# Speed of heat / quantification of heat and other magnitudes

Is the speed of heat infinite? When solving the heat equation in a semi-infinite bar, we can see that a pulse in the finite end draws an immediate change in every point of the bar. So, at any given point of the originally constant heated bar the temperature changes. That could send information at a higher speed than light rate to a distant point of the bar.

A deeper question is: Does the mathematical concept of the Real Line can model any magnitude? I mean, if every magnitude is quantified, the Real Line is "behind" reality but never gets real (sorry for the pun).

• Conduction is transfer of thermal energy via collisions of particles. The heat cannot have a higher speed than the speed of the particles, which is always less than the speed of light. Commented Apr 29, 2015 at 17:21
• I don't see how the three questions here are related. Commented Apr 29, 2015 at 17:32
• The speed of heat as calculated by a heat equation is infinite. The physical speed of heat propagation in real solids is given by the phonon spectrum and dispersion relation of the material. The discrepancy between the two is entirely the fault of the simplifying assumptions of the heat equation. Garbage in, garbage out, as they say. Having said that, the heat equation is still extremely useful and a rather good approximation for macroscopic heat flow problems on time scales that are much longer than the phonon propagation times. Commented Apr 29, 2015 at 18:36
• @CuriousOne, it'd be great if you could expand your response to an answer. I think that both of the answers up right now have kinda missed what was being asked. Commented Apr 30, 2015 at 3:09

• Actually, transient solutions to the heat equation ($\dot{T} - \alpha T_{ii})=0$ do have instantaneous global response to changing boundary conditions. Consider the error function solutions for heat conduction into a semi-infinite domain, after $t=0$ every point in the domain will have changed from the initial value. Commented Apr 30, 2015 at 3:05
• sorry, I was using pretty compact notation, the term $T_{ii}$ indicates the sum of the second spatial derivatives ($\nabla^2 T$) and $\dot{T}=\partial T/\partial t$. That is the partial differential equation commonly known as the heat equation, not a lumped model. Commented Apr 30, 2015 at 14:38