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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).

What do we know about these subjects? Are they paradoxes?

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    $\begingroup$ 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. $\endgroup$
    – pentane
    Apr 29, 2015 at 17:21
  • $\begingroup$ I don't see how the three questions here are related. $\endgroup$
    – Kyle Kanos
    Apr 29, 2015 at 17:32
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    $\begingroup$ 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. $\endgroup$
    – CuriousOne
    Apr 29, 2015 at 18:36
  • $\begingroup$ @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. $\endgroup$ Apr 30, 2015 at 3:09

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I suspect you are looking at the steady state solution of the set of partial differential equations that model heat dynamic heat conduction. The dynamic equations will show that the heat is NOT propagating at an infinite rate. Using the steady state solution to infer dynamic response is not proper.

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    $\begingroup$ 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. $\endgroup$ Apr 30, 2015 at 3:05
  • $\begingroup$ @user3823992 The equation you write is only an approximate time based model for one point in space. To closer approximate what's going on you need to also consider the dimension of space. This requires partial differential equations. Furthermore - for these equations what's going on at the boundary is also approximate. To further detail that behavior requires further modeling. In the end "[All models are wrong, some are useful"]" - George Box $\endgroup$
    – docscience
    Apr 30, 2015 at 14:09
  • $\begingroup$ 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. $\endgroup$ Apr 30, 2015 at 14:38
  • $\begingroup$ @user3823992 Thanks for the clarification. I still stand by my comment regarding the boundary. Studying, predicting what happens there requires special treatment. That's true in thermodynamics as well as fluid mechanics Mind the boundary ... $\endgroup$
    – docscience
    Apr 30, 2015 at 14:44
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Heat if you remember from 8th grade science transfers by convection, conduction and radiation. Convection is by the flow of a fluid which cannot go faster than light, conduction is caused by molecules colliding with neighboring molecules, conductive heat equations are only for after a steady flow has been established and do not treat transient effects so can not travel faster than light. Radiation is at the speed of light using the Stefan-Boltzmann equations at a rate proportional to the 4th power of absolute temperature.

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