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 dS (> 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 Conductivity [Fourier's law]; Heat Flow [mechanisms]; thermal radiation; Transport Phenomena.
> Other phenomena: see condensed matter [caloric materials]; 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 a2:= k/ (the thermal diffusivity) and f:= F/,

u, t = a22 u + f ;   more generally,   u, t = Lu ,

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.
$Def: The solution to (∇2 + ∂t) u = 0 with initial condition U0(x) = δ(x); In 1D, it is given by u(x, t) = exp{−x2/2t} / (2πt)1/2.$ Generalization: It can be generalized to the equation (Δ + ∂t) ut(g) = 0, for a function ut defined on a group G, on which the Laplacian is Δ = ∑i Xi Yi, where Xi and Yi are respectively right- and left-invariant vector fields, with solution

ut(g) = ∑r dr exp{−t λr /2} χr(g) ,

where r is an irrep of G, dr its dimension, λr the eigenvalue of Δ, and χr 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].