$$\require{cancel}$$

# 7.5: Landau Theory of Phase Transitions

$$\newcommand\bes{\begin{split}}$$
$$\newcommand\ltwid{\propto}$$
$$\newcommand\ees{\end{split}}$$
$$\newcommand\mib{\mathbf}$$
$$\newcommand\Sa{\textsf a}$$
$$\newcommand\Sb{\textsf b}$$
$$\newcommand\Sc{\textsf c}$$
$$\newcommand\Sd{\textsf d}$$
$$\newcommand\Se{\textsf e}$$
$$\newcommand\Sf{\textsf f}$$
$$\newcommand\Sg{\textsf g}$$
$$\newcommand\Sh{\textsf h}$$
$$\newcommand\Si{\textsf i}$$
$$\newcommand\Sj{\textsf j}$$
$$\newcommand\Sk{\textsf k}$$
$$\newcommand\Sl{\textsf l}$$
$$\newcommand\Sm{\textsf m}$$
$$\newcommand\Sn{\textsf n}$$
$$\newcommand\So{\textsf o}$$
$$\newcommand\Sp{\textsf p}$$
$$\newcommand\Sq{\textsf q}$$
$$\newcommand\Sr{\textsf r}$$
$$\newcommand\Ss{\textsf s}$$
$$\newcommand\St{\textsf t}$$
$$\newcommand\Su{\textsf u}$$
$$\newcommand\Sv{\textsf v}$$
$$\newcommand\Sw{\textsf w}$$
$$\newcommand\Sx{\textsf x}$$
$$\newcommand\Sy{\textsf y}$$
$$\newcommand\Sz{\textsf z}$$
$$\newcommand\SA{\textsf A}$$
$$\newcommand\SB{\textsf B}$$
$$\newcommand\SC{\textsf C}$$
$$\newcommand\SD{\textsf D}$$
$$\newcommand\SE{\textsf E}$$
$$\newcommand\SF{\textsf F}$$
$$\newcommand\SG{\textsf G}$$
$$\newcommand\SH{\textsf H}$$
$$\newcommand\SI{\textsf I}$$
$$\newcommand\SJ{\textsf J}$$
$$\newcommand\SK{\textsf K}$$
$$\newcommand\SL{\textsf L}$$
$$\newcommand\SM{\textsf M}$$
$$\newcommand\SN{\textsf N}$$
$$\newcommand\SO{\textsf O}$$
$$\newcommand\SP{\textsf P}$$
$$\newcommand\SQ{\textsf Q}$$
$$\newcommand\SR{\textsf R}$$
$$\newcommand\SS{\textsf S}$$
$$\newcommand\ST{\textsf T}$$
$$\newcommand\SU{\textsf U}$$
$$\newcommand\SV{\textsf V}$$
$$\newcommand\SW{\textsf W}$$
$$\newcommand\SX{\textsf X}$$
$$\newcommand\SY{\textsf Y}$$
$$\newcommand\SZ{\textsf Z}$$
$$\newcommand\Ha{\hat a}$$
$$\newcommand\Hb{\hat b}$$
$$\newcommand\Hc{\hat c}$$
$$\newcommand\Hd{\hat d}$$
$$\newcommand\He{\hat e}$$
$$\newcommand\Hf{\hat f}$$
$$\newcommand\Hg{\hat g}$$
$$\newcommand\Hh{\hat h}$$
$$\newcommand\Hi{\hat \imath}$$
$$\newcommand\Hj{\hat \jmath}$$
$$\newcommand\Hk{\hat k}$$
$$\newcommand\Hl{\hat l}$$
$$\newcommand\Hm{\hat m}$$
$$\newcommand\Hn{\hat n}$$
$$\newcommand\Ho{\hat o}$$
$$\newcommand\Hp{\hat p}$$
$$\newcommand\Hq{\hat q}$$
$$\newcommand\Hr{\hat r}$$
$$\newcommand\Hs{\hat s}$$
$$\newcommand\Ht{\hat t}$$
$$\newcommand\Hu{\hat u}$$
$$\newcommand\Hv{\hat v}$$
$$\newcommand\Hw{\hat w}$$
$$\newcommand\Hx{\hat x}$$
$$\newcommand\Hy{\hat y}$$
$$\newcommand\Hz{\hat z}$$
$$\newcommand\HA{\hat A}$$
$$\newcommand\HB{\hat B}$$
$$\newcommand\HC{\hat C}$$
$$\newcommand\HD{\hat D}$$
$$\newcommand\HE{\hat E}$$
$$\newcommand\HF{\hat F}$$
$$\newcommand\HG{\hat G}$$
$$\newcommand\HH{\hat H}$$
$$\newcommand\HI{\hat I}$$
$$\newcommand\HJ{\hat J}$$
$$\newcommand\HK{\hat K}$$
$$\newcommand\HL{\hat L}$$
$$\newcommand\HM{\hat M}$$
$$\newcommand\HN{\hat N}$$
$$\newcommand\HO{\hat O}$$
$$\newcommand\HP{\hat P}$$
$$\newcommand\HQ{\hat Q}$$
$$\newcommand\HR{\hat R}$$
$$\newcommand\HS{\hat S}$$
$$\newcommand\HT{\hat T}$$
$$\newcommand\HU{\hat U}$$
$$\newcommand\HV{\hat V}$$
$$\newcommand\HW{\hat W}$$
$$\newcommand\HX{\hat X}$$
$$\newcommand\HY{\hat Y}$$
$$\newcommand\HZ{\hat Z}$$
$$\newcommand\Halpha{\hat\alpha}$$
$$\newcommand\Hbeta{\hat\beta}$$
$$\newcommand\Hgamma{\hat\gamma}$$
$$\newcommand\Hdelta{\hat\delta}$$
$$\newcommand\Hepsilon{\hat\epsilon}$$
$$\newcommand\Hvarepsilon{\hat\varepsilon}$$
$$\newcommand\Hzeta{\hat\zeta}$$
$$\newcommand\Heta{\hat\eta}$$
$$\newcommand\Htheta{\hat\theta}$$
$$\newcommand\Hvartheta{\hat\vartheta}$$
$$\newcommand\Hiota{\hat\iota}$$
$$\newcommand\Hkappa{\hat\kappa}$$
$$\newcommand\Hlambda{\hat\lambda}$$
$$\newcommand\Hmu{\hat\mu}$$
$$\newcommand\Hnu{\hat\nu}$$
$$\newcommand\Hxi{\hat\xi}$$
$$\newcommand\Hom{\hat\omicron}$$
$$\newcommand\Hpi{\hat\pi}$$
$$\newcommand\Hvarpi{\hat\varpi}$$
$$\newcommand\Hrho{\hat\rho}$$
$$\newcommand\Hvarrho{\hat\varrho}$$
$$\newcommand\Hsigma{\hat\sigma}$$
$$\newcommand\Hvarsigma{\hat\varsigma}$$
$$\newcommand\Htau{\var\tau}$$
$$\newcommand\Hupsilon{\hat\upsilon}$$
$$\newcommand\Hphi{\hat\phi}$$
$$\newcommand\Hvarphi{\hat\varphi}$$
$$\newcommand\Hchi{\hat\chi}$$
$$\newcommand\Hxhi{\hat\xhi}$$
$$\newcommand\Hpsi{\hat\psi}$$
$$\newcommand\Homega{\hat\omega}$$
$$\newcommand\HGamma{\hat\Gamma}$$
$$\newcommand\HDelta{\hat\Delta}$$
$$\newcommand\HTheta{\hat\Theta}$$
$$\newcommand\HLambda{\hat\Lambda}$$
$$\newcommand\HXi{\hat\Xi}$$
$$\newcommand\HPi{\hat\Pi}$$
$$\newcommand\HSigma{\hat\Sigma}$$
$$\newcommand\HUps{\hat\Upsilon}$$
$$\newcommand\HPhi{\hat\Phi}$$
$$\newcommand\HPsi{\hat\Psi}$$
$$\newcommand\HOmega{\hat\Omega}$$
$$\newcommand\xhat{\hat\Bx}$$
$$\newcommand\yhat{\hat\By}$$
$$\newcommand\zhat{\hat\Bz}$$
$$\newcommand\ehat{\hat\Be}$$
$$\newcommand\khat{\hat\Bk}$$
$$\newcommand\nhat{\hat\Bn}$$
$$\newcommand\rhat{\hat\Br}$$
$$\newcommand\phihat{\hat\Bphi}$$
$$\newcommand\thetahat{\hat\Btheta}$$
$$\newcommand\MA{\mathbb A}$$
$$\newcommand\MB{\mathbb B}$$
$$\newcommand\MC{\mathbb C}$$
$$\newcommand\MD{\mathbb D}$$
$$\newcommand\ME{\mathbb E}$$
$$\newcommand\MF{\mathbb F}$$
$$\newcommand\MG{\mathbb G}$$
$$\newcommand\MH{\mathbb H}$$
$$\newcommand\MI{\mathbb I}$$
$$\newcommand\MJ{\mathbb J}$$
$$\newcommand\MK{\mathbb K}$$
$$\newcommand\ML{\mathbb L}$$
$$\newcommand\MM{\mathbb M}$$
$$\newcommand\MN{\mathbb N}$$
$$\newcommand\MO{\mathbb O}$$
$$\newcommand\MP{\mathbb P}$$
$$\newcommand\MQ{\mathbb Q}$$
$$\newcommand\MR{\mathbb R}$$
$$\newcommand\MS{\mathbb S}$$
$$\newcommand\MT{\mathbb T}$$
$$\newcommand\MU{\mathbb U}$$
$$\newcommand\MV{\mathbb V}$$
$$\newcommand\MW{\mathbb W}$$
$$\newcommand\MX{\mathbb X}$$
$$\newcommand\MY{\mathbb Y}$$
$$\newcommand\MZ{\mathbb Z}$$
$$\newcommand\CA{\mathcal A}$$
$$\newcommand\CB{\mathcal B}$$
$$\newcommand\CC{\mathcal C}$$
$$\newcommand\CD{\mathcal D}$$
$$\newcommand\CE{\mathcal E}$$
$$\newcommand\CF{\mathcal F}$$
$$\newcommand\CG{\mathcal G}$$
$$\newcommand\CH{\mathcal H}$$
$$\newcommand\CI{\mathcal I}$$
$$\newcommand\CJ{\mathcal J}$$
$$\newcommand\CK{\mathcal K}$$
$$\newcommand\CL{\mathcal L}$$
$$\newcommand\CM{\mathcal M}$$
$$\newcommand\CN{\mathcal N}$$
$$\newcommand\CO{\mathcal O}$$
$$\newcommand\CP{\mathcal P}$$
$$\newcommand\CQ{\mathcal Q}$$
$$\newcommand\CR{\mathcal R}$$
$$\newcommand\CS{\mathcal S}$$
$$\newcommand\CT{\mathcal T}$$
$$\newcommand\CU{\mathcal U}$$
$$\newcommand\CV{\mathcal V}$$
$$\newcommand\CW{\mathcal W}$$
$$\newcommand\CX{\mathcal X}$$
$$\newcommand\CY{\mathcal Y}$$
$$\newcommand\CZ{\mathcal Z}$$
$$\newcommand\Fa{\mathfrak a}$$
$$\newcommand\Fb{\mathfrak b}$$
$$\newcommand\Fc{\mathfrak c}$$
$$\newcommand\Fd{\mathfrak d}$$
$$\newcommand\Fe{\mathfrak e}$$
$$\newcommand\Ff{\mathfrak f}$$
$$\newcommand\Fg{\mathfrak g}$$
$$\newcommand\Fh{\mathfrak h}$$
$$\newcommand\Fi{\mathfrak i}$$
$$\newcommand\Fj{\mathfrak j}$$
$$\newcommand\Fk{\mathfrak k}$$
$$\newcommand\Fl{\mathfrak l}$$
$$\newcommand\Fm{\mathfrak m}$$
$$\newcommand\Fn{\mathfrak n}$$
$$\newcommand\Fo{\mathfrak o}$$
$$\newcommand\Fp{\mathfrak p}$$
$$\newcommand\Fq{\mathfrak q}$$
$$\newcommand\Fr{\mathfrak r}$$
$$\newcommand\Fs{\mathfrak s}$$
$$\newcommand\Ft{\mathfrak t}$$
$$\newcommand\Fu{\mathfrak u}$$
$$\newcommand\Fv{\mathfrak v}$$
$$\newcommand\Fw{\mathfrak w}$$
$$\newcommand\Fx{\mathfrak x}$$
$$\newcommand\Fy{\mathfrak y}$$
$$\newcommand\Fz{\mathfrak z}$$
$$\newcommand\FA{\mathfrak A}$$
$$\newcommand\FB{\mathfrak B}$$
$$\newcommand\FC{\mathfrak C}$$
$$\newcommand\FD{\mathfrak D}$$
$$\newcommand\FE{\mathfrak E}$$
$$\newcommand\FF{\mathfrak F}$$
$$\newcommand\FG{\mathfrak G}$$
$$\newcommand\FH{\mathfrak H}$$
$$\newcommand\FI{\mathfrak I}$$
$$\newcommand\FJ{\mathfrak J}$$
$$\newcommand\FK{\mathfrak K}$$
$$\newcommand\FL{\mathfrak L}$$
$$\newcommand\FM{\mathfrak M}$$
$$\newcommand\FN{\mathfrak N}$$
$$\newcommand\FO{\mathfrak O}$$
$$\newcommand\FP{\mathfrak P}$$
$$\newcommand\FQ{\mathfrak Q}$$
$$\newcommand\FR{\mathfrak R}$$
$$\newcommand\FS{\mathfrak S}$$
$$\newcommand\FT{\mathfrak T}$$
$$\newcommand\FU{\mathfrak U}$$
$$\newcommand\FV{\mathfrak V}$$
$$\newcommand\FW{\mathfrak W}$$
$$\newcommand\FX{\mathfrak X}$$
$$\newcommand\FY{\mathfrak Y}$$
$$\newcommand\FZ{\mathfrak Z}$$
$$\newcommand\Da{\dot a}$$
$$\newcommand\Db{\dot b}$$
$$\newcommand\Dc{\dot c}$$
$$\newcommand\Dd{\dot d}$$
$$\newcommand\De{\dot e}$$
$$\newcommand\Df{\dot f}$$
$$\newcommand\Dg{\dot g}$$
$$\newcommand\Dh{\dot h}$$
$$\newcommand\Di{\dot \imath}$$
$$\newcommand\Dj{\dot \jmath}$$
$$\newcommand\Dk{\dot k}$$
$$\newcommand\Dl{\dot l}$$
$$\newcommand\Dm{\dot m}$$
$$\newcommand\Dn{\dot n}$$
$$\newcommand\Do{\dot o}$$
$$\newcommand\Dp{\dot p}$$
$$\newcommand\Dq{\dot q}$$
$$\newcommand\Dr{\dot r}$$
$$\newcommand\Ds{\dot s}$$
$$\newcommand\Dt{\dot t}$$
$$\newcommand\Du{\dot u}$$
$$\newcommand\Dv{\dot v}$$
$$\newcommand\Dw{\dot w}$$
$$\newcommand\Dx{\dot x}$$
$$\newcommand\Dy{\dot y}$$
$$\newcommand\Dz{\dot z}$$
$$\newcommand\DA{\dot A}$$
$$\newcommand\DB{\dot B}$$
$$\newcommand\DC{\dot C}$$
$$\newcommand\DD{\dot D}$$
$$\newcommand\DE{\dot E}$$
$$\newcommand\DF{\dot F}$$
$$\newcommand\DG{\dot G}$$
$$\newcommand\DH{\dot H}$$
$$\newcommand\DI{\dot I}$$
$$\newcommand\DJ{\dot J}$$
$$\newcommand\DK{\dot K}$$
$$\newcommand\DL{\dot L}$$
$$\newcommand\DM{\dot M}$$
$$\newcommand\DN{\dot N}$$
$$\newcommand\DO{\dot O}$$
$$\newcommand\DP{\dot P}$$
$$\newcommand\DQ{\dot Q}$$
$$\newcommand\DR{\dot R}$$
$$\newcommand\DS{\dot S}$$
$$\newcommand\DT{\dot T}$$
$$\newcommand\DU{\dot U}$$
$$\newcommand\DV{\dot V}$$
$$\newcommand\DW{\dot W}$$
$$\newcommand\DX{\dot X}$$
$$\newcommand\DY{\dot Y}$$
$$\newcommand\DZ{\dot Z}$$
$$\newcommand\Dalpha ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[1], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Dbeta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[2], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Dgamma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[3], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Ddelta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[4], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Depsilon ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[5], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Dvarepsilon ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[6], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Dzeta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[7], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Deta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[8], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Dtheta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[9], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Dvartheta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[10], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Diota ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[11], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Dkappa ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[12], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Dlambda ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[13], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Dmu{\dot\mu}$$
$$\newcommand\Dnu{\dot\nu}$$
$$\newcommand\Dxi{\dot\xi}$$
$$\newcommand\Dom{\dot\omicron}$$
$$\newcommand\Dpi{\dot\pi}$$
$$\newcommand\Dvarpi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[14], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Drho{\dot\rho}$$
$$\newcommand\Dvarrho{\dot\varrho}$$
$$\newcommand\Dsigma{\dot\sigma}$$
$$\newcommand\Dvarsigma{\dot\varsigma}$$
$$\newcommand\Dtau{\var\tau}$$
$$\newcommand\Dupsilon{\dot\upsilon}$$
$$\newcommand\Dphi{\dot\phi}$$
$$\newcommand\Dvarphi{\dot\varphi}$$
$$\newcommand\Dchi{\dot\chi}$$
$$\newcommand\Dpsi{\dot\psi}$$
$$\newcommand\Domega{\dot\omega}$$
$$\newcommand\DGamma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[15], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\DDelta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\DTheta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[17], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\DLambda{\dot\Lambda}$$
$$\newcommand\DXi{\dot\Xi}$$
$$\newcommand\DPi{\dot\Pi}$$
$$\newcommand\DSigma{\dot\Sigma}$$
$$\newcommand\DUps{\dot\Upsilon}$$
$$\newcommand\DPhi{\dot\Phi}$$
$$\newcommand\DPsi{\dot\Psi}$$
$$\newcommand\DOmega{\dot\Omega}$$
$$\newcommand\Va{\vec a}$$
$$\newcommand\Vb{\vec b}$$
$$\newcommand\Vc{\vec c}$$
$$\newcommand\Vd{\vec d}$$
$$\newcommand\Ve{\vec e}$$
$$\newcommand\Vf{\vec f}$$
$$\newcommand\Vg{\vec g}$$
$$\newcommand\Vh{\vec h}$$
$$\newcommand\Vi{\vec \imath}$$
$$\newcommand\Vj{\vec \jmath}$$
$$\newcommand\Vk{\vec k}$$
$$\newcommand\Vl{\vec l}$$
$$\newcommand\Vm{\vec m}$$
$$\newcommand\Vn{\vec n}$$
$$\newcommand\Vo{\vec o}$$
$$\newcommand\Vp{\vec p}$$
$$\newcommand\Vq{\vec q}$$
$$\newcommand\Vr{\vec r}$$
$$\newcommand\Vs{\vec s}$$
$$\newcommand\Vt{\vec t}$$
$$\newcommand\Vu{\vec u}$$
$$\newcommand\Vv{\vec v}$$
$$\newcommand\Vw{\vec w}$$
$$\newcommand\Vx{\vec x}$$
$$\newcommand\Vy{\vec y}$$
$$\newcommand\Vz{\vec z}$$
$$\newcommand\VA{\vec A}$$
$$\newcommand\VB{\vec B}$$
$$\newcommand\VC{\vec C}$$
$$\newcommand\VD{\vec D}$$
$$\newcommand\VE{\vec E}$$
$$\newcommand\VF{\vec F}$$
$$\newcommand\VG{\vec G}$$
$$\newcommand\VH{\vec H}$$
$$\newcommand\VI{\vec I}$$
$$\newcommand\VJ{\vec J}$$
$$\newcommand\VK{\vec K}$$
$$\newcommand\VL{\vec L}$$
$$\newcommand\VM{\vec M}$$
$$\newcommand\VN{\vec N}$$
$$\newcommand\VO{\vec O}$$
$$\newcommand\VP{\vec P}$$
$$\newcommand\VQ{\vec Q}$$
$$\newcommand\VR{\vec R}$$
$$\newcommand\VS{\vec S}$$
$$\newcommand\VT{\vec T}$$
$$\newcommand\VU{\vec U}$$
$$\newcommand\VV{\vec V}$$
$$\newcommand\VW{\vec W}$$
$$\newcommand\VX{\vec X}$$
$$\newcommand\VY{\vec Y}$$
$$\newcommand\VZ{\vec Z}$$
$$\newcommand\Valpha{\vec\alpha}$$
$$\newcommand\Vbeta{\vec\beta}$$
$$\newcommand\Vgamma{\vec\gamma}$$
$$\newcommand\Vdelta{\vec\delta}$$
$$\newcommand\Vepsilon{\vec\epsilon}$$
$$\newcommand\Vvarepsilon{\vec\varepsilon}$$
$$\newcommand\Vzeta{\vec\zeta}$$
$$\newcommand\Veta{\vec\eta}$$
$$\newcommand\Vtheta{\vec\theta}$$
$$\newcommand\Vvartheta{\vec\vartheta}$$
$$\newcommand\Viota{\vec\iota}$$
$$\newcommand\Vkappa{\vec\kappa}$$
$$\newcommand\Vlambda{\vec\lambda}$$
$$\newcommand\Vmu ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[18], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vnu ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[19], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vxi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vom ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vpi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vvarpi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vrho ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vvarrho ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vsigma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vvarsigma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vtau ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vupsilon ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[29], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vphi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[30], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vvarphi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[31], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vchi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[32], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vpsi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[33], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vomega ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[34], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\VGamma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[35], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\VDelta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[36], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\VTheta{\vec\Theta}$$
$$\newcommand\VLambda{\vec\Lambda}$$
$$\newcommand\VXi{\vec\Xi}$$
$$\newcommand\VPi{\vec\Pi}$$
$$\newcommand\VSigma{\vec\Sigma}$$
$$\newcommand\VUps{\vec\Upsilon}$$
$$\newcommand\VPhi{\vec\Phi}$$
$$\newcommand\VPsi{\vec\Psi}$$
$$\newcommand\VOmega{\vec\Omega}$$
$$\newcommand\BA{\mib A}$$
$$\newcommand\BB{\mib B}$$
$$\newcommand\BC{\mib C}$$
$$\newcommand\BD{\mib D}$$
$$\newcommand\BE{\mib E}$$
$$\newcommand\BF{\mib F}$$
$$\newcommand\BG{\mib G}$$
$$\newcommand\BH{\mib H}$$
$$\newcommand\BI{\mib I}}$$
$$\newcommand\BJ{\mib J}$$
$$\newcommand\BK{\mib K}$$
$$\newcommand\BL{\mib L}$$
$$\newcommand\BM{\mib M}$$
$$\newcommand\BN{\mib N}$$
$$\newcommand\BO{\mib O}$$
$$\newcommand\BP{\mib P}$$
$$\newcommand\BQ{\mib Q}$$
$$\newcommand\BR{\mib R}$$
$$\newcommand\BS{\mib S}$$
$$\newcommand\BT{\mib T}$$
$$\newcommand\BU{\mib U}$$
$$\newcommand\BV{\mib V}$$
$$\newcommand\BW{\mib W}$$
$$\newcommand\BX{\mib X}$$
$$\newcommand\BY{\mib Y}$$
$$\newcommand\BZ{\mib Z}$$
$$\newcommand\Ba{\mib a}$$
$$\newcommand\Bb{\mib b}$$
$$\newcommand\Bc{\mib c}$$
$$\newcommand\Bd{\mib d}$$
$$\newcommand\Be{\mib e}$$
$$\newcommand\Bf{\mib f}$$
$$\newcommand\Bg{\mib g}$$
$$\newcommand\Bh{\mib h}$$
$$\newcommand\Bi{\mib i}$$
$$\newcommand\Bj{\mib j}$$
$$\newcommand\Bk{\mib k}$$
$$\newcommand\Bl{\mib l}$$
$$\newcommand\Bm{\mib m}$$
$$\newcommand\Bn{\mib n}$$
$$\newcommand\Bo{\mib o}$$
$$\newcommand\Bp{\mib p}$$
$$\newcommand\Bq{\mib q}$$
$$\newcommand\Br{\mib r}$$
$$\newcommand\Bs{\mib s}$$
$$\newcommand\Bt{\mib t}$$
$$\newcommand\Bu{\mib u}$$
$$\newcommand\Bv{\mib v}$$
$$\newcommand\Bw{\mib w}$$
$$\newcommand\Bx{\mib x}$$
$$\newcommand\By{\mib y}$$
$$\newcommand\Bz{\mib z}$$\)
$$\newcommand\vrh{\varrho}$$
$$\newcommand\vsig{\varsigma}$$
$$\newcommand\ups{\upsilon}$$
$$\newcommand\eps{\epsilon}$$
$$\newcommand\ve{\varepsilon}$$
$$\newcommand\vth{\vartheta}$$
$$\newcommand\vphi{\varphi}$$
$$\newcommand\xhi{\chi}$$
$$\newcommand\Ups{\Upsilon}$$
$$\newcommand\Balpha{\mib\alpha}$$
$$\newcommand\Bbeta{\mib\beta}$$
$$\newcommand\Bgamma{\mib\gamma}$$
$$\newcommand\Bdelta{\mib\delta}$$
$$\newcommand\Beps{\mib\epsilon}$$
$$\newcommand\Bve{\mib\varepsilon}$$
$$\newcommand\Bzeta{\mib\zeta}$$
$$\newcommand\Beta{\mib\eta}$$
$$\newcommand\Btheta{\mib\theta}$$
$$\newcommand\Bvth{\mib\vartheta}$$
$$\newcommand\Biota{\mib\iota}$$
$$\newcommand\Bkappa{\mib\kappa}$$
$$\newcommand\Blambda{\mib\lambda}$$
$$\newcommand\Bmu{\mib\mu}$$
$$\newcommand\Bnu{\mib\nu}$$
$$\newcommand\Bxi{\mib\xi}$$
$$\newcommand\Bom{\mib\omicron}$$
$$\newcommand\Bpi{\mib\pi}$$
$$\newcommand\Bvarpi{\mib\varpi}$$
$$\newcommand\Brho{\mib\rho}$$
$$\newcommand\Bvrh{\mib\varrho}$$
$$\newcommand\Bsigma{\mib\sigma}$$
$$\newcommand\Bvsig{\mib\varsigma}$$
$$\newcommand\Btau{\mib\tau}$$
$$\newcommand\Bups{\mib\upsilon}$$
$$\newcommand\Bphi{\mib\phi}$$
$$\newcommand\Bvphi{\mib\vphi}$$
$$\newcommand\Bchi{\mib\chi}$$
$$\newcommand\Bpsi{\mib\psi}$$
$$\newcommand\Bomega{\mib\omega}$$
$$\newcommand\BGamma{\mib\Gamma}$$
$$\newcommand\BDelta{\mib\Delta}$$
$$\newcommand\BTheta{\mib\Theta}$$
$$\newcommand\BLambda{\mib\Lambda}$$
$$\newcommand\BXi{\mib\Xi}$$
$$\newcommand\BPi{\mib\Pi}$$
$$\newcommand\BSigma{\mib\Sigma}$$
$$\newcommand\BUps{\mib\Upsilon}$$
$$\newcommand\BPhi{\mib\Phi}$$
$$\newcommand\BPsi{\mib\Psi}$$
$$\newcommand\BOmega{\mib\Omega}$$
$$\newcommand\Bxhi{\raise.35ex\hbox{\Bchi}}$$
$$\newcommand\RGamma{ \Gamma}$$
$$\newcommand\RDelta{ \Delta}$$
$$\newcommand\RTheta{ \Theta}$$
$$\newcommand\RLambda{ \Lambda}$$
$$\newcommand\RXi{ \Xi}$$
$$\newcommand\RPi{ \Pi}$$
$$\newcommand\RSigma{ \Sigma}$$
$$\newcommand\RUps{ \Upsilon}$$
$$\newcommand\RPhi{ \Phi}$$
$$\newcommand\RPsi{ \Psi}$$
$$\newcommand\ROmega{ \Omega}$$
$$\newcommand\RA{ A}$$
$$\newcommand\RB{ B}$$
$$\newcommand\RC{ C}$$
$$\newcommand\RD{ D}$$
$$\newcommand\RE{ E}$$
$$\newcommand\RF{ F}$$
$$\newcommand\RG{ G}$$
$$\newcommand\RH{ H}$$
$$\newcommand\RI{ I}$$
$$\newcommand\RJ{ J}$$
$$\newcommand\RK{ K}$$
$$\newcommand\RL{ L}$$
$$\newcommand { M}$$
$$\newcommand\RN{ N}$$
$$\newcommand\RO{ O}$$
$$\newcommand\RP{ P}$$
$$\newcommand\RQ{ Q}$$
$$\newcommand\RR{ R}$$
$$\newcommand\RS{ S}$$
$$\newcommand\RT{ T}$$
$$\newcommand\RU{ U}$$
$$\newcommand\RV{ V}$$
$$\newcommand\RW{ W}$$
$$\newcommand\RX{ X}$$
$$\newcommand\RY{ Y}$$
$$\newcommand\RZ{ Z}$$
$$\newcommand\Ra{ a}$$
$$\newcommand\Rb{ b}$$
$$\newcommand\Rc{ c}$$
$$\newcommand\Rd{ d}$$
$$\newcommand\Re{ e}$$
$$\newcommand\Rf{ f}$$
$$\newcommand\Rg{ g}$$
$$\newcommand\Rh{ h}$$
$$\newcommand\Ri{ i}$$
$$\newcommand\Rj{ j}$$
$$\newcommand\Rk{ k}$$
$$\newcommand\Rl{ l}$$
$$\newcommand { m}$$
$$\newcommand\Rn{ n}$$
$$\newcommand\Ro{ o}$$
$$\newcommand\Rp{ p}$$
$$\newcommand\Rq{ q}$$
$$\newcommand\Rr{ r}$$
$$\newcommand\Rs{ s}$$
$$\newcommand\Rt{ t}$$
$$\newcommand\Ru{ u}$$
$$\newcommand\Rv{ v}$$
$$\newcommand\Rw{ w}$$
$$\newcommand\Rx{ x}$$
$$\newcommand\Ry{ y}$$
$$\newcommand\Rz{ z}$$
$$\newcommand\BBA{\boldsymbol\RA}$$
$$\newcommand\BBB{\boldsymbol\RB}$$
$$\newcommand\BBC{\boldsymbol\RC}$$
$$\newcommand\BBD{\boldsymbol\RD}$$
$$\newcommand\BBE{\boldsymbol\RE}$$
$$\newcommand\BBF{\boldsymbol\RF}$$
$$\newcommand\BBG{\boldsymbol\RG}$$
$$\newcommand\BBH{\boldsymbol\RH}$$
$$\newcommand\BBI{\boldsymbol\RI}$$
$$\newcommand\BBJ{\boldsymbol\RJ}$$
$$\newcommand\BBK{\boldsymbol\RK}$$
$$\newcommand\BBL{\boldsymbol\RL}$$
$$\newcommand\BBM{\boldsymbol }$$
$$\newcommand\BBN{\boldsymbol\RN}$$
$$\newcommand\BBO{\boldsymbol\RO}$$
$$\newcommand\BBP{\boldsymbol\RP}$$
$$\newcommand\BBQ{\boldsymbol\RQ}$$
$$\newcommand\BBR{\boldsymbol\RR}$$
$$\newcommand\BBS{\boldsymbol\RS}$$
$$\newcommand\BBT{\boldsymbol\RT}$$
$$\newcommand\BBU{\boldsymbol\RU}$$
$$\newcommand\BBV{\boldsymbol\RV}$$
$$\newcommand\BBW{\boldsymbol\RW}$$
$$\newcommand\BBX{\boldsymbol\RX}$$
$$\newcommand\BBY{\boldsymbol\RY}$$
$$\newcommand\BBZ{\boldsymbol\RZ}$$
$$\newcommand\BBa{\boldsymbol\Ra}$$
$$\newcommand\BBb{\boldsymbol\Rb}$$
$$\newcommand\BBc{\boldsymbol\Rc}$$
$$\newcommand\BBd{\boldsymbol\Rd}$$
$$\newcommand\BBe{\boldsymbol\Re}$$
$$\newcommand\BBf{\boldsymbol\Rf}$$
$$\newcommand\BBg{\boldsymbol\Rg}$$
$$\newcommand\BBh{\boldsymbol\Rh}\}$$
$$\newcommand\BBi{\boldsymbol\Ri}$$
$$\newcommand\BBj{\boldsymbol\Rj}$$
$$\newcommand\BBk{\boldsymbol\Rk}$$
$$\newcommand\BBl{boldsymbol\Rl}$$
$$\newcommand\BBm{\boldsymbol }$$
$$\newcommand\BBn{\boldsymbol\Rn}$$
$$\newcommand\BBo{\boldsymbol\Ro}$$
$$\newcommand\BBp{\boldsymbol\Rp}$$
$$\newcommand\BBq{\boldsymbol\Rq}$$
$$\newcommand\BBr{\boldsymbol\Rr}$$
$$\newcommand\BBs{\boldsymbol\Rs}$$
$$\newcommand\BBt{\boldsymbol\Rt}$$
$$\newcommand\BBu{\boldsymbol\Ru}$$
$$\newcommand\BBv{\boldsymbol\Rv}$$
$$\newcommand\BBw{\boldsymbol\Rw}$$
$$\newcommand\BBx{\boldsymbol\Rx}$$
$$\newcommand\BBy{\boldsymbol\Ry}$$
$$\newcommand\BBz{\boldsymbol\Rz}$$
$$\newcommand\tcb{\textcolor{blue}$$
$$\newcommand\tcr{\textcolor{red}$$
$$\newcommand\bnabla{\boldsymbol{\nabla}}$$
$$\newcommand\Bell{\boldsymbol\ell}$$
$$\newcommand\dbar{\,{\mathchar'26\mkern-12mu d}}$$
$$\newcommand\ns{^\vphantom{*}}$$
$$\newcommand\uar{\uparrow}$$
$$\newcommand\dar{\downarrow}$$
$$\newcommand\impi{\int\limits_{-\infty}^{\infty}\!\!}$$
$$\newcommand\izpi{\int\limits_{0}^{\infty}\!\!}$$
$$\newcommand\etc{\it etc.\/}$$
$$\newcommand\etal{\it et al.\/}$$
$$\newcommand\opcit{\it op. cit.\/}$$
$$\newcommand\ie{\it i.e.\/}$$
$$\newcommand\Ie{\it I.e.\/}$$
$$\newcommand\viz{\it viz.\/}$$
$$\newcommand\eg{\it e.g.\/}$$
$$\newcommand\Eg{\it E.g.\/}$$
$$\newcommand\dbar{\,{\mathchar'26\mkern-12mu d}}$$
$$\def\sss#1{\scriptscriptstyle #1}$$
$$\def\ss#1{\scriptstyle #1}$$
$$\def\ssr#1{\scriptstyle #1}$$
$$\def\ssf#1{\scriptstyle #1}$$
$$\newcommand\NA{N_{\ssr{\!A}}}$$
$$\newcommand\lala{\langle\!\langle}$$
$$\newcommand\rara{\rangle\!\rangle}$$
$$\newcommand\blan{\big\langle}$$
$$\newcommand\bran{\big\rangle}$$
$$\newcommand\Blan{\Big\langle}$$
$$\newcommand\Bran{\Big\rangle}$$
$$\newcommand\intl{\int\limits}$$
$$\newcommand\half{\frac{1}{2}}$$
$$\newcommand\third{\frac{1}{3}}$$
$$\newcommand\fourth{\frac{1}{4}}$$
$$\newcommand\eighth{\frac{1}{8}}$$
$$\newcommand\uar{\uparrow}$$
$$\newcommand\dar{\downarrow}$$
$$\newcommand\undertext#1{\underline{\hbox{#1}}}$$
$$\newcommand\Tra{\mathop{\textsf{Tr}}\,}$$
$$\newcommand\det{\mathop{\textsf{det}}\,}$$
$$\def\tket#1{| #1 \rangle}$$
$$\def\tbra#1{\langle #1|}$$
$$\def\tbraket#1#2{\langle #1 | #2 \rangle}$$
$$\def\texpect#1#2#3{\langle #1 | #2 | #3 \rangle}$$
$$\def\sket#1{| \, #1 \, \rangle}$$
$$\def\sbra#1{\langle \, #1 \, |}$$
$$\def\sbraket#1#2{\langle \, #1 \, | \, #2 \, \rangle}$$
$$\def\sexpect#1#2#3{\langle \, #1 \, | \, #2 \, | \, #3 \, \rangle}$$
$$\def\ket#1{\big| \, #1\, \big\rangle}$$
$$\def\bra#1{\big\langle \, #1 \, \big|}$$
$$\def\braket#1#2{\big\langle \, #1\, \big| \,#2 \,\big\rangle}$$
$$\def\expect#1#2#3{\big\langle\, #1\, \big|\, #2\, \big| \,#3\, \big\rangle}$$
$$\newcommand\pz{\partial}$$
$$\newcommand\pzb{\bar{\partial}}$$
$$\newcommand\svph{\vphantom{\int}}$$
$$\newcommand\vph{\vphantom{\sum_i}}$$
$$\newcommand\bvph{\vphantom{\sum_N^N}}$$
$$\newcommand\nd{^{\vphantom{\dagger}}}$$
$$\newcommand\ns{^{\vphantom{*}}}$$
$$\newcommand\yd{^\dagger}$$
$$\newcommand\zb{\bar z}$$
$$\newcommand\zdot{\dot z}$$
$$\newcommand\zbdot{\dot{\bar z}}$$
$$\newcommand\kB{k_{\sss{B}}}$$
$$\newcommand\kT{k_{\sss{B}}T}$$
$$\newcommand\gtau{g_\tau}$$
$$\newcommand\Htil{\tilde H}$$
$$\newcommand\pairo{(\phi\nd_0,J\nd_0)}$$
$$\newcommand\pairm{(\phi\nd_0,J)}$$
$$\newcommand\pairob{(\Bphi\nd_0,\BJ\nd_0)}$$
$$\newcommand\pairmb{(\Bphi\nd_0,\BJ)}$$
$$\newcommand\pair{(\phi,J)}$$
$$\newcommand\Hz{H\nd_0}$$
$$\newcommand\Ho{H\nd_1}$$
$$\newcommand\Htz{\Htil\nd_0}$$
$$\newcommand\Hto{\Htil\nd_1}$$
$$\newcommand\oc{\omega_\Rc}$$

$$\newcommand \gtwid{\approx}$$

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

$$\def\pabc#1#2#3{\left({\pz #1\over\pz #2}\right)\ns_{\!\!#3}}$$
$$\def\spabc#1#2#3{\big({\pz #1\over\pz #2}\big)\ns_{\!#3}}$$
$$\def\qabc#1#2#3{\pz^2\! #1\over\pz #2\,\pz #3}$$
$$\def\rabc#1#2#3#4{(\pz #1,\pz #2)\over (\pz #3,\pz #4)}$$
$$\newcommand\subA{\ns_\ssr{A}}$$
$$\newcommand\subB{\ns_\ssr{B}}$$
$$\newcommand\subC{\ns_\ssr{C}}$$
$$\newcommand\subD{\ns_\ssr{D}}$$
$$\newcommand\subAB{\ns_\ssr{AB}}$$
$$\newcommand\subBC{\ns_\ssr{BC}}$$
$$\newcommand\subCD{\ns_\ssr{CD}}$$
$$\newcommand\subDA{\ns_\ssr{DA}}$$
$$\def\lmapright#1{\ \ \smash{\mathop{\hbox to 55pt{\rightarrowfill}}\limits^{#1}}\ \ }$$
$$\def\enth#1{\RDelta {\textsf H}^0_\Rf[{ #1}]}$$
\newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{\longrightharpoonup\joinrel}\crcr \lower1pt\hbox{\longleftharpoondown\joinrel}}}}}}
$$\newcommand\longrightharpoonup{\relbar\joinrel\rightharpoonup}$$
$$\newcommand\longleftharpoondown{\leftharpoondown\joinrel\relbar}$$
$$\newcommand\cds{\,\bullet\,}$$
$$\newcommand\ccs{\,\circ\,}$$
$$\newcommand\nsub{_{\vphantom{\dagger}}}$$
$$\newcommand\rhohat{\hat\rho}$$
$$\newcommand\vrhhat{\hat\vrh}$$
$$\newcommand\impi{\int\limits_{-\infty}^\infty\!\!\!}$$
$$\newcommand\brangle{\big\rangle}$$
$$\newcommand\blangle{\big\langle}$$
$$\newcommand\vet{\tilde\ve}$$
$$\newcommand\zbar{\bar z}$$
$$\newcommand\ftil{\tilde f}$$
$$\newcommand\XBE{\RXi\ns_\ssr{BE}}$$
$$\newcommand\XFD{\RXi\ns_\ssr{FD}}$$
$$\newcommand\OBE{\Omega\ns_\ssr{BE}}$$
$$\newcommand\OFD{\Omega\ns_\ssr{FD}}$$
$$\newcommand\veF{\ve\ns_\RF}$$
$$\newcommand\kF{k\ns_\RF}$$
$$\newcommand\kFu{k\ns_{\RF\uar}}$$
$$\newcommand\muB{\mu\ns_\ssr{B}}$$
$$\newcommand\mutB{\tilde\mu}\ns_\ssr{B}$$
$$\newcommand\xoN{\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N}$$
$$\newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}$$
$$\newcommand\xhiOZ{\xhi^\ssr{OZ}}$$
$$\newcommand\xhihOZ ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/span[1], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\jhz{\HJ(0)}$$
$$\newcommand\nda{\nd_\alpha}$$
$$\newcommand\ndap{\nd_{\alpha'}}$$
$$\newcommand\labar ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/span[2], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.05:_Landau_Theory_of_Phase_Transitions), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\msa{m\ns_\ssr{A}}$$
$$\newcommand\msb{m\ns_\ssr{B}}$$
$$\newcommand\mss{m\ns_\Rs}$$
$$\newcommand\HBx{\hat\Bx}$$
$$\newcommand\HBy{\hat\By}$$
$$\newcommand\HBz{\hat\Bz}$$
$$\newcommand\thm{\theta\ns_m}$$
$$\newcommand\thp{\theta\ns_\phi}$$
$$\newcommand\mtil{\widetilde m}$$
$$\newcommand\phitil{\widetilde\phi}$$
$$\newcommand\delf{\delta\! f}$$
$$\newcommand\coll{\bigg({\pz f\over\pz t}\bigg)\nd_{\! coll}}$$
$$\newcommand\stre{\bigg({\pz f\over\pz t}\bigg)\nd_{\! str}}$$
$$\newcommand\idrp{\int\!\!{d^3\!r\,d^3\!p\over h^3}\>}$$
$$\newcommand\vbar{\bar v}$$
$$\newcommand\BCE{\mbox{\boldmath{\CE}}\!}$$
$$\newcommand\BCR{\mbox{\boldmath{\CR}}\!}$$
$$\newcommand\gla{g\nd_{\RLambda\nd}}$$
$$\newcommand\TA{T\ns_\ssr{A}}$$
$$\newcommand\TB{T\ns_\ssr{B}}$$
$$\newcommand\ncdot{\!\cdot\!}$$
$$\newcommand\NS{N\ns_{\textsf S}}$$

Landau’s theory of phase transitions is based on an expansion of the free energy of a thermodynamic system in terms of an order parameter, which is nonzero in an ordered phase and zero in a disordered phase. For example, the magnetization $$M$$ of a ferromagnet in zero external field but at finite temperature typically vanishes for temperatures $$T>T\nd_\Rc$$, where $$T\nd_\Rc$$ is the critical temperature, also called the Curie temperature in a ferromagnet. A low order expansion in powers of the order parameter is appropriate sufficiently close to the phase transition, at temperatures such that the order parameter, if nonzero, is still small.

## Quartic free energy with Ising symmetry

The simplest example is the quartic free energy, $f(m,h=0,\theta)=f_0+\half a m^2 + \fourth b m^4 \ ,$ where $$f_0=f_0(\theta)$$, $$a=a(\theta)$$, and $$b=b(\theta)$$. Here, $$\theta$$ is a dimensionless measure of the temperature. If for example the local exchange energy in the ferromagnet is $$J$$, then we might define $$\theta=\kT/zJ$$, as before. Let us assume $$b>0$$, which is necessary if the free energy is to be bounded from below16. The equation of state , ${\pz f\over\pz m}=0 = am + b m^3\ ,$ has three solutions in the complex $$m$$ plane: (i) $$m=0$$, (ii) $$m=\sqrt{-a/b\,}$$, and (iii) $$m=-\sqrt{-a/b\,}$$. The latter two solutions lie along the (physical) real axis if $$a<0$$. We assume that there exists a unique temperature $$\theta_\Rc$$ where $$a(\theta_\Rc)=0$$. Minimizing $$f$$, we find $\begin{split} \theta < \theta_\Rc \quad &\colon \quad f(\theta)=f_0 - {a^2\over 4b} \\ \theta > \theta_\Rc \quad &\colon \quad f(\theta)=f_0 \ . \end{split}$ The free energy is continuous at $$\theta_\Rc$$ since $$a(\theta_\Rc)=0$$. The specific heat, however, is discontinuous across the transition, with $c\big(\theta_\Rc^+\big)- c\big(\theta_\Rc^-\big) = -\theta_\Rc\,{\pz^2\over\pz \theta^2}\bigg|_{\theta=\theta_\Rc} \bigg({a^2\over 4b}\bigg) = -{\theta_\Rc\,\big[a'(\theta_\Rc)\big]^2\over 2 b(\theta_\Rc)}\ .$

The presence of a magnetic field $$h$$ breaks the $${\mathbb Z}\ns_2$$ symmetry of $$m\to -m$$. The free energy becomes $f(m,h,\theta)=f_0+\half a m^2 + \fourth b m^4 -hm \ ,$ and the mean field equation is $b m^3 + am - h=0\ .$ This is a cubic equation for $$m$$ with real coefficients, and as such it can either have three real solutions or one real solution and two complex solutions related by complex conjugation. Clearly we must have $$a<0$$ in order to have three real roots, since $$bm^3 + am$$ is monotonically increasing otherwise. The boundary between these two classes of solution sets occurs when two roots coincide, which means $$f''(m)=0$$ as well as $$f'(m)=0$$. Simultaneously solving these two equations, we find $h^*(a)=\pm {2\over 3^{3/2}}\,{ (-a)^{3/2}\over b^{1/2}}\ ,$ or, equivalently, $a^*(h)=-{3\over 2^{2/3}}\,b^{1/3}\,|h|^{2/3} .$ If, for fixed $$h$$, we have $$a<a^*(h)$$, then there will be three real solutions to the mean field equation $$f'(m)=0$$, one of which is a global minimum (the one for which $$m\cdot h>0$$). For $$a>a^*(h)$$ there is only a single global minimum, at which $$m$$ also has the same sign as $$h$$. If we solve the mean field equation perturbatively in $$h/a$$, we find $\begin{split} m(a,h)&={h\over a} -{b\over a^4}\,h^3 + \CO(h^5)\hskip1.45in (a>0)\\ &=\pm{|a|^{1/2}\over b^{1/2}} + {h\over 2\,|a|} \pm{3\,b^{1/2}\over 8\, |a|^{5/2}}\>h^2 + \CO(h^3)\qquad (a<0)\ .\bvph \end{split}$

## Cubic terms in Landau theory : first order transitions

Next, consider a free energy with a cubic term, $f=f_0 + \half a m^2 -\frac{1}{3} y m^3 + \fourth b m^4\ ,$ with $$b>0$$ for stability. Without loss of generality, we may assume $$y>0$$ (else send $$m\to -m$$). Note that we no longer have $$m\to -m$$ ( $$\MZ\nd_2$$) symmetry. The cubic term favors positive $$m$$. What is the phase diagram in the $$(a,y)$$ plane?

Extremizing the free energy with respect to $$m$$, we obtain ${\pz f\over\pz m}=0=am -ym^2 + bm^3\ .$ This cubic equation factorizes into a linear and quadratic piece, and hence may be solved simply. The three solutions are $$m=0$$ and $m=m\ns_\pm\equiv{y\over 2b}\pm\sqrt{\Big({y\over 2b}\Big)^2 - {a\over b}}\ .$ We now see that for $$y^2<4ab$$ there is only one real solution, at $$m=0$$, while for $$y^2>4ab$$ there are three real solutions. Which solution has lowest free energy? To find out, we compare the energy $$f(0)$$ with $$f(m\ns_+)$$17. Thus, we set $f(m) = f(0) \quad \Longrightarrow \quad \half a m^2 - \frac{1}{3} y m^3 + \fourth b m^4 = 0\ ,$ and we now have two quadratic equations to solve simultaneously: $\begin{split} 0&=a-ym+bm^2\\ 0&=\half a -\frac{1}{3} y m + \fourth b m^2=0\ . \end{split}$

Eliminating the quadratic term gives $$m=3a/y$$. Finally, substituting $$m=m\ns_+$$ gives us a relation between $$a$$, $$b$$, and $$y$$: $y^2=\frac{9}{2}\,ab\ .$ Thus, we have the following: $\begin{split} a > {y^2\over 4b} \quad &\colon \quad \hbox{1 real root } m=0\\ {y^2\over 4b}>a>{2y^2\over 9b}\quad &\colon\quad \hbox{3 real roots; minimum at } m=0 \\ {2y^2\over 9b} > a \quad &\colon \quad \hbox{3 real roots; minimum at } m={y\over 2b}+\sqrt{\Big({y\over 2b}\Big)^2 - {a\over b}} \end{split}$ The solution $$m=0$$ lies at a local minimum of the free energy for $$a>0$$ and at a local maximum for $$a<0$$. Over the range $${y^2\over 4b}>a>{2y^2\over 9b}$$, then, there is a global minimum at $$m=0$$, a local minimum at $$m=m\ns_+$$, and a local maximum at $$m=m\ns_-$$, with $$m\ns_+>m\ns_- > 0$$. For $${2y^2\over 9b} > a > 0$$, there is a local minimum at $$a=0$$, a global minimum at $$m=m\ns_+$$, and a local maximum at $$m=m\ns_-$$, again with $$m\ns_+ >m\ns_->0$$. For $$a<0$$, there is a local maximum at $$m=0$$, a local minimum at $$m=m\ns_-$$, and a global minimum at $$m=m\ns_+$$, with $$m\ns_+ > 0 > m\ns_-$$. See Figure [quartic].

With $$y=0$$, we have a second order transition at $$a=0$$. With $$y\ne 0$$, there is a discontinuous (first order) transition at $$a\ns_\Rc=2y^2/9b>0$$ and $$m\ns_\Rc=2y/3b$$ . This occurs before $$a$$ reaches the value $$a=0$$ where the curvature at $$m=0$$ turns negative. If we write $$a=\alpha (T-T\ns_0)$$, then the expected second order transition at $$T=T\ns_0$$ is preempted by a first order transition at $$T\ns_\Rc=T\ns_0+2y^2/9\alpha b$$.

## Magnetization dynamics

Suppose we now impose some dynamics on the system, of the simple relaxational type ${\pz m\over\pz t}=-\Gamma\>{\pz f\over\pz m}\ ,$ where $$\Gamma$$ is a phenomenological kinetic coefficient. Assuming $$y>0$$ and $$b>0$$, it is convenient to adimensionalize by writing $m\equiv {y\over b}\cdot u \qquad,\qquad a\equiv {y^2\over b}\cdot r\qquad,\qquad t\equiv {b\over \Gamma y^2}\cdot s\ .$ Then we obtain ${\pz u\over\pz s}=-{\pz \vphi\over\pz u}\ , \label{LBdyn}$ where the dimensionless free energy function is $\vphi(u)=\half r u^2 - \third u^3 + \fourth u^4\ .$ We see that there is a single control parameter, $$r$$. The fixed points of the dynamics are then the stationary points of $$\vphi(u)$$, where $$\vphi'(u)=0$$, with $\vphi'(u)=u\,(r-u+u^2)\ .$ The solutions to $$\vphi'(u)=0$$ are then given by $u^*=0\qquad,\qquad u^*=\half\pm\sqrt{\fourth-r}\ .$ For $$r>\fourth$$ there is one fixed point at $$u=0$$, which is attractive under the dynamics $${\dot u}=-\vphi'(u)$$ since $$\vphi''(0)=r$$. At $$r=\fourth$$ there occurs a saddle-node bifurcation and a pair of fixed points is generated, one stable and one unstable. As we see from Figure [Landau_a], the interior fixed point is always unstable and the two exterior fixed points are always stable. At $$r=0$$ there is a transcritical bifurcation where two fixed points of opposite stability collide and bounce off one another (metaphorically speaking).

At the saddle-node bifurcation, $$r=\fourth$$ and $$u=\half$$, and we find $$\vphi(u=\half;r=\fourth)=\frac{1}{192}$$, which is positive. Thus, the thermodynamic state of the system remains at $$u=0$$ until the value of $$\vphi(u\ns_+)$$ crosses zero. This occurs when $$\vphi(u)=0$$ and $$\vphi'(u)=0$$, the simultaneous solution of which yields $$r=\frac{2}{9}$$ and $$u=\frac{2}{3}$$.

Suppose we slowly ramp the control parameter $$r$$ up and down as a function of the dimensionless time $$s$$. Under the dynamics of Equation [LBdyn], $$u(s)$$ flows to the first stable fixed point encountered – this is always the case for a dynamical system with a one-dimensional phase space. Then as $$r$$ is further varied, $$u$$ follows the position of whatever locally stable fixed point it initially encountered. Thus, $$u\big(r(s)\big)$$ evolves smoothly until a bifurcation is encountered. The situation is depicted by the arrows in Figure [Landau_b]. The equilibrium thermodynamic value for $$u(r)$$ is discontinuous; there is a first order phase transition at $$r=\frac{2}{9}$$, as we’ve already seen. As $$r$$ is increased, $$u(r)$$ follows a trajectory indicated by the magenta arrows. For an negative initial value of $$u$$, the evolution as a function of $$r$$ will be reversible. However, if $$u(0)$$ is initially positive, then the system exhibits hysteresis, as shown. Starting with a large positive value of $$r$$, $$u(s)$$ quickly evolves to $$u=0^+$$, which means a positive infinitesimal value. Then as $$r$$ is decreased, the system remains at $$u=0^+$$ even through the first order transition, because $$u=0$$ is an attractive fixed point. However, once $$r$$ begins to go negative, the $$u=0$$ fixed point becomes repulsive, and $$u(s)$$ quickly flows to the stable fixed point $$u\ns_+=\half + \sqrt{\fourth-r}$$. Further decreasing $$r$$, the system remains on this branch. If $$r$$ is later increased, then $$u(s)$$ remains on the upper branch past $$r=0$$, until the $$u\ns_+$$ fixed point annihilates with the unstable fixed point at $$u\ns_-=\half - \sqrt{\fourth-r}$$, at which time $$u(s)$$ quickly flows down to $$u=0^+$$ again.

## Sixth order Landau theory: tricritical point

Finally, consider a model with $$\MZ\nd_2$$ symmetry, with the Landau free energy $f=f_0 + \half a m^2 + \fourth b m^4 + \frac{1}{6} c m^6\ , \label{sextic}$ with $$c>0$$ for stability. We seek the phase diagram in the $$(a,b)$$ plane. Extremizing $$f$$ with respect to $$m$$, we obtain ${\pz f\over\pz m}=0=m\,(a + bm^2 + cm^4)\ , \label{quintic}$ which is a quintic with five solutions over the complex $$m$$ plane. One solution is obviously $$m=0$$. The other four are $m=\pm\sqrt{-{b\over 2c}\pm\sqrt{\bigg({b\over 2c}\bigg)^2 -{a\over c}}}\ .$ For each $$\pm$$ symbol in the above equation, there are two options, hence four roots in all.

If $$a>0$$ and $$b>0$$, then four of the roots are imaginary and there is a unique minimum at $$m=0$$.

For $$a<0$$, there are only three solutions to $$f'(m)=0$$ for real $$m$$, since the $$-$$ choice for the $$\pm$$ sign under the radical leads to imaginary roots. One of the solutions is $$m=0$$. The other two are $m=\pm\sqrt{-{b\over 2c}+\sqrt{\Big({b\over 2c}\Big)^2 -{a\over c}}}\ .$

The most interesting situation is $$a>0$$ and $$b<0$$. If $$a>0$$ and $$b<-2\sqrt{ac}$$, all five roots are real. There must be three minima, separated by two local maxima. Clearly if $$m^*$$ is a solution, then so is $$-m^*$$. Thus, the only question is whether the outer minima are of lower energy than the minimum at $$m=0$$. We assess this by demanding $$f(m^*)=f(0)$$, where $$m^*$$ is the position of the largest root ( the rightmost minimum). This gives a second quadratic equation, $0= \half a + \fourth b m^2 + \frac{1}{6} cm^4\ ,$ which together with equation [quintic] gives $b=-\frac{4}{\sqrt{3}}\,\sqrt{ac}\ .$ Thus, we have the following, for fixed $$a>0$$: \begin{aligned} b > -2\sqrt{ac} \quad &\colon \quad \hbox{1 real root } m=0\nonumber\\ -2\sqrt{ac} >b>-\frac{4}{\sqrt{3}}\,\sqrt{ac}\quad &\colon\quad \hbox{5 real roots; minimum at } m=0 \\ -\frac{4}{\sqrt{3}}\,\sqrt{ac} > b \quad &\colon \quad \hbox{5 real roots; minima at } m=\pm\sqrt{-{b\over 2c}+\sqrt{\Big({b\over 2c}\Big)^2 -{a\over c}}}\nonumber\end{aligned} The point $$(a,b)=(0,0)$$, which lies at the confluence of a first order line and a second order line, is known as a tricritical point.

## Hysteresis for the sextic potential

Once again, we consider the dissipative dynamics $${\dot m}=-\Gamma\,f'(m)$$. We adimensionalize by writing $m\equiv \sqrt{{|b|\over c}\,}\cdot u \qquad,\qquad a\equiv {b^2\over c}\cdot r \qquad,\qquad t\equiv {c\over\Gamma\,b^2}\cdot s\ .$ Then we obtain once again the dimensionless equation ${\pz u\over\pz s}=-{\pz\vphi\over\pz u}\ ,$ where $\vphi(u)=\half r u^2 \pm \fourth u^4 + \frac{1}{6} u^6\ .$ In the above equation, the coefficient of the quartic term is positive if $$b>0$$ and negative if $$b<0$$. That is, the coefficient is $$\sgn\!(b)$$. When $$b>0$$ we can ignore the sextic term for sufficiently small $$u$$, and we recover the quartic free energy studied earlier. There is then a second order transition at $$r=0$$. .

New and interesting behavior occurs for $$b>0$$. The fixed points of the dynamics are obtained by setting $$\vphi'(u)=0$$. We have $\begin{split} \vphi(u)&=\half r u^2 - \fourth u^4 + \frac{1}{6} u^6\\ \vphi'(u)&=u\,(r-u^2 + u^4)\ . \end{split}$ Thus, the equation $$\vphi'(u)=0$$ factorizes into a linear factor $$u$$ and a quartic factor $$u^4-u^2+r$$ which is quadratic in $$u^2$$. Thus, we can easily obtain the roots: $\begin{split} r<0 \quad &\colon\quad u^*=0 \ ,\ u^*=\pm\sqrt{\half+\sqrt{\fourth - r\>}\>} \\ 0 < r < \fourth \quad &\colon\quad u^*=0 \ ,\ u^*=\pm\sqrt{\half+\sqrt{\fourth - r\>}\>} \ , \ u^*=\pm\sqrt{\half-\sqrt{\fourth - r\>}\>}\bvph \\ r> \fourth \quad &\colon \quad u^*=0\ . \end{split}$

In Figure [Landau_c], we plot the fixed points and the hysteresis loops for this system. At $$r=\fourth$$, there are two symmetrically located saddle-node bifurcations at $$u=\pm\frac{1}{\sqrt{2}}$$. We find $$\vphi(u=\pm\frac{1}{\sqrt{2}} , r=\fourth)=\frac{1}{48}$$, which is positive, indicating that the stable fixed point $$u^*=0$$ remains the thermodynamic minimum for the free energy $$\vphi(u)$$ as $$r$$ is decreased through $$r=\fourth$$. Setting $$\vphi(u)=0$$ and $$\vphi'(u)=0$$ simultaneously, we obtain $$r=\frac{3}{16}$$ and $$u=\pm\frac{\sqrt{3}}{2}$$. The thermodynamic value for $$u$$ therefore jumps discontinuously from $$u=0$$ to $$u=\pm\frac{\sqrt{3}}{2}$$ (either branch) at $$r=\frac{3}{16}$$; this is a first order transition.

Under the dissipative dynamics considered here, the system exhibits hysteresis, as indicated in the figure, where the arrows show the evolution of $$u(s)$$ for very slowly varying $$r(s)$$. When the control parameter $$r$$ is large and positive, the flow is toward the sole fixed point at $$u^*=0$$. At $$r=\fourth$$, two simultaneous saddle-node bifurcations take place at $$u^*=\pm\frac{1}{\sqrt{2}}$$; the outer branch is stable and the inner branch unstable in both cases. At $$r=0$$ there is a subcritical pitchfork bifurcation, and the fixed point at $$u^*=0$$ becomes unstable.

Suppose one starts off with $$r\gg \frac{1}{4}$$ with some value $$u>0$$. The flow $${\dot u}=-\vphi'(u)$$ then rapidly results in $$u\to 0^+$$. This is the ‘high temperature phase’ in which there is no magnetization. Now let $$r$$ increase slowly, using $$s$$ as the dimensionless time variable. The scaled magnetization $$u(s)=u^*\big(r(s)\big)$$ will remain pinned at the fixed point $$u^*=0^+$$. As $$r$$ passes through $$r=\fourth$$, two new stable values of $$u^*$$ appear, but our system remains at $$u=0^+$$, since $$u^*=0$$ is a stable fixed point. But after the subcritical pitchfork, $$u^*=0$$ becomes unstable. The magnetization $$u(s)$$ then flows rapidly to the stable fixed point at $$u^*=\frac{1}{\sqrt{2}}$$, and follows the curve $$u^*(r)=\big(\half+(\fourth-r)^{1/2}\big)^{1/2}$$ for all $$r<0$$.

Now suppose we start increasing $$r$$ ( increasing temperature). The magnetization follows the stable fixed point $$u^*(r)=\big(\half+(\fourth-r)^{1/2}\big)^{1/2}$$ past $$r=0$$, beyond the first order phase transition point at $$r=\frac{3}{16}$$, and all the way up to $$r=\fourth$$, at which point this fixed point is annihilated at a saddle-node bifurcation. The flow then rapidly takes $$u\to u^*=0^+$$, where it remains as $$r$$ continues to be increased further.

Within the region $$r\in \big[0,\fourth\big]$$ of control parameter space, the dynamics are said to be irreversible and the behavior of $$u(s)$$ is said to be hysteretic.