Heat| 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 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/cρ (the thermal diffusivity) and
f:= F/cρ,
u, t = a2 ∇2 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].
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