Has the property $\int\Phi =1$ (of the fundamental solution $\Phi$ of the heat equation) a physical interpretation?

Question

For each $t>0$ the fundamental solution of the heat equation is given by $$\Phi(x,t)=\frac{1}{(4\pi t)^{n/2}}\exp\left(-\frac{|x|^2}{4t} \right )$$ and satisfies $$\int_{\mathbb{R}^n}\Phi(x,t)\,dx=1.$$ Is there any physical interpretation for this property?

Thanks.

Answer 1

Yes, since $\Phi(t,x)\mathrm dx$ represents the amount of heat contained in the volume $\mathrm dx$, this is saying that the total amount of heat contained in the whole $\mathbb R^n$ does not change with time.