Specific Heat / Heat Capacity  

In General > s.a. heat; Heat Transfer.
* History: Einstein's 1907 article on the specific heat of solids introduced for the first time the effect of lattice vibrations in the thermodynamic properties of crystals; The next important step was the introduction of Debye's model.
* Heat capacity: The quantity C:= ∂U/∂T, calculated either at constant volume or at constant pressure, if appropriate; When the two are different, Cp is greater than CV because of the extra work the system does in the expansion, but for a solid there is only one notion.
* Specific heat: The heat capacity per mole c = C/n, with n the number of moles, or per unit mass, c' = C/m.
> Online resources: see Wikipedia page.

For Solids
* Dulong-Petit law: The universality of specific heats of solids at high temperature, stating that C = 5.94 cal/K per mole; It breaks down at low T (say, below 600 K or so, but it depends on the material), when equipartition no longer holds; > s.a. energy; Wikipedia page.
* Einstein model: Atoms are treated as non-interacting harmonic oscillators, so the phonon density of states is a delta function at a single frequency; > s.a. Wikipedia page.
* Debye model: The density of states for atomic vibrations is modeled as g(ω) = const ω2, up to some ωH; This corresponds to a constant speed of sound, and gives C proportional to T 3; > s.a. scienceworld page; Wikipedia page.
@ References: Einstein AdP(07); Shubin & Sunada mp/05 [geometric approach]; Grabowski et al PRB(09) + Grimvall Phy(09) [ab initio, up to melting point]; Mahmood et al AJP(11)nov [experimental determination].

Other Systems > s.a. ising model; non-extensive statistics.
* Classical gas: From the equipartition principle, CV = (3/2) Nk [monatomic], (5/2) Nk or (5/2) Nk [diatomic].
* Liquids: A general theory of the heat capacity of liquids has always remained elusive, in part because the relevant interactions in a liquid are both strong and specific to that liquid; 2012, The "phonon theory of liquid thermodynamics" has successfully predicted the heat capacity of 21 different liquids.
* Black hole: It is negative (as is typical for a self-gravitating system, since there can be no equilibrium with an infinite thermal bath), and given by

CS = T (∂S/∂T) = (∂M/∂T) = –8πM 2 = –TH2/8π .

@ Black hole: Gibbons & Perry PRS(78) [thermal Green's functions]; Górski & Mazur ht/97 [quantum effects, positive].
@ Boson system: Wang AJP(04)sep [above condensation T]; Ramakumar & Das PLA(06) [on a lattice].
@ Self-gravitating: Lynden-Bell & Wood MNRAS(67), Lynden-Bell PhyA(99)cm/98-proc.
@ Other systems: Albuquerque et al PhyA(04) [quasi-periodic structures, oscillatory c(T)]; Moreira & Oliveira PRA(06)gq [relativistic particle on a cone]; Bolmatov et al SciRep(12) + news pw(12)jun [liquids].

Special Concepts and Results > s.a. sound [speed].
* Negative: In addition to gravitating systems, it can happen in systems with small numbers of particles, or some non-ergodic systems.
@ Negative: Antoni et al cm/99-proc [N-body]; Schmidt et al PRL(01) + pn(01)feb [Na clusters]; Thirring et al PRL(03) [non-ergodic]; Einarsson PLA(04)gq [conditions]; Posch & Thirring PRL(05) [and stellar stability]; Rao et al AP(08) [particles in box with potential well]; Staniscia et al PRL(10) [in the canonical statistical ensemble]; Serra et al EPL(13)-a1305 [finite quantum systems].
@ In non-extensive statistics: Lenzi et al PLA(02); Álvarez-Ramírez et al PLA(05).
@ Related topics: Pizarro et al AJP(96)jun; Gearhart AJP(96)aug [and equipartition]; Filardo Bassalo et al NCB(01) [dissipative]; Fraundorf AJP(03)nov; Behringer et al JPA(05) [microcanonical, finite size]; Starikov a1007 [from Bayesian approach].

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