Heat| Heat |
In General, Thermodynamics
* History: 1842, Robert
Meyer, heat is not a fluid that can penetrate material bodies but a form of
energy; 1943–1849, James Joule, experiments on the equivalence of heat and
energy.
@ Finite heat bath: Potiguar & Costa PhyA(04)
[thermodynamic relations]; Gemmer
& Michel EPL(06)qp/05
[thermalization]; Ford & O'Connell PhyE(05)qp/06 [different
models, and quantum
oscillator]; Nechita & Pellegrini a0908 [statistical model].
@ Quantum heat engine: Scully PRL(02);
Kieu PRL(04)qp/05 [second
law, Maxwell's demon].
>
Related concepts: see specific
heat [heat capacity].
Heat Flow / Thermal Conductivity > s.a. Kinetic
Theory; Lennard-Jones Fluid; Transport.
* Idea: Governed by
Fourier's law J = –
T,
with J = heat flux, =
coefficient of thermal/heat conductivity, T = temperature.
* Status: Empirically
well tested for fluids and crystals, but there is little theoretical
understanding.
@ References: Desloge AJP(62)dec
[thermal conductivity for a gas]; Bonetto et al mp/00;
Bertola & Cafaro PLA(07)
[speed of propagation]; Komatsu et al PRL(08)
[microscopic derivation]; Collet & Eckmann CMP(09) [model].
Heat Equation / Operator > s.a. spectral
geometry.
* Idea: The diffusion
equation, applied to the temperature in a heat conductor; When the density 
,
specific heat c and thermal
conductivity k are
constant, and setting a2:= k/c and
f:= F/c,
u, t = a2 2 u + f ; more
generally, u, t = Lu ,
with L a second-order differential operator; Related
with the Schrödinger equation by analytic contiinuation in t.
@ General references: Widder 75; in Gilkey 84.
@ Related topics: Gilkey et al NPPS(02)mp/01-in
[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 mp/06 [discrete,
and Toda hierarchy]; Hall a0710 [in
infinite dimensions]; He & Lee PLA(09)
[constrained variational principle].
Heat Kernel > s.a. effective
action in quantum field theory.
$ Def: The solution to
(2 +
t) u =
0 with initial condition U0(x)
= 
(x); In
1D, it is given by u(x, t) = exp{–x2/2t}/(2
t)1/2.
$ Generalization: It
can be generalized to the equation (
+
t)
ut(g)
= 0, for a function ut defined
on a group G, on which the Laplacian
is
= 
i Xi Yi,
where Xi and
Yi are
respectively right- and left-invariant vector fields, with solution
ut(g)
= r dr exp{–t 
r /2}
r(g)
,
where r is an irrep of G, dr its
dimension, r the
eigenvalue of , and 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-in
[on compact Lie groups].
@ In curved spacetime: Martin & McKeon ht/96;
Salcedo PRD(07)-a0706 [to
4th order in derivative expansion].
@ 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-mp/05 [and
KdV hierarchy].
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aug 2009