Heat conduction, like any other conduction process, is a non-equilibrium process, even though it may be a steady-state one. To explain the process, one will have to calculate the non-equilibrium distribution function of the particles. Microscopically, one can employ the Keldysh formalism. In doing so, one will have to solve a boundary-value partial differential equation (for describing the transient processes, an initial condition is to be added to the formulation of the problem). The relevant boundary conditions are provided by the equilibrium distribution functions of the heat baths (each characterised by its own temperature and chemical potential) with which the system under investigation is brought into contact. For some relevant details, you may wish to consult the following review article by Rammer and Smith:

http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.58.323

Incidentally, in describing heat conduction, in general both the electronic and ionic degrees of freedom (i.e. electronic excitations as well as vibrational excitations - phonons) have to be taken into account. Depending on the temperatures involved, for wide-gap insulators electronic excitations may be neglected as a reasonable approximation.

Heat conduction, like any other conduction process, is a non-equilibrium process, even though it may be a steady-state one. To explain the process, one will have to calculate the non-equilibrium distribution function of the particles. Microscopically, one can employ the Keldysh formalism. In doing so, one will have to solve a boundary-value partial differential equation (for describing the transient processes, an initial condition is to be added to the formulation of the problem). The relevant boundary conditions are provided by the equilibrium distribution functions of the heat baths (each characterised by its own temperature and chemical potential) with which the system under investigation is brought into contact. For some relevant details, you may wish to consult the following review article by Rammer and Smith:

http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.58.323

Incidentally, in describing heat conduction, in general both the electronic and ionic degrees of freedom (i.e. electronic excitations as well as vibrational excitations - phonons) have to be taken into account. Depending on the temperatures involved, for wide-gap insulators electronic excitations may be neglected as a reasonable approximation.

I am not asking how to solve the problem. If you read the question, we are talking about physics of heat conduction in solids....

I think I understood the question as described: "Can anyone suggest what to do when faced with a Heat Conduction problem?". The question explicitly asks "what to do", and what I described above amounts to what I would "do" when faced with this problem.

Incidentally, my earlier explicit reference to 'non-equilibrium' hints at the fact that viewing temperature as a thermodynamic variable in the system where heat conduction takes place is not justified; in the steady state, one can of course define a local temperature (for which a coarse-graining becomes necessary), but this "temperature" has no physical significance; it is just a by-product and considering it as a relevant physical quantity amounts to over-specifying the problem. And in my view the main question ascribes too much significance to temperature (that its value describes a linear interpolation between two temperatures, etc; of course, the temperatures of the heat baths are very relevant). In short, the "physics" of heat conduction in a solid is contained in the non-equilibrium distribution function of the eigenstates of the Hamiltonian describing that solid. Once that function has been calculated, one can know what takes place at the physical level. Actually, what I proposed above is very doable for a one-dimensional toy model.

More simply, I think the question can be answered by recognizing that the conduction heat flux is proportional to the gradient of the “energy of the particles”, not to the energy itself. In other words, it is proportional to the temperature gradient, and not the temperature itself (Fourier law)…

I don’t know if that answers your question?

Perhaps I could point out that the interested may wish to consult the books by Ziman (Electrons and Phonons: The Theory of Transport Phenomena in Solids, Oxford University Press, 1960), Sec. 9.9, and Ashcroft and Mermin (Solid State Physics), pp. 499-505, where a simplified kinetic theory of heat conduction in solids is described, a la Drude. Naturally, the Drude model is incorrect, nonetheless the underlying ideas are instructive. It should however be noted that here one is interested in the calculation of the thermal conductivity (as well as the electrical conductivity -- the context is that despite the Drude model being erroneous, it correctly explains the empirical Wiedemann-Franz law, which incidentally under certain circumstances breaks down), so that one considers infinitesimal temperature differences (necessary for evaluating the underlying derivative with respect to temperature). This is significant, since here, in contrast to the general case, one remains close to thermal equilibrium.

According to fourier's law of heat conduction the rate of heat flux transferred through a  wall is directly proportional to temperature gradient. From this equation we can get one constant that is thermal conductivity a property of material. It is only parameter effects the rate of heat flux for fixed temperature  difference between two points

I think Behman's suggestion is the good one. There is statement says: If you want to learn new things, read old books...

For those who mentioned Fourier's law, which is phenomenological equation, does add to our knowledge on the physics of the problem...

Thank you Abdulmajeed. Yes indeed, in particular when those old books are by the people like the late John Ziman.

Dear Mohamad,

I would point to the carrier effect in heat conduction. 'Fundamentals of Heat and Mass Transfer" 7th Edition Incropera, has a chapter 2 item 2.2, which introduces something about heat conduction in materials.

I won't feel any strange feeling in say that even in low KE, the layer would transfer the same rate due to the fact that I see "heat rate" as a "wave of vibrations". I would imagine the following picture: This wave pass through the hot and cold points with same amplitude and frequency. Heat rate is "-k (dT/dx)" and it give an idea of the amount of temperature gradient around the point of interest. Therefore, if we state that the temperature gradient is the same in both points, what the temperature of each would have to do with the heat rate? For me it has nothing to do!

Nevertheless, if we account for nano-scale effects, then I would start to feel strange! But than, I think this would be a different problem, more info about it nanohub.org.

Best Regards

lb

Part one is fully understood, continuum approach, Fourier's phenomenological law and wave theory. I well aware of Incropera et al. (my former teacher) book.

The second part (last statement, molecular level) is the puzzling part. I have some explanation for that but I would like to bring to attention of heat transfer community that some simple classical problem is not easy to explain on the fundamental bases...

Regardless of material the temperature! is linearly changes from hight to low temperature, of course provided that the so called thermal conductivity (man made) is constant and area perpendicular to the heat flow! is constant...

You can bridge both scales (molecular and macro) by considering phonons move describing random walks. If that is the case then heat flow follows Fick´s law of diffusive flow (Fourier law). This will lead you, eventually, to the macroscopic diffusion equation.   

By the way, if you do the random walk simulation, a bunch of effects not anticipated by the macroscopic diffusion equation will show up in the numerical results (specifically at internal boundaries).

JC

"Transporting" (of energy or anything else) is something different from "storing" (energy or so). It is as simple as you think of the amount of money you have saved in your account (at the moment) and the amount of money traded through your account in a period of time. If you look at two accounts and record the balance just for a moment, you will not recognize how much has been traded so far. When we look at a particle the vibrations show the stored kinetic energy, but the amount of interactions with other particles is a time averaged quantity and can not be calculated just from the vibration of one particle.