{smfl}\]

    where

    \[\begin{aligned} \dbar W&=-\sum_n P\ns_n\,dE\ns_n\\ \dbar Q&=\sum_n E\ns_n\,dP\ns_n\ .\end{aligned}\]

    Finally, from \(P\ns_n=Z^{-1}\,e^{-E\ns_n/k\ns_\RB T}\), we can write

    \[E\ns_n=-\kT\ln Z - \kT\ln P\ns_n\ ,\]

    with which we obtain

    \[\begin{split} \dbar Q&=\sum_n E\ns_n\,dP\ns_n\\ &=-\kT\ln Z\sum_n dP\ns_n - \kT\sum_n \ln P\ns_n\>dP\ns_n\\ &=T \,d\Big(\!-\kB\sum_n P\ns_n\ln P\ns_n\Big)=T\,dS\ . \end{split}\]

    Note also that

    \[\begin{align} \dbar W&=-\sum_n P\ns_n \, dE\ns_n \\ &=-\sum_nP\ns_n \Bigg(\!\sum_i {\pz E\ns_n\over\pz X\ns_i}\>dX\ns_i\Bigg)\\ &=-\sum_{n,i} P\ns_n\,\expect{n}{\pz \HH\over\pz X\ns_i}{n}\>dX\ns_i \equiv\sum_i F\ns_i\,dX\ns_i\ , \end{align} \label{workeqn}\]

    so the generalized force \(F\ns_i\) conjugate to the generalized displacement \(dX\ns_i\) is

    \[F\ns_i=-\sum_n P\ns_n\,{\pz E\ns_n\over\pz X\ns_i}=-\,\bigg\langle {\pz\HH\over\pz X\ns_i}\bigg\rangle\ . \label{thermforce}\]

    This is the force acting on the system8. In the chapter on thermodynamics, we defined the generalized force conjugate to \(X\ns_i\) as \(y\ns_i\equiv - F\ns_i\).

    [SMfirst] Microscopic, statistical interpretation of the First Law of Thermodynamics.
    Figure \(\PageIndex{1}\): Microscopic, statistical interpretation of the First Law of Thermodynamics.

    Thus we see from Equation \ref{smfl} that there are two ways that the average energy can change; these are depicted in the sketch of Figure \(\PageIndex{1}\). Starting from a set of energy levels \(\{E\ns_n\}\) and probabilities \(\{P\ns_n\}\), we can shift the energies to \(\{E'_n\}\). The resulting change in energy \((\RDelta E)\ns_{\ssr{I}}=-W\) is identified with the work done on the system. We could also modify the probabilities to \(\{P'_n\}\) without changing the energies. The energy change in this case is the heat absorbed by the system: \((\RDelta E)\ns_{\ssr{II}} = Q\). This provides us with a statistical and microscopic interpretation of the First Law of Thermodynamics.

    Generalized Susceptibilities

    Suppose our Hamiltonian is of the form

    \[\HH=\HH(\lambda)=\HH\ns_0-\lambda\,{\hat Q}\ ,\]

    where \(\lambda\) is an intensive parameter, such as magnetic field. Then

    \[Z(\lambda)=\Tra e^{-\beta(\HH\ns_0-\lambda{\hat Q})}\]

    and

    \[{1\over Z}\,{\pz Z\over\pz \lambda}=\beta\cdot{1\over Z}\Tra\Big( {\hat Q}\,e^{-\beta\HH(\lambda)}\Big)=\beta\>\langle{\hat Q}\rangle\ .\]

    But then from \(Z=e^{-\beta F}\) we have

    \[Q(\lambda,T)=\langle\,{\hat Q}\,\rangle=-\pabc{F}{\lambda}{T}\ .\]

    Typically we will take \(Q\) to be an extensive quantity. We can now define the susceptibility \(\xhi\) as

    \[\xhi={1\over V}{\pz Q\over\pz\lambda}=-{1\over V}\,{\pz^2\!F\over\pz\lambda^2}\ .\]

    The volume factor in the denominator ensures that \(\xhi\) is intensive.

    It is important to realize that we have assumed here that \(\big[\HH\ns_0\,,\,{\hat Q}\big]=0\), the ‘bare’ Hamiltonian \(\HH\ns_0\) and the operator \({\hat Q}\) commute. If they do not commute, then the response functions must be computed within a proper quantum mechanical formalism, which we shall not discuss here.

    Note also that we can imagine an entire family of observables \(\big\{{\hat Q}\ns_k\big\}\) satisfying \(\big[{\hat Q}\ns_{k\ns}\,,\,{\hat Q}\ns_{k'}\big]=0\) and \(\big[\HH\ns_0\,,\,{\hat Q}\ns_k\big]=0\), for all \(k\) and \(k'\). Then for the Hamiltonian

    \[\HH\ns(\Vlambda)=\HH\ns_0-\sum_k\lambda\ns_k\,{\hat Q}\ns_k\ ,\]

    we have that

    \[Q\ns_k({\Vlambda},T)=\langle\,{\hat Q}\ns_k\,\rangle=-\pabc{F}{\lambda\ns_k}{T,\,N\ns_a,\,\lambda\ns_{k'\ne k}}\]

    and we may define an entire matrix of susceptibilities,

    \[\xhi\ns_{kl}={1\over V}{\pz Q\ns_k\over\pz\lambda\ns_l}=-{1\over V}\,{\pz^2\!F\over\pz\lambda\ns_k\,\pz\lambda\ns_l}\ .\]