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- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07%3A_Mean_Field_Theory_of_Phase_Transitions/7.09%3A_Appendix_I-_Equivalence_of_the_Mean_Field_DescriptionsWe cannot solve these nonlinear equations analytically, but they may be recast, by exponentiating them, as \[x\ns_\alpha={1\over Z}\,\exp\Bigg\{{1\over \theta}\,\bigg[\sum_p\sum_{\alpha'} A_p\, \lambd...We cannot solve these nonlinear equations analytically, but they may be recast, by exponentiating them, as x\nsα=1Zexp{1θ[∑p∑α′Apλp(s\nsα)λp(s\nsα′)x\nsα′+\vphi(s\nsα)]} , with \[Z=e^{(\zeta/\theta)+1}=\sum_\alpha\exp\Bigg\{{1\over \theta}\,\bigg[\sum_p\sum_{\alpha'} A_p\, \lambda_p(s\ns_\alpha)\,\lambda_p(s\ns_{\alpha'})\,x\ns_{\alpha'}+ \vphi(s\ns_\alph…
- https://phys.libretexts.org/Under_Construction/Arovas_Texts/Book%3A_Superconductivity_(Arovas)Thumbnail: A magnet levitating above a high-temperature superconductor, cooled with liquid nitrogen. Persistent electric current flows on the surface of the superconductor, acting to exclude the magne...Thumbnail: A magnet levitating above a high-temperature superconductor, cooled with liquid nitrogen. Persistent electric current flows on the surface of the superconductor, acting to exclude the magnetic field of the magnet (Faraday's law of induction). This current effectively forms an electromagnet that repels the magnet. (CC BY-SA 2.0 Generic; Mai-Linh Doan via Wikipedia)
- https://phys.libretexts.org/Under_Construction/Arovas_Texts/07%3A_Electronic_Materials
- https://phys.libretexts.org/Under_Construction/Arovas_Texts/Book%3A_Nonlinear_Dynamics_(Arovas)/05%3A_HAMILTONIAN_MECHANICS
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04%3A_Statistical_Ensembles/4.05%3A_Grand_Canonical_Ensemble_(GCE)\[\begin{split} \ln P\ns_n&=\ln D\ns_{\ssr{W}} (E\ns_{\ssr{U}}-E\ns_n\,,\, V\ns_{\ssr{U}}-V\ns_{\!n}) - \ln D\ns_{\ssr{U}} (E\ns_{\ssr{U}},V\ns_{\ssr{U}})\\ &=\ln D\ns_{\ssr{W}} (E\ns_{\ssr{U}},V\ns_{...\[\begin{split} \ln P\ns_n&=\ln D\ns_{\ssr{W}} (E\ns_{\ssr{U}}-E\ns_n\,,\, V\ns_{\ssr{U}}-V\ns_{\!n}) - \ln D\ns_{\ssr{U}} (E\ns_{\ssr{U}},V\ns_{\ssr{U}})\\ &=\ln D\ns_{\ssr{W}} (E\ns_{\ssr{U}},V\ns_{\ssr{U}})- \ln D\ns_{\ssr{U}}(E\ns_{\ssr{U}},V\ns_{\ssr{U}})\bvph\\ &\qquad - E\ns_n\>{\pz\ln D\ns_{\ssr{W}}(E,V)\over\pz E}\bigg|\nd_{E=E\ns_{\ssr{U}}\atop V=V\ns_{\ssr{U}}} - V\ns_{\!n}\>{\pz\ln D\ns_{\ssr{W}}(E,V)\over\pz V}\bigg|\nd_{E=E\ns_{\ssr{U}}\atop V=V\ns_{\ssr{U}}} \!\! + \ldots\\ &\equ…
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09%3A_RenormalizationThumbnail: RG flow for the Ising model with nearest and nearest neighbor interactions.
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08%3A_Nonequilibrium_Phenomena/8.10%3A_Appendix_I-_Boltzmann_Equation_and_Collisional_InvariantsPlugging the conveniently parameterized form of ψ(\Bv,t) into \CL, we have \[\begin{split} \CL\psi&=-\gamma\sum_{r\ell m} a_{r\ell m}(t)\>S^r_{\ell +\half}(x) \>x^{\ell/2}\>Y^\ell_m(\nhat)\...Plugging the conveniently parameterized form of ψ(\Bv,t) into \CL, we have \[\begin{split} \CL\psi&=-\gamma\sum_{r\ell m} a_{r\ell m}(t)\>S^r_{\ell +\half}(x) \>x^{\ell/2}\>Y^\ell_m(\nhat)\ +\ {\gamma\over2\pi^{3/2}}\,\sum_{r\ell m}a_{r\ell m}(t)\!\! \int\limits_0^\infty\!\!dx\nd_1\,x_1^{1/2}\,e^{-x\nd_1}\\ &\qquad\times\int\!\!d\nhat\ns_1 \Big[1+2\,x^{1/2} x_1^{1/2}\,\nhat\!\cdot\!\nhat\ns_1+ \frac{2}{3}\big(x-\frac{3}{2}\big)\big(x\nd_1-\frac{3}{2}\big)\Big]\,S^r_{\ell +\half}(x\nd…
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09%3A_Renormalization/9.S%3A_Summary\newcommand\Sa{\textsf a} \newcommand\Sd{\textsf d} \newcommand\Sz{\textsf z} \newcommand\SA{\textsf A} \newcommand\SD{\textsf D} \newcommand\SZ{\textsf Z} \( \newcommand... \newcommand\Sa{\textsf a} \newcommand\Sd{\textsf d} \newcommand\Sz{\textsf z} \newcommand\SA{\textsf A} \newcommand\SD{\textsf D} \newcommand\SZ{\textsf Z} \newcommand\Da{\dot a} \newcommand\Dz{\dot z} \newcommand\DA{\dot A} \newcommand\zbdot{\dot{\bar z}} \def\enth#1{\RDelta {\textsf H}^0_\Rf[{ #1}]} \newcommand\SZ{\textsf Z}} \( \newcommand\kFd{k\ns_{\RF\dar}\) \newcommand\idrp{\int\!\!{d^3\!r\,d^3\!p\over h^3}\>}
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04%3A_Statistical_Ensembles/4.S%3A_SummaryThe differential of E is defined to be dE=T\,dS - p\,dV + \mu\,dN, thus T=\pabc{E}{S}{\,V,N} is the temperature, p=-\pabc{E}{V}{\,S,N} is the pressure, and \mu=\pabc{E}{N}{\,S,V} i...The differential of E is defined to be dE=T\,dS - p\,dV + \mu\,dN, thus T=\pabc{E}{S}{\,V,N} is the temperature, p=-\pabc{E}{V}{\,S,N} is the pressure, and \mu=\pabc{E}{N}{\,S,V} is the chemical potential. In applying Equation [EminusTS] to the denominator of Equation [PEOCE], we shift \CE' by E and integrate over the difference \delta\CE'\equiv\CE'-E, retaining terms up to quadratic order in \delta\CE' in the argument of the exponent.↩
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/01%3A_Fundamentals_of_Probability/1.02%3A_Basic_Concepts_in_Probability_TheoryHere we recite the basics of probability theory.
- https://phys.libretexts.org/Under_Construction/Arovas_Texts/04%3A_Nonequilibrium_Statistical_Physics
