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, *C*_{p} is
greater than *C*_{V} 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 *ω*^{2},
up to some *ω*_{H};
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, *C*_{V} =
(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

*C*_{S} = *T* (∂*S*/∂*T*)
= (∂*M*/∂*T*)
= –8π*M*^{ 2}
= –*T*_{H}^{2}/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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jan 2016