Re:Mathematicians try to understand how CS proof solves Conn...
by shoo on Friday April 17, 2020 @05:16AM
(#22897443)
> Physics do not allow you to simply modify the temperature of a point. Thermodynamics prevent that.
gotcha - in terms of the analogy, the temperature distribution cannot be any arbitrary distribution with a finite average (i.e. a finite L2 norm) --- including those pathological cases e.g. the temperature is 0 everywhere except at one point x where it takes the value 1 ---- no, the temperature distribution has to have some minimum smoothness structure, meaning that if you adjust the temperature at a single point, you have to also locally adjust the temperature of neighbouring points enough for it to make a difference to the overall global average temperature over the entire space.
gotcha - in terms of the analogy, the temperature distribution cannot be any arbitrary distribution with a finite average (i.e. a finite L2 norm) --- including those pathological cases e.g. the temperature is 0 everywhere except at one point x where it takes the value 1 ---- no, the temperature distribution has to have some minimum smoothness structure, meaning that if you adjust the temperature at a single point, you have to also locally adjust the temperature of neighbouring points enough for it to make a difference to the overall global average temperature over the entire space.
for intuition, the animation on the wikipedia "heat equation" page is a good start: https://en.wikipedia.org/wiki/Heat_equation
When you plug initial conditions into the heat equation, over time it evolves to be smoother.