Heat |

**In General: Thermodynamics and Heat**

* __Idea__: Heat is energy transferred
between two systems purely as the result of a difference in temperature (no work done).

* __Value__: In an infinitesimal reversible
transformation the heat absorbed by a system can be expressed as δQ = *T* d*S*
(> see Heat Theorem).

* __History__: Late 1600s, Guillaume Amontons
intuitively surmised that heat flowed in solids in the direction of decreasing temperature;
Early 1700s, Daniel Fahrenheit invented a mercury thermometer capable of reproducible
measurements; 1761, Joseph Black introduced the concepts of latent heat and specific heat;
1780s, Invention of the calorimeter by Antoine Lavoisier and Pierre Simon Laplace, and use
of the latent heat of melting ice as a standard for quantifying heat; The nature of what had
been quantified, though, would elude comprehension throughout the 19th century; 1807–1811,
Joseph Fourier conducted experiments and devised mathematical techniques that together yielded
the first estimate of a material's thermal conductivity; 1842, Robert Meyer, heat is not a
fluid that can penetrate material bodies but a form of energy; 1843–1849, James Joule,
experiments on the equivalence of heat and energy.

* __Status of theory__: It is empirically
well tested for fluids and crystals, but there is little theoretical understanding.

@ __Historical / conceptual__: Narasimhan PT(10)aug-a1005 [measurement, historical];
Votsis & Schurz SHPSA(12) [caloric and kinetic theories, and structural realism];
Hari Dass a1306 [caloric theory and Carnot's work].

@ __Related topics__:
Salazar et al EJP(10) [thin plate vs thick slab];
Schittny et al PRL(13)
+ news bbc(13)may [heat cloaking].

> __Related concepts__: see inertia;
Kinetic Theory; specific heat [heat capacity];
Thermal Bath.

> __Heat transport__: see Heat Flow
[mechanisms – conduction, convection, radiation]; thermal radiation;
Transport Phenomena.

> __Other related phenomena__: see Heat Engine;
Thermal Expansion; Elasticity [elastocaloric effect].

**Mathematical Theory of Heat Transfer: Heat Equation / Operator**
> s.a. spectral geometry;
QED phenomenology [for nanoscale objects].

* __Idea__: The heat equation is the diffusion
equation, applied to the temperature in a heat conductor; When the density *ρ*,
specific heat *c* and thermal conductivity *k* are
constant, and setting *a*^{2}:=
*k*/*c**ρ* (the thermal diffusivity) and
*f*:= *F*/*cρ*,

*u*_{, t} = *a*^{2}
∇^{2} *u* + *f* ; more generally,
*u*_{, t} = *L**u* ,

with *L* a second-order differential operator; It is related to the
Schrödinger equation by an analytic continuation in *t*.

@ __General references__: Widder 75;
in Gilkey 84.

@ __Relativistic__: López-Monsalvo & Andersson PRS(11)-a1006;
Mendez & García-Perciante AIP(10)-a1010 [kinetic-theory approach];
Andersson & López-Monsalvo CQG(11)-a1107 [consistent first-order model];
López-Monsalvo PhD(11)-a1107;
Govender & Thirukkanesh MPLA(14)-a1404 [causal heat flow in Bianchi V spacetimes];
Duong PhyA(15)-a1501 [and relativistic kinetic Fokker-Planck equation].

@ __Related topics__: Gilkey et al NPPS(02)mp/01 [asymptotics];
Bustamante & Hojman mp/01/JPA [Lagrangian, Hamiltonian, Poisson brackets];
Hall in(06)m.DG/04 [range of time-*t* heat operator];
Gibou & Fedkiw JCP(05) [Dirichlet boundary conditions, 4th-order discretization];
Iliev SelMath(08)mp/06 [discrete, and Toda hierarchy];
Hall in(08)-a0710 [in infinite dimensions];
He & Lee PLA(09) [constrained variational principle];
Smerlak EPJP(12)-a1202 [thermal diffusivity as space-dependent];
Deconinck et al PRS(14) [non-steady-state heat flow];
Ramm a1601 [in a complex medium].

**Heat Kernel** > s.a. effective action in quantum field theory.

$ __Def__: The solution to
(∇^{2} +
∂_{t}) *u* = 0 with
initial condition *U*_{0}(*x*)
= δ(*x*); In 1D, it is given by *u*(*x*,* t*)
= exp{−*x*^{2}/2*t*}
/ (2π*t*)^{1/2}.

$ __Generalization__: It can be
generalized to the equation (Δ + ∂_{t})
*u*_{t}(*g*) = 0,
for a function *u*_{t}
defined on a group *G*, on which the Laplacian is Δ
= ∑_{i} *X*_{i}
*Y*_{i},
where *X*_{i}
and *Y*_{i} are
respectively right- and left-invariant vector fields, with solution

*u*_{t}(*g*)
= ∑_{r}* d*_{r}
exp{−*t λ*_{r} /2}
*χ*_{r}(*g*) ,

where *r* is an irrep of *G*, *d*_{r}
its dimension, *λ*_{r} the eigenvalue of
Δ, and *χ*_{r} the characteristic.

* __Applications__: It is used in
the coherent state transformation for quantum theory, and is a convenient tool
for studying one-loop divergences and renormalization, anomalies and various
asymptotics of the effective action; It is also the transition density of a
Brownian motion.

@ __General references__: Fulling ed-95 [and quantum gravity];
Booth ht/98
[heat kernel coefficients with Mathematica];
Moss & Naylor CQG(99)gq/01 [diagrammatic expansion technique];
Maher m.RT/06-proc [on compact Lie groups].

@ __In curved spacetime__: Martin & McKeon ht/96;
Salcedo PRD(07)-a0706 [to 4th order in derivative expansion];
> s.a. effective field theories.

@ __Calculation of coefficients__: Nesterenko et al CQG(03) [corner contributions];
Vassilevich PRP(03) [tools];
Bordag & Vassilevich PRD(04)ht [with discontinuous backgrounds];
Iliev AIF(05)mp,
PAMS(07)mp/05 [and KdV hierarchy].

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