Consider a material that is in local equilibrium: this means that we consider a situation where the system is composed of spatial regions that are large enough for thermodynamics to apply and for them to be in equilibrium, but small enough that the thermodynamic variables are constant over each region. We can then talk about a temperature field T(x,y,z,t). When two nearby regions are at different temperatures, energy flows from the hot one to the cold, so q/t = J where q is the added heat density and J = _T T for some constant _T . You may assume that the pressure is uniform and does not depend on time.

i) If the material has constant specific heat cp and density , show that for small variations in temperature, T satisfies the diffusion equation:

T/t = D_T²T

What is D_T ? Hint: Since the temperature variations are small, we can take the specific heat to be constant, so how are and T related?

ii) Putting the material in a microwave cavity, we heat it with a sinusoidal pattern

T = T_offset + T₀ sin(kx) at time t = 0,

and then we turn the microwaves off. Show that the temperature modulation stays sinusoidal, and that its amplitude decays exponentially in time. How does the amplitude decay rate depend on the wavevector k?