Phase Transitions

In General > s.a. Catastrophe; complexity; critical phenomena; 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 a thermodynamic potential (for example, in the susceptibility for a ferromagnet); The transition happens all at once, with no coexistence or latent heat, energy density is C⁰ in T, not C¹; 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.
* Examples: Evaporation; Melting; Sublimation (shrinking of ice cubes in freezer, dry ice, ink in printing).
* 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.
@ General references: Ruelle CMP(77) [heuristic theory]; Binder RPP(87) [first-order]; Lebowitz RMP(99)mp/00; Tobochnik AJP(01)RL [critical phenomena, renormalization].
@ Critical exponents: Brout PRP(74); Cardy JPA(99)cm/98 [near fractal boundary].
@ Non-equilibrium: Wattis & Coveney JPA(01), JPA(01) [and renormalization]; Koverda & Skokov PhyA(05) [fluctuations]; Hinrichsen PhyA(06) [intro].
@ First-order: 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).
@ 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-in [microscopic chaos]; Oppenheim et al PRL(03)qp/02 [and info]; Franzosi & Pettini PRL(04)cm/03 [origin], NPB(07)mp/05; Janke et al NPB(06) [higher-order]; Kholodenko & Ballard PhyA(07) [Ginzburg-Landau equations from Hilbert–Einstein action].

Examples and Analogs > s.a. Crumpling; event horizons; lattice field theory; magnetism; posets; regge calculus; spheres [packings].
* Mermin-Wagner theorem: A 2D continuous system cannot undergo an order-disorder phase transition at finite T.
@ Fluids: Wilding AJP(01) [numerical]; Donth 01 [liquid-glass, r PT(02)dec]; Kitamura PRP(03) [liquid-glass]; Barmatz et al RMP(07) [experiments in microgravity]; Arinshtein TMP(07) [liquid-crystal]; > s.a. Superfluids.
@ Liquid crystals: Verma PLA(96) [MonteCarlo]; Singh PRP(00).
@ Spin systems: Costin et al JSP(90) [infinite-order]; > s.a. ising model, Potts Model.
@ 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]; > s.a. graphs, networks.
@ In 2D: Mermin & Wagner PRL(66); Barber PRP(80); Naumovets CP(89); Antoni et al cm/99-in [N-body].
@ Small systems: Borrmann et al PRL(00) [classification]; Gross 01; Dunkel & Hilbert PhyA(06) [canonical and microcanonical].
@ Other examples: Stanley in(82) [geometric analog]; Fletcher AJP(97) [mechanical analog]; Drouffe et al JPA(98) [condensation]; Eggers PRL(99) [in heated sand]; Fendley & Tchernyshyov NPB(02)cm [1D]; Biskup & Chayes CMP(03) [discontinuous]; Messager & Molchanov JSP(06) [non-linear XY model with first and second-order phase transition]; Velasco & Fernández-Pineda AJP(07) [triple point]; English EJP(08) [spontaneous synchronization of oscillators].
> Other: see Cellular Automata; differentiable manifolds; Gross-Neveu Model; quantum phase transitions [including ft and early universe].

Related Topics > see casimir effect [critical]; Clausius-Clapeyron Equation; Hysteresis; scale invariance.
@ And symmetry breaking: Gill CP(98); Baroni & Casetti JPA(06) [topological conditions]; Del Giudice & Vitiello PRA(06)cm [electromagnetic field and matter, phase locking].

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