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| In the [[mathematics|mathematical]] study of [[heat conduction]] and [[diffusion]], a '''heat kernel''' is the [[fundamental solution]] to the [[heat equation]] on a specified domain with appropriate [[boundary conditions]]. It is also one of the main tools in the study of the [[spectral theory|spectrum]] of the [[Laplace operator]], and is thus of some auxiliary importance throughout [[mathematical physics]]. The heat kernel represents the evolution of [[temperature]] in a region whose boundary is held fixed at a particular temperature (typically zero), such that an initial unit of heat energy is placed at a point at time ''t'' = 0.
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| The most well-known heat kernel is the heat kernel of ''d''-dimensional [[Euclidean space]] '''R'''<sup>''d''</sup>, which has the form
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| :<math>K(t,x,y) = \frac{1}{(4\pi t)^{d/2}} e^{-|x-y|^2/4t}\,</math>
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| This solves the heat equation
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| :<math>\frac{\partial K}{\partial t}(t,x,y) = \Delta_x K(t,x,y)\,</math>
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| for all ''t'' > 0 and ''x'',''y'' ∈ '''R'''<sup>''d''</sup>, with the initial condition
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| :<math>\lim_{t \to 0} K(t,x,y) = \delta(x-y)=\delta_x(y)</math>
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| where δ is a [[Dirac delta distribution]] and the limit is taken in the sense of [[distribution (mathematics)|distributions]]. To wit, for every smooth function φ of [[compact support]],
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| :<math>\lim_{t \to 0}\int_{\mathbf{R}^d} K(t,x,y)\phi(y)\,dy = \phi(x).</math>
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| On a more general domain Ω in '''R'''<sup>''d''</sup>, such an explicit formula is not generally possible. The next simplest cases of a disc or square involve, respectively, [[Bessel functions]] and [[Jacobi theta function]]s. Nevertheless, the heat kernel (for, say, the [[Dirichlet problem]]) still exists and is [[smooth function|smooth]] for ''t'' > 0 on arbitrary domains and indeed on any [[Riemannian manifold]] [[manifold with boundary|with boundary]], provided the boundary is sufficiently regular. More precisely, in these more general domains, the heat kernel for the Dirichlet problem is the solution of the initial boundary value problem
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| :<math>\frac{\partial K}{\partial t}(t,x,y) = \Delta K(t,x,y) \rm{\ \ for\ all\ } t>0 \rm{\ and\ } x,y\in\Omega</math>
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| :<math>\lim_{t \to 0} K(t,x,y) = \delta_x(y)\rm{\ \ for\ all\ } x,y\in\Omega</math> | |
| :<math>K(t,x,y) = 0, \quad x\in\partial\Omega \rm{\ or\ } y\in\partial\Omega.</math>
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| It is not difficult to derive a formal expression for the heat kernel on an arbitrary domain. Consider the Dirichlet problem in a connected domain (or manifold with boundary) ''U''. Let λ<sub>''n''</sub> be the [[eigenvalue]]s for the Dirichlet problem of the Laplacian
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| :<math>\left\{ | |
| \begin{array}{ll}
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| \Delta \phi + \lambda \phi = 0 & \mathrm{in\ }\ U\\
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| \phi=0 & \mathrm{on\ }\ \partial U.
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| \end{array}\right.
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| </math>
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| Let φ<sub>''n''</sub> denote the associated [[eigenfunction]]s, normalized to be orthonormal in [[Lp space|L<sup>2</sup>(''U'')]]. The inverse Dirichlet Laplacian Δ<sup>−1</sup> is a [[compact operator|compact]] and [[selfadjoint operator]], and so the [[spectral theorem]] implies that the eigenvalues satisfy
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| :<math>0 < \lambda_1 < \lambda_2\le \lambda_3\le\cdots,\quad \lambda_n\to\infty.</math>
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| The heat kernel has the following expression:
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| {{NumBlk|:|<math>K(t,x,y) = \sum_{n=0}^\infty e^{-\lambda_n t}\phi_n(x)\phi_n(y).</math>|{{EquationRef|1}}}}
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| Formally differentiating the series under the sign of the summation shows that this should satisfy the heat equation. However, convergence and regularity of the series are quite delicate.
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| The heat kernel is also sometimes identified with the associated [[integral transform]], defined for compactly supported smooth φ by
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| :<math>T\phi = \int_\Omega K(t,x,y)\phi(y)\,dy.</math>
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| The [[spectral mapping theorem]] gives a representation of ''T'' in the form
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| :<math>T = e^{t\Delta}.</math>
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| ==See also==
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| *[[Heat kernel signature]]
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| *[[Minakshisundaram–Pleijel zeta function]]
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| *[[Mehler kernel]]
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| ==References==
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| * {{Citation | last1=Berline | first1=Nicole | last2=Getzler | first2=E. | last3=Vergne | first3=Michèle | title=Heat Kernels and Dirac Operators | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2004}}
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| * {{Citation | last1=Chavel | first1=Isaac | title=Eigenvalues in Riemannian geometry | publisher=[[Academic Press]] | location=Boston, MA | series=Pure and Applied Mathematics | isbn=978-0-12-170640-1 | mr=768584 | year=1984 | volume=115}}.
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| * {{Citation | last1=Evans | first1=Lawrence C. | title=Partial differential equations | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-0772-9 | year=1998}}
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| * {{Citation | last1=Gilkey | first1=Peter B. | title=Invariance Theory, the Heat Equation, and the Atiyah–Singer Theorem | url=http://www.emis.de/monographs/gilkey/ | isbn=978-0-8493-7874-4 | year=1994}}
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| *{{Citation | last1=Grigor'yan | first1=Alexander | title=Heat kernel and analysis on manifolds | url=http://books.google.com/books?id=X7QQcVa2EWsC | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=AMS/IP Studies in Advanced Mathematics | isbn=978-0-8218-4935-4 | mr=2569498 | year=2009 | volume=47}}
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| {{DEFAULTSORT:Heat Kernel}}
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| [[Category:Heat conduction]]
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| [[Category:Spectral theory]]
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| [[Category:Parabolic partial differential equations]]
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