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; history of physics;
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];
González et al a1908 [Debye function].

**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 proc(00)cm/99 [*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].

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 26 aug 2019