2.9: Equilibrium and Stability
- Page ID
- 18861
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)
\( \newcommand\bes{\begin{equation}\begin{split}}\)
\( \newcommand\ltwid{\propto}\)
\( \newcommand\ees{\end{split}\end{equation}}\)
\( \newcommand\mib{\mathbf}\)
\( \newcommand\Sa{\textsf a}\)
\( \newcommand\Sb{\textsf b}\)
\( \newcommand\Sc{\textsf c}\)
\( \newcommand\Sd{\textsf d}\)
\( \newcommand\Se{\textsf e}\)
\( \newcommand\Sf{\textsf f}\)
\( \newcommand\Sg{\textsf g}\)
\( \newcommand\Sh{\textsf h}\)
\( \newcommand\Si{\textsf i}\)
\( \newcommand\Sj{\textsf j}\)
\( \newcommand\Sk{\textsf k}\)
\( \newcommand\Sl{\textsf l}\)
\( \newcommand\Sm{\textsf m}\)
\( \newcommand\Sn{\textsf n}\)
\( \newcommand\So{\textsf o}\)
\( \newcommand\Sp{\textsf p}\)
\( \newcommand\Sq{\textsf q}\)
\( \newcommand\Sr{\textsf r}\)
\( \newcommand\Ss{\textsf s}\)
\( \newcommand\St{\textsf t}\)
\( \newcommand\Su{\textsf u}\)
\( \newcommand\Sv{\textsf v}\)
\( \newcommand\Sw{\textsf w}\)
\( \newcommand\Sx{\textsf x}\)
\( \newcommand\Sy{\textsf y}\)
\( \newcommand\Sz{\textsf z}\)
\( \newcommand\SA{\textsf A}\)
\( \newcommand\SB{\textsf B}\)
\( \newcommand\SC{\textsf C}\)
\( \newcommand\SD{\textsf D}\)
\( \newcommand\SE{\textsf E}\)
\( \newcommand\SF{\textsf F}\)
\( \newcommand\SG{\textsf G}\)
\( \newcommand\SH{\textsf H}\)
\( \newcommand\SI{\textsf I}\)
\( \newcommand\SJ{\textsf J}\)
\( \newcommand\SK{\textsf K}\)
\( \newcommand\SL{\textsf L}\)
\( \newcommand\SM{\textsf M}\)
\( \newcommand\SN{\textsf N}\)
\( \newcommand\SO{\textsf O}\)
\( \newcommand\SP{\textsf P}\)
\( \newcommand\SQ{\textsf Q}\)
\( \newcommand\SR{\textsf R}\)
\( \newcommand\SS{\textsf S}\)
\( \newcommand\ST{\textsf T}\)
\( \newcommand\SU{\textsf U}\)
\( \newcommand\SV{\textsf V}\)
\( \newcommand\SW{\textsf W}\)
\( \newcommand\SX{\textsf X}\)
\( \newcommand\SY{\textsf Y}\)
\( \newcommand\SZ{\textsf Z}\)
\( \newcommand\Ha{\hat a}\)
\( \newcommand\Hb{\hat b}\)
\( \newcommand\Hc{\hat c}\)
\( \newcommand\Hd{\hat d}\)
\( \newcommand\He{\hat e}\)
\( \newcommand\Hf{\hat f}\)
\( \newcommand\Hg{\hat g}\)
\( \newcommand\Hh{\hat h}\)
\( \newcommand\Hi{\hat \imath}\)
\( \newcommand\Hj{\hat \jmath}\)
\( \newcommand\Hk{\hat k}\)
\( \newcommand\Hl{\hat l}\)
\( \newcommand\Hm{\hat m}\)
\( \newcommand\Hn{\hat n}\)
\( \newcommand\Ho{\hat o}\)
\( \newcommand\Hp{\hat p}\)
\( \newcommand\Hq{\hat q}\)
\( \newcommand\Hr{\hat r}\)
\( \newcommand\Hs{\hat s}\)
\( \newcommand\Ht{\hat t}\)
\( \newcommand\Hu{\hat u}\)
\( \newcommand\Hv{\hat v}\)
\( \newcommand\Hw{\hat w}\)
\( \newcommand\Hx{\hat x}\)
\( \newcommand\Hy{\hat y}\)
\( \newcommand\Hz{\hat z}\)
\( \newcommand\HA{\hat A}\)
\( \newcommand\HB{\hat B}\)
\( \newcommand\HC{\hat C}\)
\( \newcommand\HD{\hat D}\)
\( \newcommand\HE{\hat E}\)
\( \newcommand\HF{\hat F}\)
\( \newcommand\HG{\hat G}\)
\( \newcommand\HH{\hat H}\)
\( \newcommand\HI{\hat I}\)
\( \newcommand\HJ{\hat J}\)
\( \newcommand\HK{\hat K}\)
\( \newcommand\HL{\hat L}\)
\( \newcommand\HM{\hat M}\)
\( \newcommand\HN{\hat N}\)
\( \newcommand\HO{\hat O}\)
\( \newcommand\HP{\hat P}\)
\( \newcommand\HQ{\hat Q}\)
\( \newcommand\HR{\hat R}\)
\( \newcommand\HS{\hat S}\)
\( \newcommand\HT{\hat T}\)
\( \newcommand\HU{\hat U}\)
\( \newcommand\HV{\hat V}\)
\( \newcommand\HW{\hat W}\)
\( \newcommand\HX{\hat X}\)
\( \newcommand\HY{\hat Y}\)
\( \newcommand\HZ{\hat Z}\)
\( \newcommand\Halpha{\hat\alpha}\)
\( \newcommand\Hbeta{\hat\beta}\)
\( \newcommand\Hgamma{\hat\gamma}\)
\( \newcommand\Hdelta{\hat\delta}\)
\( \newcommand\Hepsilon{\hat\epsilon}\)
\( \newcommand\Hvarepsilon{\hat\varepsilon}\)
\( \newcommand\Hzeta{\hat\zeta}\)
\( \newcommand\Heta{\hat\eta}\)
\( \newcommand\Htheta{\hat\theta}\)
\( \newcommand\Hvartheta{\hat\vartheta}\)
\( \newcommand\Hiota{\hat\iota}\)
\( \newcommand\Hkappa{\hat\kappa}\)
\( \newcommand\Hlambda{\hat\lambda}\)
\( \newcommand\Hmu{\hat\mu}\)
\( \newcommand\Hnu{\hat\nu}\)
\( \newcommand\Hxi{\hat\xi}\)
\( \newcommand\Hom{\hat\omicron}\)
\( \newcommand\Hpi{\hat\pi}\)
\( \newcommand\Hvarpi{\hat\varpi}\)
\( \newcommand\Hrho{\hat\rho}\)
\( \newcommand\Hvarrho{\hat\varrho}\)
\( \newcommand\Hsigma{\hat\sigma}\)
\( \newcommand\Hvarsigma{\hat\varsigma}\)
\( \newcommand\Htau{\var\tau}\)
\( \newcommand\Hupsilon{\hat\upsilon}\)
\( \newcommand\Hphi{\hat\phi}\)
\( \newcommand\Hvarphi{\hat\varphi}\)
\( \newcommand\Hchi{\hat\chi}\)
\( \newcommand\Hxhi{\hat\xhi}\)
\( \newcommand\Hpsi{\hat\psi}\)
\( \newcommand\Homega{\hat\omega}\)
\( \newcommand\HGamma{\hat\Gamma}\)
\( \newcommand\HDelta{\hat\Delta}\)
\( \newcommand\HTheta{\hat\Theta}\)
\( \newcommand\HLambda{\hat\Lambda}\)
\( \newcommand\HXi{\hat\Xi}\)
\( \newcommand\HPi{\hat\Pi}\)
\( \newcommand\HSigma{\hat\Sigma}\)
\( \newcommand\HUps{\hat\Upsilon}\)
\( \newcommand\HPhi{\hat\Phi}\)
\( \newcommand\HPsi{\hat\Psi}\)
\( \newcommand\HOmega{\hat\Omega}\)
\( \newcommand\xhat{\hat\Bx}\)
\( \newcommand\yhat{\hat\By}\)
\( \newcommand\zhat{\hat\Bz}\)
\( \newcommand\ehat{\hat\Be}\)
\( \newcommand\khat{\hat\Bk}\)
\( \newcommand\nhat{\hat\Bn}\)
\( \newcommand\rhat{\hat\Br}\)
\( \newcommand\phihat{\hat\Bphi}\)
\( \newcommand\thetahat{\hat\Btheta}\)
\( \newcommand\MA{\mathbb A}\)
\( \newcommand\MB{\mathbb B}\)
\( \newcommand\MC{\mathbb C}\)
\( \newcommand\MD{\mathbb D}\)
\( \newcommand\ME{\mathbb E}\)
\( \newcommand\MF{\mathbb F}\)
\( \newcommand\MG{\mathbb G}\)
\( \newcommand\MH{\mathbb H}\)
\( \newcommand\MI{\mathbb I}\)
\( \newcommand\MJ{\mathbb J}\)
\( \newcommand\MK{\mathbb K}\)
\( \newcommand\ML{\mathbb L}\)
\( \newcommand\MM{\mathbb M}\)
\( \newcommand\MN{\mathbb N}\)
\( \newcommand\MO{\mathbb O}\)
\( \newcommand\MP{\mathbb P}\)
\( \newcommand\MQ{\mathbb Q}\)
\( \newcommand\MR{\mathbb R}\)
\( \newcommand\MS{\mathbb S}\)
\( \newcommand\MT{\mathbb T}\)
\( \newcommand\MU{\mathbb U}\)
\( \newcommand\MV{\mathbb V}\)
\( \newcommand\MW{\mathbb W}\)
\( \newcommand\MX{\mathbb X}\)
\( \newcommand\MY{\mathbb Y}\)
\( \newcommand\MZ{\mathbb Z}\)
\( \newcommand\CA{\mathcal A}\)
\( \newcommand\CB{\mathcal B}\)
\( \newcommand\CC{\mathcal C}\)
\( \newcommand\CD{\mathcal D}\)
\( \newcommand\CE{\mathcal E}\)
\( \newcommand\CF{\mathcal F}\)
\( \newcommand\CG{\mathcal G}\)
\( \newcommand\CH{\mathcal H}\)
\( \newcommand\CI{\mathcal I}\)
\( \newcommand\CJ{\mathcal J}\)
\( \newcommand\CK{\mathcal K}\)
\( \newcommand\CL{\mathcal L}\)
\( \newcommand\CM{\mathcal M}\)
\( \newcommand\CN{\mathcal N}\)
\( \newcommand\CO{\mathcal O}\)
\( \newcommand\CP{\mathcal P}\)
\( \newcommand\CQ{\mathcal Q}\)
\( \newcommand\CR{\mathcal R}\)
\( \newcommand\CS{\mathcal S}\)
\( \newcommand\CT{\mathcal T}\)
\( \newcommand\CU{\mathcal U}\)
\( \newcommand\CV{\mathcal V}\)
\( \newcommand\CW{\mathcal W}\)
\( \newcommand\CX{\mathcal X}\)
\( \newcommand\CY{\mathcal Y}\)
\( \newcommand\CZ{\mathcal Z}\)
\( \newcommand\Fa{\mathfrak a}\)
\( \newcommand\Fb{\mathfrak b}\)
\( \newcommand\Fc{\mathfrak c}\)
\( \newcommand\Fd{\mathfrak d}\)
\( \newcommand\Fe{\mathfrak e}\)
\( \newcommand\Ff{\mathfrak f}\)
\( \newcommand\Fg{\mathfrak g}\)
\( \newcommand\Fh{\mathfrak h}\)
\( \newcommand\Fi{\mathfrak i}\)
\( \newcommand\Fj{\mathfrak j}\)
\( \newcommand\Fk{\mathfrak k}\)
\( \newcommand\Fl{\mathfrak l}\)
\( \newcommand\Fm{\mathfrak m}\)
\( \newcommand\Fn{\mathfrak n}\)
\( \newcommand\Fo{\mathfrak o}\)
\( \newcommand\Fp{\mathfrak p}\)
\( \newcommand\Fq{\mathfrak q}\)
\( \newcommand\Fr{\mathfrak r}\)
\( \newcommand\Fs{\mathfrak s}\)
\( \newcommand\Ft{\mathfrak t}\)
\( \newcommand\Fu{\mathfrak u}\)
\( \newcommand\Fv{\mathfrak v}\)
\( \newcommand\Fw{\mathfrak w}\)
\( \newcommand\Fx{\mathfrak x}\)
\( \newcommand\Fy{\mathfrak y}\)
\( \newcommand\Fz{\mathfrak z}\)
\( \newcommand\FA{\mathfrak A}\)
\( \newcommand\FB{\mathfrak B}\)
\( \newcommand\FC{\mathfrak C}\)
\( \newcommand\FD{\mathfrak D}\)
\( \newcommand\FE{\mathfrak E}\)
\( \newcommand\FF{\mathfrak F}\)
\( \newcommand\FG{\mathfrak G}\)
\( \newcommand\FH{\mathfrak H}\)
\( \newcommand\FI{\mathfrak I}\)
\( \newcommand\FJ{\mathfrak J}\)
\( \newcommand\FK{\mathfrak K}\)
\( \newcommand\FL{\mathfrak L}\)
\( \newcommand\FM{\mathfrak M}\)
\( \newcommand\FN{\mathfrak N}\)
\( \newcommand\FO{\mathfrak O}\)
\( \newcommand\FP{\mathfrak P}\)
\( \newcommand\FQ{\mathfrak Q}\)
\( \newcommand\FR{\mathfrak R}\)
\( \newcommand\FS{\mathfrak S}\)
\( \newcommand\FT{\mathfrak T}\)
\( \newcommand\FU{\mathfrak U}\)
\( \newcommand\FV{\mathfrak V}\)
\( \newcommand\FW{\mathfrak W}\)
\( \newcommand\FX{\mathfrak X}\)
\( \newcommand\FY{\mathfrak Y}\)
\( \newcommand\FZ{\mathfrak Z}\)
\( \newcommand\Da{\dot a}\)
\( \newcommand\Db{\dot b}\)
\( \newcommand\Dc{\dot c}\)
\( \newcommand\Dd{\dot d}\)
\( \newcommand\De{\dot e}\)
\( \newcommand\Df{\dot f}\)
\( \newcommand\Dg{\dot g}\)
\( \newcommand\Dh{\dot h}\)
\( \newcommand\Di{\dot \imath}\)
\( \newcommand\Dj{\dot \jmath}\)
\( \newcommand\Dk{\dot k}\)
\( \newcommand\Dl{\dot l}\)
\( \newcommand\Dm{\dot m}\)
\( \newcommand\Dn{\dot n}\)
\( \newcommand\Do{\dot o}\)
\( \newcommand\Dp{\dot p}\)
\( \newcommand\Dq{\dot q}\)
\( \newcommand\Dr{\dot r}\)
\( \newcommand\Ds{\dot s}\)
\( \newcommand\Dt{\dot t}\)
\( \newcommand\Du{\dot u}\)
\( \newcommand\Dv{\dot v}\)
\( \newcommand\Dw{\dot w}\)
\( \newcommand\Dx{\dot x}\)
\( \newcommand\Dy{\dot y}\)
\( \newcommand\Dz{\dot z}\)
\( \newcommand\DA{\dot A}\)
\( \newcommand\DB{\dot B}\)
\( \newcommand\DC{\dot C}\)
\( \newcommand\DD{\dot D}\)
\( \newcommand\DE{\dot E}\)
\( \newcommand\DF{\dot F}\)
\( \newcommand\DG{\dot G}\)
\( \newcommand\DH{\dot H}\)
\( \newcommand\DI{\dot I}\)
\( \newcommand\DJ{\dot J}\)
\( \newcommand\DK{\dot K}\)
\( \newcommand\DL{\dot L}\)
\( \newcommand\DM{\dot M}\)
\( \newcommand\DN{\dot N}\)
\( \newcommand\DO{\dot O}\)
\( \newcommand\DP{\dot P}\)
\( \newcommand\DQ{\dot Q}\)
\( \newcommand\DR{\dot R}\)
\( \newcommand\DS{\dot S}\)
\( \newcommand\DT{\dot T}\)
\( \newcommand\DU{\dot U}\)
\( \newcommand\DV{\dot V}\)
\( \newcommand\DW{\dot W}\)
\( \newcommand\DX{\dot X}\)
\( \newcommand\DY{\dot Y}\)
\( \newcommand\DZ{\dot Z}\)
\( \newcommand\Dalpha
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Dbeta
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Dgamma
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Ddelta
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Depsilon
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Dvarepsilon
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Dzeta
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Deta
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Dtheta
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Dvartheta
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Diota
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Dkappa
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Dlambda
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /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
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /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
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\DDelta
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\DTheta
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /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
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vnu
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vxi
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vom
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vpi
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarpi
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vrho
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarrho
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vsigma
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarsigma
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vtau
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vupsilon
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vphi
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarphi
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vchi
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vpsi
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\Vomega
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\VGamma
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\VDelta
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\VTheta{\vec\Theta}\)
\( \newcommand\VLambda{\vec\Lambda}\)
\( \newcommand\VXi{\vec\Xi}\)
\( \newcommand\VPi{\vec\Pi}\)
\( \newcommand\VSigma{\vec\Sigma}\)
\( \newcommand\VUps{\vec\Upsilon}\)
\( \newcommand\VPhi{\vec\Phi}\)
\( \newcommand\VPsi{\vec\Psi}\)
\( \newcommand\VOmega{\vec\Omega}\)
\( \newcommand\BA{\mib A}\)
\( \newcommand\BB{\mib B}\)
\( \newcommand\BC{\mib C}\)
\( \newcommand\BD{\mib D}\)
\( \newcommand\BE{\mib E}\)
\( \newcommand\BF{\mib F}\)
\( \newcommand\BG{\mib G}\)
\( \newcommand\BH{\mib H}\)
\( \newcommand\BI{\mib I}}\)
\( \newcommand\BJ{\mib J}\)
\( \newcommand\BK{\mib K}\)
\( \newcommand\BL{\mib L}\)
\( \newcommand\BM{\mib M}\)
\( \newcommand\BN{\mib N}\)
\( \newcommand\BO{\mib O}\)
\( \newcommand\BP{\mib P}\)
\( \newcommand\BQ{\mib Q}\)
\( \newcommand\BR{\mib R}\)
\( \newcommand\BS{\mib S}\)
\( \newcommand\BT{\mib T}\)
\( \newcommand\BU{\mib U}\)
\( \newcommand\BV{\mib V}\)
\( \newcommand\BW{\mib W}\)
\( \newcommand\BX{\mib X}\)
\( \newcommand\BY{\mib Y}\)
\( \newcommand\BZ{\mib Z}\)
\( \newcommand\Ba{\mib a}\)
\( \newcommand\Bb{\mib b}\)
\( \newcommand\Bc{\mib c}\)
\( \newcommand\Bd{\mib d}\)
\( \newcommand\Be{\mib e}\)
\( \newcommand\Bf{\mib f}\)
\( \newcommand\Bg{\mib g}\)
\( \newcommand\Bh{\mib h}\)
\( \newcommand\Bi{\mib i}\)
\( \newcommand\Bj{\mib j}\)
\( \newcommand\Bk{\mib k}\)
\( \newcommand\Bl{\mib l}\)
\( \newcommand\Bm{\mib m}\)
\( \newcommand\Bn{\mib n}\)
\( \newcommand\Bo{\mib o}\)
\( \newcommand\Bp{\mib p}\)
\( \newcommand\Bq{\mib q}\)
\( \newcommand\Br{\mib r}\)
\( \newcommand\Bs{\mib s}\)
\( \newcommand\Bt{\mib t}\)
\( \newcommand\Bu{\mib u}\)
\( \newcommand\Bv{\mib v}\)
\( \newcommand\Bw{\mib w}\)
\( \newcommand\Bx{\mib x}\)
\( \newcommand\By{\mib y}\)
\( \newcommand\Bz{\mib z}\)\)
\( \newcommand\vrh{\varrho}\)
\( \newcommand\vsig{\varsigma}\)
\( \newcommand\ups{\upsilon}\)
\( \newcommand\eps{\epsilon}\)
\( \newcommand\ve{\varepsilon}\)
\( \newcommand\vth{\vartheta}\)
\( \newcommand\vphi{\varphi}\)
\( \newcommand\xhi{\chi}\)
\( \newcommand\Ups{\Upsilon}\)
\( \newcommand\Balpha{\mib\alpha}\)
\( \newcommand\Bbeta{\mib\beta}\)
\( \newcommand\Bgamma{\mib\gamma}\)
\( \newcommand\Bdelta{\mib\delta}\)
\( \newcommand\Beps{\mib\epsilon}\)
\( \newcommand\Bve{\mib\varepsilon}\)
\( \newcommand\Bzeta{\mib\zeta}\)
\( \newcommand\Beta{\mib\eta}\)
\( \newcommand\Btheta{\mib\theta}\)
\( \newcommand\Bvth{\mib\vartheta}\)
\( \newcommand\Biota{\mib\iota}\)
\( \newcommand\Bkappa{\mib\kappa}\)
\( \newcommand\Blambda{\mib\lambda}\)
\( \newcommand\Bmu{\mib\mu}\)
\( \newcommand\Bnu{\mib\nu}\)
\( \newcommand\Bxi{\mib\xi}\)
\( \newcommand\Bom{\mib\omicron}\)
\( \newcommand\Bpi{\mib\pi}\)
\( \newcommand\Bvarpi{\mib\varpi}\)
\( \newcommand\Brho{\mib\rho}\)
\( \newcommand\Bvrh{\mib\varrho}\)
\( \newcommand\Bsigma{\mib\sigma}\)
\( \newcommand\Bvsig{\mib\varsigma}\)
\( \newcommand\Btau{\mib\tau}\)
\( \newcommand\Bups{\mib\upsilon}\)
\( \newcommand\Bphi{\mib\phi}\)
\( \newcommand\Bvphi{\mib\vphi}\)
\( \newcommand\Bchi{\mib\chi}\)
\( \newcommand\Bpsi{\mib\psi}\)
\( \newcommand\Bomega{\mib\omega}\)
\( \newcommand\BGamma{\mib\Gamma}\)
\( \newcommand\BDelta{\mib\Delta}\)
\( \newcommand\BTheta{\mib\Theta}\)
\( \newcommand\BLambda{\mib\Lambda}\)
\( \newcommand\BXi{\mib\Xi}\)
\( \newcommand\BPi{\mib\Pi}\)
\( \newcommand\BSigma{\mib\Sigma}\)
\( \newcommand\BUps{\mib\Upsilon}\)
\( \newcommand\BPhi{\mib\Phi}\)
\( \newcommand\BPsi{\mib\Psi}\)
\( \newcommand\BOmega{\mib\Omega}\)
\( \newcommand\Bxhi{\raise.35ex\hbox{$\Bchi$}}\)
\( \newcommand\RGamma{ \Gamma}\)
\( \newcommand\RDelta{ \Delta}\)
\( \newcommand\RTheta{ \Theta}\)
\( \newcommand\RLambda{ \Lambda}\)
\( \newcommand\RXi{ \Xi}\)
\( \newcommand\RPi{ \Pi}\)
\( \newcommand\RSigma{ \Sigma}\)
\( \newcommand\RUps{ \Upsilon}\)
\( \newcommand\RPhi{ \Phi}\)
\( \newcommand\RPsi{ \Psi}\)
\( \newcommand\ROmega{ \Omega}\)
\( \newcommand\RA{ A}\)
\( \newcommand\RB{ B}\)
\( \newcommand\RC{ C}\)
\( \newcommand\RD{ D}\)
\( \newcommand\RE{ E}\)
\( \newcommand\RF{ F}\)
\( \newcommand\RG{ G}\)
\( \newcommand\RH{ H}\)
\( \newcommand\RI{ I}\)
\( \newcommand\RJ{ J}\)
\( \newcommand\RK{ K}\)
\( \newcommand\RL{ L}\)
\( \newcommand { M}\)
\( \newcommand\RN{ N}\)
\( \newcommand\RO{ O}\)
\( \newcommand\RP{ P}\)
\( \newcommand\RQ{ Q}\)
\( \newcommand\RR{ R}\)
\( \newcommand\RS{ S}\)
\( \newcommand\RT{ T}\)
\( \newcommand\RU{ U}\)
\( \newcommand\RV{ V}\)
\( \newcommand\RW{ W}\)
\( \newcommand\RX{ X}\)
\( \newcommand\RY{ Y}\)
\( \newcommand\RZ{ Z}\)
\( \newcommand\Ra{ a}\)
\( \newcommand\Rb{ b}\)
\( \newcommand\Rc{ c}\)
\( \newcommand\Rd{ d}\)
\( \newcommand\Re{ e}\)
\( \newcommand\Rf{ f}\)
\( \newcommand\Rg{ g}\)
\( \newcommand\Rh{ h}\)
\( \newcommand\Ri{ i}\)
\( \newcommand\Rj{ j}\)
\( \newcommand\Rk{ k}\)
\( \newcommand\Rl{ l}\)
\( \newcommand { m}\)
\( \newcommand\Rn{ n}\)
\( \newcommand\Ro{ o}\)
\( \newcommand\Rp{ p}\)
\( \newcommand\Rq{ q}\)
\( \newcommand\Rr{ r}\)
\( \newcommand\Rs{ s}\)
\( \newcommand\Rt{ t}\)
\( \newcommand\Ru{ u}\)
\( \newcommand\Rv{ v}\)
\( \newcommand\Rw{ w}\)
\( \newcommand\Rx{ x}\)
\( \newcommand\Ry{ y}\)
\( \newcommand\Rz{ z}\)
\( \newcommand\BBA{\boldsymbol\RA}\)
\( \newcommand\BBB{\boldsymbol\RB}\)
\( \newcommand\BBC{\boldsymbol\RC}\)
\( \newcommand\BBD{\boldsymbol\RD}\)
\( \newcommand\BBE{\boldsymbol\RE}\)
\( \newcommand\BBF{\boldsymbol\RF}\)
\( \newcommand\BBG{\boldsymbol\RG}\)
\( \newcommand\BBH{\boldsymbol\RH}\)
\( \newcommand\BBI{\boldsymbol\RI}\)
\( \newcommand\BBJ{\boldsymbol\RJ}\)
\( \newcommand\BBK{\boldsymbol\RK}\)
\( \newcommand\BBL{\boldsymbol\RL}\)
\( \newcommand\BBM{\boldsymbol }\)
\( \newcommand\BBN{\boldsymbol\RN}\)
\( \newcommand\BBO{\boldsymbol\RO}\)
\( \newcommand\BBP{\boldsymbol\RP}\)
\( \newcommand\BBQ{\boldsymbol\RQ}\)
\( \newcommand\BBR{\boldsymbol\RR}\)
\( \newcommand\BBS{\boldsymbol\RS}\)
\( \newcommand\BBT{\boldsymbol\RT}\)
\( \newcommand\BBU{\boldsymbol\RU}\)
\( \newcommand\BBV{\boldsymbol\RV}\)
\( \newcommand\BBW{\boldsymbol\RW}\)
\( \newcommand\BBX{\boldsymbol\RX}\)
\( \newcommand\BBY{\boldsymbol\RY}\)
\( \newcommand\BBZ{\boldsymbol\RZ}\)
\( \newcommand\BBa{\boldsymbol\Ra}\)
\( \newcommand\BBb{\boldsymbol\Rb}\)
\( \newcommand\BBc{\boldsymbol\Rc}\)
\( \newcommand\BBd{\boldsymbol\Rd}\)
\( \newcommand\BBe{\boldsymbol\Re}\)
\( \newcommand\BBf{\boldsymbol\Rf}\)
\( \newcommand\BBg{\boldsymbol\Rg}\)
\( \newcommand\BBh{\boldsymbol\Rh}\}\)
\( \newcommand\BBi{\boldsymbol\Ri}\)
\( \newcommand\BBj{\boldsymbol\Rj}\)
\( \newcommand\BBk{\boldsymbol\Rk}\)
\( \newcommand\BBl{boldsymbol\Rl}\)
\( \newcommand\BBm{\boldsymbol }\)
\( \newcommand\BBn{\boldsymbol\Rn}\)
\( \newcommand\BBo{\boldsymbol\Ro}\)
\( \newcommand\BBp{\boldsymbol\Rp}\)
\( \newcommand\BBq{\boldsymbol\Rq}\)
\( \newcommand\BBr{\boldsymbol\Rr}\)
\( \newcommand\BBs{\boldsymbol\Rs}\)
\( \newcommand\BBt{\boldsymbol\Rt}\)
\( \newcommand\BBu{\boldsymbol\Ru}\)
\( \newcommand\BBv{\boldsymbol\Rv}\)
\( \newcommand\BBw{\boldsymbol\Rw}\)
\( \newcommand\BBx{\boldsymbol\Rx}\)
\( \newcommand\BBy{\boldsymbol\Ry}\)
\( \newcommand\BBz{\boldsymbol\Rz}\)
\( \newcommand\tcb{\textcolor{blue}\)
\( \newcommand\tcr{\textcolor{red}\)
\( \newcommand\bnabla{\boldsymbol{\nabla}}\)
\( \newcommand\Bell{\boldsymbol\ell}\)
\( \newcommand\dbar{\,{\mathchar'26\mkern-12mu d}} \)
\( \newcommand\ns{^\vphantom{*}}\)
\( \newcommand\uar{\uparrow}\)
\( \newcommand\dar{\downarrow}\)
\( \newcommand\impi{\int\limits_{-\infty}^{\infty}\!\!}\)
\( \newcommand\izpi{\int\limits_{0}^{\infty}\!\!}\)
\( \newcommand\etc{\it etc.\/}\)
\( \newcommand\etal{\it et al.\/}\)
\( \newcommand\opcit{\it op. cit.\/}\)
\( \newcommand\ie{\it i.e.\/}\)
\( \newcommand\Ie{\it I.e.\/}\)
\( \newcommand\viz{\it viz.\/}\)
\( \newcommand\eg{\it e.g.\/}\)
\( \newcommand\Eg{\it E.g.\/}\)
\( \newcommand\dbar{\,{\mathchar'26\mkern-12mu d}} \)
\( \def\sss#1{\scriptscriptstyle #1}\)
\( \def\ss#1{\scriptstyle #1}\)
\( \def\ssr#1{\scriptstyle #1}\)
\( \def\ssf#1{\scriptstyle #1}\)
\( \newcommand\NA{N_{\ssr{\!A}}}\)
\( \newcommand\lala{\langle\!\langle}\)
\( \newcommand\rara{\rangle\!\rangle}\)
\( \newcommand\blan{\big\langle}\)
\( \newcommand\bran{\big\rangle}\)
\( \newcommand\Blan{\Big\langle}\)
\( \newcommand\Bran{\Big\rangle}\)
\( \newcommand\intl{\int\limits}\)
\( \newcommand\half{\frac{1}{2}}\)
\( \newcommand\third{\frac{1}{3}}\)
\( \newcommand\fourth{\frac{1}{4}}\)
\( \newcommand\eighth{\frac{1}{8}}\)
\( \newcommand\uar{\uparrow}\)
\( \newcommand\dar{\downarrow}\)
\( \newcommand\undertext#1{$\underline{\hbox{#1}}$}\)
\( \newcommand\Tra{\mathop{\textsf{Tr}}\,}\)
\( \newcommand\det{\mathop{\textsf{det}}\,}\)
\( \def\tket#1{| #1 \rangle}\)
\( \def\tbra#1{\langle #1|}\)
\( \def\tbraket#1#2{\langle #1 | #2 \rangle}\)
\( \def\texpect#1#2#3{\langle #1 | #2 | #3 \rangle}\)
\( \def\sket#1{| \, #1 \, \rangle}\)
\( \def\sbra#1{\langle \, #1 \, |}\)
\( \def\sbraket#1#2{\langle \, #1 \, | \, #2 \, \rangle}\)
\( \def\sexpect#1#2#3{\langle \, #1 \, | \, #2 \, | \, #3 \, \rangle}\)
\(\def\ket#1{\big| \, #1\, \big\rangle}\)
\( \def\bra#1{\big\langle \, #1 \, \big|}\)
\( \def\braket#1#2{\big\langle \, #1\, \big| \,#2 \,\big\rangle}\)
\( \def\expect#1#2#3{\big\langle\, #1\, \big|\, #2\, \big| \,#3\, \big\rangle}\)
\( \newcommand\pz{\partial}\)
\( \newcommand\pzb{\bar{\partial}}\)
\( \newcommand\svph{\vphantom{\int}}\)
\( \newcommand\vph{\vphantom{\sum_i}}\)
\( \newcommand\bvph{\vphantom{\sum_N^N}}\)
\( \newcommand\nd{^{\vphantom{\dagger}}}\)
\( \newcommand\ns{^{\vphantom{*}}}\)
\( \newcommand\yd{^\dagger}\)
\( \newcommand\zb{\bar z}\)
\( \newcommand\zdot{\dot z}\)
\( \newcommand\zbdot{\dot{\bar z}}\)
\( \newcommand\kB{k_{\sss{B}}}\)
\( \newcommand\kT{k_{\sss{B}}T}\)
\( \newcommand\gtau{g_\tau}\)
\( \newcommand\Htil{\tilde H}\)
\( \newcommand\pairo{(\phi\nd_0,J\nd_0)}\)
\( \newcommand\pairm{(\phi\nd_0,J)}\)
\( \newcommand\pairob{(\Bphi\nd_0,\BJ\nd_0)}\)
\( \newcommand\pairmb{(\Bphi\nd_0,\BJ)}\)
\( \newcommand\pair{(\phi,J)}\)
\( \newcommand\Hz{H\nd_0}\)
\( \newcommand\Ho{H\nd_1}\)
\( \newcommand\Htz{\Htil\nd_0}\)
\( \newcommand\Hto{\Htil\nd_1}\)
\( \newcommand\oc{\omega_\Rc}\)
\(\newcommand \gtwid{\approx}\)
\( \newcommand\index{\textsf{ind}}\)
\( \newcommand\csch{\,{ csch\,}}\)
\( \newcommand\ctnh{\,{ ctnh\,}}\)
\( \newcommand\ctn{\,{ ctn\,}}\)
\( \newcommand\sgn{\,{ sgn\,}}\)
\( \def\tmapright#1{\xrightarrow \limits^{#1}}\)
\( \def\bmapright#1{\xrightarrow\limits_{#1}}\)
\( \newcommand\hfb{\hfill\break}\)
\( \newcommand\Rep{\textsf{Re}\,}\)
\( \newcommand\Imp{\textsf{Im}\,}\)
\( \newcommand\ncdot{\!\cdot\!}\)
\( \def\tmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits^{#1}}\ }\)
\( \def\bmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits_{#1}}\ }\)
\( \newcommand\bsqcap{\mbox{\boldmath{$\sqcap$}}}\)
\( \def\spabc#1#2#3{\big({\pz #1\over\pz #2}\big)\ns_{\!#3}}\)
\( \def\qabc#1#2#3{\pz^2\! #1\over\pz #2\,\pz #3}\)
\( \def\rabc#1#2#3#4{(\pz #1,\pz #2)\over (\pz #3,\pz #4)}\)
\( \newcommand\subA{\ns_\ssr{A}}\)
\( \newcommand\subB{\ns_\ssr{B}}\)
\( \newcommand\subC{\ns_\ssr{C}}\)
\( \newcommand\subD{\ns_\ssr{D}}\)
\( \newcommand\subAB{\ns_\ssr{AB}}\)
\( \newcommand\subBC{\ns_\ssr{BC}}\)
\( \newcommand\subCD{\ns_\ssr{CD}}\)
\( \newcommand\subDA{\ns_\ssr{DA}}\)
\( \def\lmapright#1{\ \ \smash{\mathop{\hbox to 55pt{\rightarrowfill}}\limits^{#1}}\ \ }\)
\( \def\enth#1{\RDelta {\textsf H}^0_\Rf[{ #1}]}\)
\( \newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{$\longrightharpoonup\joinrel$}\crcr \lower1pt\hbox{$\longleftharpoondown\joinrel$}}}}}}\)
\( \newcommand\longrightharpoonup{\relbar\joinrel\rightharpoonup}\)
\( \newcommand\longleftharpoondown{\leftharpoondown\joinrel\relbar}\)
\( \newcommand\cds{\,\bullet\,}\)
\( \newcommand\ccs{\,\circ\,}\)
\( \newcommand\nsub{_{\vphantom{\dagger}}}\)
\( \newcommand\rhohat{\hat\rho}\)
\( \newcommand\vrhhat{\hat\vrh}\)
\( \newcommand\impi{\int\limits_{-\infty}^\infty\!\!\!}\)
\( \newcommand\brangle{\big\rangle}\)
\( \newcommand\blangle{\big\langle}\)
\( \newcommand\vet{\tilde\ve}\)
\( \newcommand\zbar{\bar z}\)
\( \newcommand\ftil{\tilde f}\)
\( \newcommand\XBE{\RXi\ns_\ssr{BE}}\)
\( \newcommand\XFD{\RXi\ns_\ssr{FD}}\)
\( \newcommand\OBE{\Omega\ns_\ssr{BE}}\)
\( \newcommand\OFD{\Omega\ns_\ssr{FD}}\)
\( \newcommand\veF{\ve\ns_\RF}\)
\( \newcommand\kF{k\ns_\RF}\)
\( \newcommand\kFu{k\ns_{\RF\uar}}\)
\( \newcommand\SZ{\textsf Z}}\) \( \newcommand\kFd{k\ns_{\RF\dar}\)
\( \newcommand\muB{\mu\ns_\ssr{B}}\)
\( \newcommand\mutB{\tilde\mu}\ns_\ssr{B}\)
\( \newcommand\xoN{\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N}\)
\( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
\( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
\( \newcommand\xhihOZ
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\jhz{\HJ(0)}\)
\( \newcommand\nda{\nd_\alpha}\)
\( \newcommand\ndap{\nd_{\alpha'}}\)
\( \newcommand\labar
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)/02:_Thermodynamics/2.09:_Equilibrium_and_Stability), /content/body/p/span, line 1, column 23
\( \newcommand\msa{m\ns_\ssr{A}}\)
\( \newcommand\msb{m\ns_\ssr{B}}\)
\( \newcommand\mss{m\ns_\Rs}\)
\( \newcommand\HBx{\hat\Bx}\)
\( \newcommand\HBy{\hat\By}\)
\( \newcommand\HBz{\hat\Bz}\)
\( \newcommand\thm{\theta\ns_m}\)
\( \newcommand\thp{\theta\ns_\phi}\)
\( \newcommand\mtil{\widetilde m}\)
\( \newcommand\phitil{\widetilde\phi}\)
\( \newcommand\delf{\delta\! f}\)
\( \newcommand\coll{\bigg({\pz f\over\pz t}\bigg)\nd_{\! coll}}\)
\( \newcommand\stre{\bigg({\pz f\over\pz t}\bigg)\nd_{\! str}}\)
\( \newcommand\idrp{\int\!\!{d^3\!r\,d^3\!p\over h^3}\>}\)
\( \newcommand\vbar{\bar v}\)
\( \newcommand\BCE{\mbox{\boldmath{$\CE$}}\!}\)
\( \newcommand\BCR{\mbox{\boldmath{$\CR$}}\!}\)
\( \newcommand\gla{g\nd_{\RLambda\nd}}\)
\( \newcommand\TA{T\ns_\ssr{A}}\)
\( \newcommand\TB{T\ns_\ssr{B}}\)
\( \newcommand\ncdot{\!\cdot\!}\)
\( \newcommand\NS{N\ns_{\textsf S}}\)
Equilibrium
Suppose we have two systems, A and B, which are free to exchange energy, volume, and particle number, subject to overall conservation rules
\[E_A+E_B=E\quad,\quad V_A+V_B=V\quad,\quad N_A+N_B=N\ ,\]
where \(E\), \(V\), and \(N\) are fixed. Now let us compute the change in the total entropy of the combined systems when they are allowed to exchange energy, volume, or particle number. We assume that the entropy is additive,
\[\begin{aligned} dS&=\left[\pabc{S_A}{E_A}{V_A,N_A}- \pabc{S_B}{E_B}{V_B,N_B}\right]dE_A + \left[\pabc{S_A}{V_A}{E_A,N_A}- \pabc{S_B}{V_B}{E_B,N_B}\right]dV_A \nonumber\\ &\qquad\qquad\qquad + \left[\pabc{S_A}{N_A}{E_A,V_A}- \pabc{S_B}{N_B}{E_B,V_B}\right]dN_A\ .\end{aligned}\]
Note that we have used \(dE_B=-dE_A\), \(dV_B=-dV_A\), and \(dN_B=-dN_A\). Now we know from the Second Law that spontaneous processes result in \(T\,dS>0\), which means that \(S\) tends to a maximum. If \(S\) is a maximum, it must be that the coefficients of \(dE_A\), \(dV_A\), and \(dN_A\) all vanish, else we could increase the total entropy of the system by a judicious choice of these three differentials. From \(T\,dS=dE + p\,dV - \mu,dN\), we have
\[{1\over T}=\pabc{S}{E}{V,N} \quad,\quad {p\over T}=\pabc{S}{V}{E,N}\quad,\quad {\mu\over T}=-\pabc{S}{N}{E,V}\ .\]
Thus, we conclude that in order for the system to be in equilibrium, so that \(S\) is maximized and can increase no further under spontaneous processes, we must have
\[\begin{aligned} && T_A&=T_B & & \hbox{(thermal equilibrium)} && \\ && {p_A\over T_A}&={p_B\over T_B} & & \hbox{(mechanical equilibrium)} && \\ && {\mu_A\over T_A}&={\mu_B\over T_B} & & \hbox{(chemical equilibrium)} && \end{aligned}\]
Stability
Next, consider a uniform system with energy \(E'=2E\), volume \(V'=2V\), and particle number \(N'=2N\). We wish to check that this system is not unstable with respect to spontaneously becoming inhomogeneous. To that end, we imagine dividing the system in half. Each half would have energy \(E\), volume \(V\), and particle number \(N\). But suppose we divided up these quantities differently, so that the left half had slightly different energy, volume, and particle number than the right, as depicted in Figure \(\PageIndex{1}\) . Does the entropy increase or decrease? We have
\[\begin{aligned} \RDelta S &=S(E+\RDelta E, V+\RDelta V, N+\RDelta N) + S(E-\RDelta E, V-\RDelta V, N-\RDelta N)-S(2E,2V,2N)\nonumber\\ &= {\pz^2 \!S\over \pz E^2}\,(\RDelta E)^2 +{\pz^2 \!S\over \pz V^2}\,(\RDelta V)^2 +{\pz^2 \!S\over \pz N^2}\,(\RDelta N)^2 \\ &\qquad\qquad\qquad +2\,{\pz^2\!S\over\pz E\,\pz V}\>\RDelta E\>\RDelta V +2\,{\pz^2\!S\over\pz E\,\pz N}\>\RDelta E\>\RDelta N + 2\,{\pz^2\!S\over\pz V\,\pz N}\>\RDelta V\,\RDelta N\ .\nonumber \end{aligned}\]
Thus, we can write
\[\RDelta S=\sum_{i,j} Q\ns_{ij}\,\Psi\ns_i\,\Psi\ns_j\ ,\]
where
\[Q=\begin{pmatrix} {\pz^2 \!S\over \pz E^2} & {\pz^2\!S\over\pz E\,\pz V} & {\pz^2\!S\over\pz E\,\pz N} \\ && \\ {\pz^2\!S\over\pz E\,\pz V} & {\pz^2 \!S\over \pz V^2} & {\pz^2\!S\over\pz V\,\pz N} \\ && \\ {\pz^2\!S\over\pz E\,\pz N} & {\pz^2\!S\over\pz V\,\pz N} & {\pz^2 \!S\over \pz N^2} \end{pmatrix}\]
is the matrix of second derivatives, known in mathematical parlance as the Hessian, and \(\BPsi=(\RDelta E,\RDelta V,\RDelta N)\). Note that \(Q\) is a symmetric matrix.
Since \(S\) must be a maximum in order for the system to be in equilibrium, we are tempted to conclude that the homogeneous system is stable if and only if all three eigenvalues of \(Q\) are negative. If one or more of the eigenvalues is positive, then it is possible to choose a set of variations \(\BPsi\) such that \(\RDelta S>0\), which would contradict the assumption that the homogeneous state is one of maximum entropy. A matrix with this restriction is said to be negative definite. While it is true that \(Q\) can have no positive eigenvalues, it is clear from homogeneity of \(S(E,V,N)\) that one of the three eigenvalues must be zero, corresponding to the eigenvector \(\BPsi=(E,V,N)\). Homogeneity means \(S(\lambda E,\lambda V,\lambda N)=\lambda S(E,V,N)\). Now let us take \(\lambda=1+\eta\), where \(\eta\) is infinitesimal. Then \(\RDelta E = \eta E\), \(\RDelta V = \eta V\), and \(\RDelta N = \eta N\), and homogeneity says \(S(E\pm\RDelta E,V\pm\RDelta V,N\pm\RDelta N)=(1\pm\eta) \,S(E,V,N)\) and \(\RDelta S=(1+\eta)S+(1-\eta)S-2S=0\). We then have a slightly weaker characterization of \(Q\) as negative semidefinite.
However, if we fix one of the components of \((\RDelta E,\RDelta V, \RDelta N)\) to be zero, then \(\BPsi\) must have some component orthogonal to the zero eigenvector, in which case \(\RDelta S < 0\). Suppose we set \(\RDelta N=0\) and we just examine the stability with respect to inhomogeneities in energy and volume. We then restrict our attention to the upper left \(2\times 2\) submatrix of \(Q\). A general symmetric \(2\times 2\) matrix may be written
\[Q=\begin{pmatrix} a & b \\ b & c \end{pmatrix}\]
It is easy to solve for the eigenvalues of \(Q\). One finds
\[\lambda\ns_\pm=\left({a+c\over 2}\right) \pm\sqrt{\left({a-c\over 2}\right)^{\!\!2} + b^2}\ .\]
In order for \(Q\) to be negative definite, we require \(\lambda\ns_+ < 0\) and \(\lambda\ns_- < 0\). Thus, \(\Tra Q = a+c = \lambda\ns_+ + \lambda\ns_- < 0\) and \(\det Q=ac-b^2=\lambda\ns_+\,\lambda\ns_- > 0\). Taken together, these conditions require
\[a<0 \qquad,\qquad c<0 \qquad,\qquad ac>b^2\ .\]
Going back to thermodynamic variables, this requires
\[{\pz^2\!S\over\pz E^2}<0 \qquad,\qquad {\pz^2\!S\over\pz V^2}<0 \qquad,\qquad {\pz^2\!S\over\pz E^2} \cdot {\pz^2\!S\over\pz V^2}>\bigg({\pz^2\!S\over\pz E\,\pz V}\bigg)^{\!\!2}\ .\]
Thus the entropy is a concave function of \(E\) and \(V\) at fixed \(N\). Had we set \(\RDelta E=0\) and considered the lower right \(2\times 2\) submatrix of \(Q\), we’d have concluded that \(S(V,N)\) is concave at fixed \(E\). Since \(\big({\pz S\over\pz E}\big)\ns_V=T^{-1}\), we have \({\pz^2 \!S\over\pz E^2} = -{1\over T^2}\big({\pz T\over \pz E}\big)\ns_{V}=-{C\ns_V \over T^2}<0\) and we conclude \(C\ns_V>0\) for stability.
Many thermodynamic systems are held at fixed \((T,p,N)\), which suggests we examine the stability criteria for \(G(T,p,N)\). Suppose our system is in equilibrium with a reservoir at temperature \(T\ns_0\) and pressure \(p\ns_0\). Then, suppressing \(N\) (which is assumed constant), we have
\[G(T\ns_0,p\ns_0)=E-T\ns_0\,S + p\ns_0\,V\ .\]
Now suppose there is a fluctuation in the entropy and the volume of our system, which is held at fixed particle number. Going to second order in \(\RDelta S\) and \(\RDelta V\), we have
\[\begin{split} \RDelta G&=\Bigg[\pabc{E}{S}{V} - T\ns_0\Bigg]\,\RDelta S + \Bigg[\pabc{E}{V}{S} + p\ns_0\Bigg]\,\RDelta V\\ &\qquad\qquad + {1\over 2}\Bigg[ {\pz^2\! E\over\pz S^2}\,(\RDelta S)^2 + 2\,{\pz^2\! E\over\pz S\,\pz V}\,\RDelta S\,\RDelta V +{\pz^2\! E\over\pz V^2}\,(\RDelta V)^2\Bigg] + \ldots\ . \end{split}\]
Equilibrium requires that the coefficients of \(\RDelta S\) and \(\RDelta V\) both vanish, that \(T=\big({\pz E\over\pz S}\big)\ns_{V,N}=T\ns_0\) and \(p=-\big({\pz E\over\pz V}\big)\ns_{S,N}=p\ns_0\) . The condition for stability is that \(\RDelta G > 0\) for all \((\RDelta S,\RDelta V)\). Stability therefore requires that the Hessian matrix \(Q\) be positive definite, with
\[Q=\begin{pmatrix} {\pz^2\! E\over\pz S^2} & {\pz^2\! E\over\pz S\,\pz V} \\ & \\ {\pz^2\! E\over\pz S\,\pz V} & {\pz^2\! E\over\pz V^2} \end{pmatrix}\ .\]
Thus, we have the following three conditions:
\[\begin{aligned} {\pz^2\! E\over\pz S^2} = \pabc{T}{S}{V} = {T\over C\ns_V} & > 0\\ {\pz^2\! E\over\pz V^2} = -\pabc{p}{V}{S} = {1\over V \kappa\ns_S} & > 0 \bvph \\ {\pz^2\! E\over\pz S^2}\cdot {\pz^2\! E\over\pz V^2} - \left({\pz^2\! E\over\pz S\,\pz V}\right)^{\!\!2} = {T\over V\kappa\ns_S\,C\ns_V} - \left({\pz T\over\pz V}\right)^{\!\!2}_{\!\!S} & >0\ .\end{aligned}\]
As we shall discuss below, the quantity \(\alpha\ns_S\equiv{1\over V}\big({\pz V\over\pz T}\big)\ns_{S,N}\) is the adiabatic thermal expansivity coefficient. We therefore conclude that stability of any thermodynamic system requires
\[{C\ns_V\over T} > 0 \qquad,\qquad \kappa\ns_S > 0 \qquad,\qquad \alpha\ns_S > \sqrt{\kappa\ns_S\,C\ns_V\over VT}.\]