Talk:Heat kernel

Puzzling expression in introduction

The Introduction begins as follows:

"In the 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 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.

"The most well-known heat kernel is the heat kernel of d-dimensional Euclidean space R^d, which has the form of a time-varying Gaussian function,

$K(t,x,y)=\exp \left(t\Delta \right)(x,y)={\frac {1}{(4\pi t)^{d/2}}}e^{-\|x-y\|^{2}/4t}\qquad (x,y\in \mathbb {R} ^{d},t>0)\,$"

What does exp(∆t) (x,y) mean here?

As a mathematician, my understanding is that since ∆ is an operator, then so is exp(∆t) also an operator.

So, what does it mean to apply an operator on a function space to the ordered pair (x,y) of points in R^d ???

2601:200:C000:1A0:DD41:F29D:81CE:BF6E (talk) 16:25, 3 September 2022 (UTC)

Unexplained symbol and other problems with this article

In the section Definition, the symbol for the Laplacian has a subscript of x.

This symbol is never explained.

This is particularly confusing, since I strongly suspect that it means the space Laplacian, the sum of the second derivatives with respect to each spatial variable comprising the point x in euclidean n-space.

I hope that someone knowledgeable about this subject will explain what this symbol means.

Also it would be helpful if the subscripted Dirac delta function symbol were also explained, instead of sending readers to that article so they could search that article for where the subscripted Dirac delta symbol might occur (which is very far down in the article).

Also since the Laplacian is often defined to be the negative sum of the second derivatives instead of the positive sum of second derivatives: It would be enormously helpful if the article did not assume that the reader is telepathic and knows which form of the Laplacian the writer is thinking of ... by simply stating which definition of the Laplacian is being used here.

Also further down in the article the Laplacian symbol is used without any subscript. Can someone explain the difference between this symbol with and without its subscript?

So, it would be a Very Good Idea to clarify what the reason for the two kinds of symbols is.

Furthermore: The first sentence of the article reads as follows:

"In the 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."

But in the section Definition, no boundary conditions are mentioned. It is necessary in any coherent account of the heat equation and its heat kernels to mention that this is the case where there are no boundary conditions — if that is indeed the intention here. It is ridiculous to describe the problem with boundary conditions and then leave that entirely unmentioned in the definition of the subject of the article. 2601:204:F181:9410:7893:ECDB:85D3:1CD3 (talk) 14:49, 9 May 2025 (UTC)

You're right that $\Delta _{x}$ refers to the Laplacian with respect to the x variables, treating y and t as constants. The $\delta _{x}$ is just alternative notation for $\delta _{x}(y)=\delta (x-y),$ it could be deleted without major loss. The Laplacian here is the (positive) sum of second derivatives. Further down the page, $\Delta$ appears without a subscript since there's no ambiguity; one just has $\Delta \phi (x)$ without any extra t or y variables.

Overall I agree with your criticisms, this article is in a pretty bad state. If I didn't already know the material it would be pretty confusing. Gumshoe2 (talk) 02:36, 10 May 2025 (UTC)

Some recommendations for slightly expanding the article

• It would seem worthwhile to mention explicitly — not merely with mathematical notation – that the heat kernel describes the time-evolution of an infinitesimal unit-heat impulse at time zero.

• It also seems extremely worthwhile to mention that the heat kernel also leads very quickly to the time-evolution for a large variety of other initial conditions, by integrating it against the initial condition.

• Finally, it also seems worth mentioning at least something about what the limit of the heat evolution is as t → ∞. (Such as the equation that this limit is the solution of.)

• These things belong in the article before anything about a "more general domain" than euclidean space is discussed, because they are fundamental to the subject.

Since I am just a beginner at this subject, I hope that someone knowledgeable on this topic will fill in this information to improve the article. 2601:204:F181:9410:A083:8DBE:A302:8BB0 (talk) 16:59, 22 May 2025 (UTC)

Please do not add comments with signature lines or remove existing signature lines per WP:SIGN.

The key ingredient in expanding any article is finding suitable reliable sources. Top sources in science articles are reviews or textbooks focused on the article topic or directly discussing the topic. Post sources if you have them. Johnjbarton (talk) 17:08, 22 May 2025 (UTC)