Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 1.62.
The concepts of thermal conductivity and specific heat capacity can be combined to derive the heat equation, which governs how heat spreads through an object with a non-uniform temperature distribution. We’ll derive the one-dimensional version of the heat equation.
Suppose we have a bar of material with some initial temperature distribution , where is a function of position along the bar and time , so is the initial state of the bar at . Consider two adjacent slices in the bar, each of width . The first slice is bounded by and and the second slice by and . According to heat conduction equation, the amount of heat flowing into the second slice from the first slice in time interval is
where is the thermal conductivity and is the cross-sectional area of the bar.
Similarly, the amount of heat flowing out of the second slice on the other side is
The difference is stored in the second slice and will cause a change in temperature within the slice. If the heat capacity of the material is and its mass density is then
Plugging in the values for and we get
In the limit , the quantity in brackets goes to .
If you haven’t seen this form before, the argument goes like this. For some function , the second derivative is defined as
This has the same form as 4. Thus in the limit, we get the heat equation
One solution of this equation is
This can be verified by taking the derivatives, and we find that
The function 9 with and is actually a delta function in the limit . Using Maple, we find that
For any , so in this limit, the function has an infinitely high spike at and an integral of 1, which are the conditions of a delta function. As time increases, the function becomes a standard Gaussian curve which gradually spreads out until as , it becomes zero, so . This is what we’d expect since the heat will gradually diffuse over the bar until everywhere is at the same background temperature. Here are some plots of for various values of , with , and :