Phase Transitions  

In General > s.a. Catastrophe; complexity; critical phenomena; history of physics; symmetry breaking.
* Idea: A thermodynamic state such that a small change around it causes a thermodynamic quantity to vary discontinuously.
* First-order: The discontinuity happens in a first derivative of a thermodynamic potential (for example, in the volume V = ∂G/∂p for the water-vapor transition); In the real physical process, the transition doesn't happen all at once, there is a latent heat and phases may coexist.
* Second-order: The discontinuity happens in a second derivative of some thermodynamic potential (for example, in the susceptibility χ for a ferromagnet); The transition happens all at once, with no coexistence or latent heat, and the energy density is C0 in T, not C1; Fluctuations occur at all scales and correlations are scale invariant; When such a phase transition occurs at absolute zero, quantum mechanics comes into play, giving a quantum phase transition (and a quantum critical point) where the interactions have infinite range not just in space but also in time.
* Continuous: Like second-order, but smoother; The energy density is C1 in T.
* Remark: If one uses the canonical ensemble, a true phase transition can only be defined in the thermodynamic limit of infinite system size.
* Cause: Most phase transitions are a result of thermal fluctuations; Quantum ones are different in that they are caused by fluctuations allowed by the Heisenberg uncertainty principle and can happen at or near 0 K.
@ Texts: Brout 65; Careri 84; Stanley 87; Yeomans 92; Goldenfeld 93; Kadanoff 00; Onuki 02; Gitterman & Halpern 04; Hillert 07; Zinn-Justin 07 [and renormalization group]; Uzunov 10; Nishimori & Ortiz 11; Gitterman 13; Fultz 14 [III].
@ General references: Ruelle CMP(77) [heuristic]; Lebowitz RMP(99)mp/00; Tobochnik AJP(01)mar [critical phenomena, renormalization, RL]; Kadanoff JSP(09); Bangu PhSc(09)oct [conceptual]; Kadanoff a1002; Nogueira a1009-ln [field-theoretic methods]; Medved' et al EJP(13) [do-it-yourself modeling]; Singh a1402 [and mean-field theories and renormalization, pedagogical]; Hendi et al a1706 [new approach, motivated by quantum gravity].
@ Critical exponents: Brout PRP(74); Cardy JPA(99)cm/98 [near fractal boundary]; Kumar & Sarkar PRE(14)-a1405 [new geometric critical exponents].
@ Non-equilibrium: Wattis & Coveney JPA(01), JPA(01) [and renormalization]; Koverda & Skokov PhyA(05) [fluctuations]; Hinrichsen PhyA(06) [intro]; Henkel et al 08, Henkel & Pleimling 10; > s.a. Percolation.
@ First-order: Binder RPP(87); Zheng JPA(02) [short-time dynamics]; Gross cm/05 [and microcanonical statistics].
@ And configuration-space topology: Franzosi et al mp/03, NPB(07)mp/05; Kastner PhyA(06), RMP(08); Gori et al a1706; > s.a. XY Chain.
@ Continuous: Schwartz JPA(03) [Fokker-Planck operator]; Sarig CMP(06) [for dynamical systems].
@ Other types: Tolédano & Dmitriev 96 [reconstructive phase transitions].
@ Related topics: Brokate & Sprekels 96 [hysteresis]; Latora et al PhyD(99)cd/98-conf [microscopic chaos]; Oppenheim et al PRL(03)qp/02 [and information]; Franzosi & Pettini PRL(04)cm/03 [origin], NPB(07)mp/05; Kholodenko & Ballard PhyA(07) [Ginzburg-Landau equations from Hilbert–Einstein action]; Franco et al PRD(10)-a0911 [holographic approach]; Maslov TMP(10) [and superfluid transition]; Alhambra et al PRX(16)-a1504 [probability of a thermodynamically forbidden transition].
> And configuration-space metric: see thermodynamics [thermodynamic curvature]; types of metrics [information geometry].

Examples and Analogs > s.a. Crumpling; Freezing; Glass; lattice field theory; magnetism; posets; spheres [packings]; water.
* First examples: Evaporation; Melting; Sublimation (shrinking of ice cubes in freezer, dry ice, ink in printing).
* Mermin-Wagner theorem: A 2D continuous system cannot undergo an order-disorder phase transition at finite T.
@ Fluids, other: Wilding AJP(01)nov [numerical]; Barmatz et al RMP(07) [experiments in microgravity]; Arinshtein TMP(07) [liquid-crystal]; Fabrizio JMP(08) [ice-water and liquid-vapor, in Ginzburg-Landau model]; Radin NAMS-a1209 [fluid-solid transition, for mathematicians]; Brazhkin & Trachenko PT(12)nov [liquid-gas distinction, microscopic]; Shimizu et al PRL(14) [liquid-to-liquid phase transition in triphenyl phosphate]; > s.a. Superfluids.
@ Liquid crystals: Verma PLA(96) [Monte Carlo]; Singh PRP(00).
@ Spin systems: Costin et al JSP(90) [infinite-order]; Biskup in(09)mp/06; Sadhukhan et al PRE(15)-a1412 [fluctuations and order]; > s.a. ising model; Potts Model.
@ Superconductivity: Watanabe FJMS(09)-a0808 [BCS-Bogoliubov theory, second-order nature].
@ On graphs, networks: Lyons JMP(00)m.PR/99 [graphs]; Goltsev et al PRE(03)cm/02 [networks]; Hartmann & Weigt 05 [statistical mechanics of combinatorial optimization]; Andrecut & Kauffman PLA(08) [random Boolean networks, order-disorder]; Radin a1601 [large combinatorial systems, graphons and permutons]; > s.a. graphs and graph types; networks; XY Chain.
@ In 2D: Mermin & Wagner PRL(66); Barber PRP(80); Naumovets CP(89); Antoni et al cm/99-proc [N-body]; Koibuchi PhyA(11) [triangulated surfaces on a spherical core].
@ Small systems: Borrmann et al PRL(00) [classification]; Gross 01; Dunkel & Hilbert PhyA(06) [canonical and microcanonical].
@ Higher-order: Janke et al NPB(06); Stošić et al PhyA(09) [Ising model on Cayley tree].
@ Condensed-matter systems: Drouffe et al JPA(98) [condensation]; Eggers PRL(99) [in heated sand]; Mayorga et al PhyA(09) [precursors of order and disorder in colloids]; Williams & Ackland PRE(12)-a1212 [sudoku as a model frustrated glassy system]; Fultz 14 [in materials]; Kitagawa Phy(14) [re new phase in solid oxygen]; > s.a. Disordered Systems; Metals [metal-insulator].
@ Other examples: Stanley in(82) [geometric analog]; Fletcher AJP(97)jan [mechanical analog]; Fendley & Tchernyshyov NPB(02)cm [1D]; Biskup & Chayes CMP(03) [discontinuous]; Velasco & Fernández-Pineda AJP(07)dec [triple point]; English EJP(08) [spontaneous synchronization of oscillators]; Meshcherov AP(08) [conducting filament burnout]; Zweig et al PhyA(10) [random k-SAT problems]; Caldarelli Phy(12) [longevity/volatility of rankings]; Fläschner et al a1608 [dynamical topological phase transition].
> Examples in gravity: see black holes; event horizons; inflation; lovelock gravity; regge calculus.
> Other examples: see cellular automata; differentiable manifolds; elements [Si melts when cooled]; Gross-Neveu Model; ideal gas [relativistic]; metamaterials [jamming transition]; molecular physics [polymers]; quantum phase transitions [field theory and early-universe cosmology]; random tilings; topological defects; Van der Waals fluid.

Related Topics > s.a. casimir effect [critical]; Lee-Yang Theory; scale invariance.
* Applications: A phase transition is the basis for the operation of a Cloud Chamber; For a quantum analog, see Huang et al PRA(09)-a0902.
@ And symmetry breaking: Gill CP(98); Baroni & Casetti JPA(06) [topological conditions]; Del Giudice & Vitiello PRA(06)cm [electromagnetic field and matter, phase locking]; Wen ISRN(13)-a1210 [topological order and phases]; del Campo & Zurek IJMPA(14)-a1310 [Kibble-Zurek mechanism and density of defects].
> Other related topics: see Clausius-Clapeyron Equation; Critical Points; Hysteresis; Order Parameter; Topological Materials; Universality.

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