In Japan, days from about April 29 until about May 5 are successive holidays as you know.
So today i.e., April 30, I could have enough time to read through a book entitled "Fourier Series" written by Georgi. Tolstov and translated into English by R.A.Silverman and published from Dover Publication. Reading former part was a teadious work because almost of it is the repeatition of "Theorem of something, The prooof of it", "The theorem of next thing, The proof of it,",,, and so on. But the last part of the book mentioned some physical applications of the Forurier theory, like string viblation, rectangular membrane viblation, and heat transfer in a cylinderical rod.
What has astounded me was a explanation about the heat propagation equation in an infinite length rod of which lateral surface was surrounded by thermal insulator. I would like to copy the part from the book that "formula(25.10)(which expesses the solution u(x,t)) shows that as time increases, u(x,t) approaches 0, i.e., the heat "spreads out" along the rod. On the other hand, (25.10) shows that the heat is "transmitted" instantaneously along the rod. In fact, let the initial temperature be positive for x0<=v<=x1 and zero outside this interval. Then it is clear that (an intgration equation is written and ) from which it is clear that u(x,t)>0 for arbitrarily small t>0 and arbitrarily large x."
This portion is explaning what happens in the temperature distribution u(x,t) along the infinite length rod after setting an initial condition of rectanguler temperature distribution. The analytical solution compells an "instantaneous" spread of temperature rise instantaneously at even far position x, that is, in no time. "In no time," so that the heat transfer has infinite speed much faster than light speed! Does this really happen? Can this really happen?
The starting base is the diffusion equation. The book does not take into account the microscopic interpretation of thermal phenomenon. The heat transfer may be interpreted microscopically as the statistical propergation of excess kinetic energy from one side to the lower side via the atomic contacts. Therefore I suppose that the heat tranfer speed cannot exceed the sonic speed of the rod. Or is the author Tolstov actually right? Does the heat run faster than light speed in the straight rod which is thermally insulated laterally? Are the contact processes between the atoms quantum mechanical and faster than the usual sonic speed? What is the truth? Pleae someone tell me the truth and resolve this puzzle!
A stray sheep
April 30, 2012
All models are approximations of reality. The heat equation assumes a continuum, ignoring the actual microscopic mechanisms which transfer energy. On a large enough scale, the continuum assumption is fairly good. However, as you have noted, there are situations where the assumptions break down. When you get down to the atomic-scale contacts, the continuum assumption is no longer valid. We need to modify the model to account for reality at this scale, or find a different model to use.
It may be useful to think of models and equations as descriptions of reality. A useful model gives us insight into the underlying rules, but one should not confuse the model for reality itself! The power of science lies in our willingness to test our models against reality through experiments and let the data guide us to an improved understanding.
Thank you Dr. Hsieh and Dr. Pudasaini, for your responses.
Yes, regarding the rod as ideal continuum was my mistake.
Then how do you think about the thermal conduction phenomenon from view points of
(A) microscopic scale where quantum mechanical interpretations of each atomic and electronic motions are required, and
(B) macroscopic scale where calculations by classical statistical mechanics are valid?
Will the solutions from both viewpionts be deterministic?
How will the complicated wave function that involves many,many variables behave?
I'd think that the thermal conduction through a metal rod is a coupled phenomenon of vibrational motions of the atoms and electronic excitations or phonon (due to extremely small band gap). Quantum wave nature of heat transfer is debatable, while relativistic effects have been used to avoid infinite speed of heat transfer (see. Relativistic heat conduction), but the theory itself has some issues.
While electrons in conducting band also contribute to heat energy transfer, I speculate the transfer rate is limited to some degree by the vibrational lifetimes of each atom in the lattice. Thus, in principle, heat transfer to could simply be perturbations (stark effect) on each atom due to the motions (vibrations) of the adjacent atoms.
Finally, I must point out that, at this point, I do not know whether the viewpoint is already been addressed or is valid.
Thank you very much, yes indeed, Dr. Pudasaini for your detailed explanations about heat conduction in metal rod.
As you say, our discussion has lost the focus regarding the speed of heat conduction. But please be relaxed. This is not the place for fighting, but I only wished to provide a place for everyone to join to exchange various views peacefully. And if you are too busy, don't bother the obligation to reply this my oipinion.
Dr. Pudasaini, your model of metal rod is fundamentally based on a spring and ball model. And the electrons are merely accessaries. Your view of metal crystal is a mixture of classic and quantum mechanical view. If a rod is consisted from a single giant crystal, then the entire crystal body must be hold by the whole valance of the all of nucleus and electrons described by the pure many many variables time dependent Schroedinger wave function.
In actual metal rod, it is consisted from many crystal domains. They are connected by irregulary boundary surfaces. What should we expect from this irregularities? Does the cleal phonon notion still hold? Is the band theory hold valid in spite of the existence of the grain boundaries? Are the electron moving in the plane wave fashion irrespectrive of the existence of these irregular surface planes?
What I would like to emphasize is to promote a thinking from the ab in itio Schroedinger equation that is applied to the macroscopic body. And how do you think the classical notion of heat corresponds to the modern quantum mechanistic terminology? What I am asking is this. If N nucleoei (e.g. iron nuclei) and ZN electrons are left in nature, how do they assemble together? What and how will their eintitre quantum mechanical wave function be? And, if a heat is applied to one portion of that crystal nugget, how should we use the quantum mechanical terminology for the heat transfer? What do you think the quantum mechanistic term corresponds to the notion of entropy?
Like I said .... My best guess is to include stark-shift Hamiltonian in the time dependent Schrodinger wave equation on the model I mentioned. Macroscopic bodies aren't wave-like; they are more particle like. Therefore, treating them with probability theory isn't accurate. While heat transfer is atomistic, I believe it requires various levels of theories (classical, quantum and relativistic) simply because heat can be produced various ways. Electricity passing through a conductor produces heat, so does microwaves heating the water molecules. These processes involve different types of excitations, either at translational, rotational, vibrational or electronic levels (or all). Computational chemists often use such excitations as ways to define temperature of the system.
"If N nucleoei (e.g. iron nuclei) and ZN electrons are left in nature, how do they assemble together?"; I really don't any ideas on this. Thank you for bringing it up. My initial guess is that Born-Oppenheimer Approximation is no longer valid as nucleons are probably quantum particles, making the problem beyond our computational capabilities. (Physicists claim to have some answer to the aftermath of Big Bang, which is a similar problem). Inform me if you make any progress on this.
Sadly, no modern quantum theory or post Hartree-Fock methods accurately describe quantum many body problems. All of them use variational principle which an upper bound energy of the system and rely on substantial systematic error cancellation relative to reference systems.
I am sorry to be so late to respond you, Dr. Pudasaini.
Thank you for your kind responses sent thus far.
To say candidly, I do not know the stark shift Hamiltonian effect yet.
I am an old student. But someday I will study it.
I really agree with your view on the Born-Oppenheimer approximation.
From when I was young college student, I have been suspecting that the Born-Oppenheimer approximation must inhibit all chemical reaction to occur theoretically. And you also mentioned the many-body problem quoting the Hartree-Vock method. It is just what I really wanted to express to everyone when to address the physics of real matter microscopically rigorously.
Now all has been discussed. You have described exactly all what I really wanted to express on behalf of me.
Let us close this discussion, shall we?
I really thank you again, Dr. Pudasaini.
June 10, 2012
The main difference lies in the propagation of energy that is light actually photons requires a medium for its propagation in its surroundings or( its polarizing power) and passes that quantum while heat as we know flows through higher temp.point to the lower temp.point via a process known as conduction until the temp. equals for both the points.
I was thinking that this discussion was over.
Then, you have appeared! Welcome to this Blog!
I think that the more person participates in a discussion, the more flowers of various opinions will bloom.
I am sorry, but I do not see the gist of your assertion. Your assertion itself is not wrong at all.
What is the curious or strange point about the difference in the propagation fashions between photon and heat? Each of which is governed by its own equation, the heat dissipation being described by the heat diffusion equation, while the light travelling being described by the ordinary wave equation.(I do not yet know strangely what equation desribes the spatial flight track of a photon, except that the photon system is described by harmonic oscillator Hamiltonian.) Of course being aware of the governing equation is not equivalent to being aware of the underlying final physical rules or physical laws as Dr. Hsieh said. Going back to the present subject, each of photon or a pulse of heat has starting point and ending point in space or in matter. As to the traveling of photon, I think Dr. William Aton's Blog "There is no such thing as pure energy, only energy currencies such as ATP in cells in biology" must interest you.
I graduated in physic about 35 years ago. In that time, we were only taught about the first steps in quantum mechanics and thermodynamics and electromagnetism. Now having become an old age, I began to ponder if we treat every atoms or molecules in a gas or in a liquid or in a solid matter, how will we look at the thermal quantities quantum mechnically? If the matter is an iron rod, the iron atom 26Fe is itself astonishing object because it has 26 electrons! We must solve 3 times 26 variables Schroedinger equation first. The thing is much more. Since we are addreesing a macroscopic object in which every atoms are neighboring with and contacting each other, so that we must also take into account the tremendously many ways of thermal perturbations which each atom feels via the electrons-nuclei interaction. What will we get as the whole solution that must describe the whole matter quantum mechanically rigourously? What is the classical notion of temperature in this framework of consideration based on quantum mechanics? What will a heat pulse which is located somewhere be interpreted quantum mechanically? I would like to emphasize here that rigourously speaking, every macroscopic matter is composed many many electrons and nuclei!( Dr. Pudasaini said that for that case things behave more particle like than wave like.)
Anyway, thank you for joining in this discussion session.
Let us enjoy talking each other!
June 16, 2012