Dear Karol :

Thank you for your reply and contribution to this discussion :)

As you said; the effects of considering a finite speed for Heat transport may be negligible in most of the cases when working with conduction's steady-state heat transfer... at the scales down to a few micrometers though.

The reason I'm interesting in this topic is that the field of research of my PhD is Heat Transfer at the Nanoscale.

So I know of 3 particular cases where the classical Fourier's Law si not valid anymore, but might be more:

- When near absolute zero-temperatures

- When extreme Temperature Gradients

- When experimenting with very short periods of time.

I don't understand the reasons why these particular cases are so, yet. The logics behind these.

I know what you are trying to say about the relation between the description of the transport of a physical phenomenon (driven by a physical potential) and a wave-like behaviour.

It ss true that transport phenomena of these forms doen't have necessarly to be assosiated with a wave-like behaviour of transport...

I'm going to take a read to the articles you attached for my, I will leave a reference ("Relativistic Heat Conduction") attached to you too, you can find here about the approach of a possible wave-like description for Heat.

Regarding of the nature of reality (whether is particle like or wave like) I think we can build a lot on this topic with the risk of getting away fast from the original point of the discussion jejejej... But we can comment about this too if you want ...

although I invite others to join the discussion too.

Best Regards ! :)

I am not an expert of relativity theory but I don't agree about your statement that the heat equation governs a propagation at infinite speed. This is true if you model heat using the steady state equation that is elliptic. For the time dependent equation the characteristic lines exist in real space. Just as an example, consider the 1D case of a propagation of heat flux prescribed as BC at x=0 over a medium having initial uniform temperature. You will see the time evolution of the temperature gradient along x until a steady state is reached.

Hi Filippo,

You are partially right: the heat pulse from BC will, of course, evolve in time until reach the steady-state profile. But before this happens, small perturbations from the heat pulse will be felt instantaneously at any distance from the heat source. So, formally, the solution to the Fourier equation is not a wave (there is no wavefront). This is what people call "paradox of the heat conduction". Practically, such perturbations decay exponentially at the room temperature and the Fourier model gives a good approximation to many engineering problems. However, in the relativistic applications, such a pathology makes either Fourier or Navier-Stokes unusable.

Roughly speaking, the Fourier or Navier-Stokes constitutive laws are equilibrium laws and not applicable to non-equilibrium phenomena (small scales, fast processes, etc.). But in relativistic settings, they cannot be used even for equilibrium flows because of the causality issue. 

Hi Franklin ( Franklin Uriel Parás Hernández ),

relativistic heat conduction or actually any dissipative transport phenomena still don't have a conventional formulation. The whole bunch of various hyperbolic formulations has been proposed since the 1950s or something. Please have a look at works of these authors, for example:

David Jou , Georgy Lebon , P. Ván

they contributed a lot in the subject.

Best,

Hi Ilya,

yes, you are correct, a perturbation that has intensity depending on the diffusion coefficient is asymptotically present along x (we get a similar problem in the Prandtl equations for the boundary layer, x being the time-like axis, the BL extending up to y->+Inf). I addressed the fact that parabolic equations have different nature in the characteristic lines, more similar to the nature of hyperbolic equations than elliptic equations. This latter really are equilibrium equations that model an infinite velocity propagation of the disturbance in any direction.

I think that the key is in the Fourier closure model (the counterpart of the Newton model for the stress in fluid), the heat flux is linked by the temperature gradient only by a coefficient that models globally the "heat velocity", that is the way in which the kinetic energy of the molecule is transmitted to other molecules at a finite speed. In other words, as happens in defining a macroscopic velocity of the fluid in the momentum, I suppose one can define a macroscopic "heat velocity" that should be subtracted to the conduction coefficient, isn't that?

Just my two cents for an approximate evaluation of the heat speed with the Fourier model. Consider the analogy with the thermal boundary layer thickness delta(x). For the heat equations we could define in a similar approximate way the front displacement delta(t) and, by derivation in time, the heat velocity. Of course, the approximation is still in considering a finite length instead of the asymptotic extension.

Hi Franklin, in addition to some of three cases you have mentioned (low temperatures, high temperature gradients, short periods of time) I would add, in general, fast phase transitions. Probably three of attached files may somehow assist you in estimations of heat speeds or some concepts for processes with memory.

All the Best,

Peter

Hi Filippo,

>> I suppose one can define a macroscopic "heat velocity" that should be subtracted to the conduction coefficient, isn't that?

Exactly, the heat conductivity \kappa and the characteristic velocity ch of heat propagation canbe related as \kappa = cV \rho \tau ch^2, where \tau is a characteristic dissipation time (it can be related to the microscopic length scale of the heat carriers).

Franklin Uriel Parás Hernández

Also, please keep in mind that at small length scales, or any other situations mentioned by Peter, dispersive effects might be important. In some situations this can be addresed with the Guyer-Krumhansl model but, I think, it also should have causality issues and should be "hyperbolized".

Thank you all,

I see. The three of you have gave me more complementary points to this approach. I am gonna give a read to the papers attached.

By the way, the infinite speed of propagation and non-considerations of relativity description of Heat is one of the issues of the Fourier model of conduction ,

A second point is the inability of this equation to describe the cases mentioned above ( near absolute zero-temperatures, extreme Temperature Gradients, very short periods of time and fast phase transitions).

What I would like to know is; what of these cases are in direct relation (can be viewed also from the point of view) with the size of the system at very small scales? Since I know there is the particular case of heat transport at very tiny scales (nanoscale), where the Fourier's Law to model the conduction heat flux is not valid anymore. is this in function of the Knudsen number (Kn). Kn = lamdba/L, where lamdab is the mean free path of phonons or electrons and L is a characteristic lenght of the system.

If so, there may exists three different regimes of conduction in matter;

when Kn >>1, Kn >= 1 and Kn < 1

where the case when when Kn >>1 being the one when Fourier equation stops working ... , Am I telling the things more or less right ?

Dear Ilya M. Peshkov:

Does the expression for the velocity of heat propagation in the non-parabolic model is n't: \kappa = cV \tau ch^2, where \tau ?

( I wrote your expression without the Density (rho)) , I believe this is what I am finding in various references .

Best Regards all !

Hi Franklin,

>> Does the expression for the velocity of heat propagation in the non-parabolic model is n't: \kappa = cV \tau ch^2?

sorry, I didn't mentioned that cV in my message is the specific heat capacity with units J M-1 K-1=L2T-2K-1. So, the density should be present.

Just a curiosity, in case the heat is also trasported by a macroscopic velocity, for example in a fluid, the heat equation has also a hyperbolic term for the convection. Is this term somehow associated to a finite "heat velocity" propagation?

Of course, I am not considering a characteristic lenght scale of the problem below the requirement of the continuum model.

velocity turbulence can use a hyperbolic term for the convection example water on boiler .

There are three types of models for continuum media: hyperbolic, parabolic and elliptic ones. (Under a model, we mean a system of equations in particle derivatives.) According to a hyperbolic model, perturbations propagate at a finite speed. According to a parabolic one, they propagate at an infinite speed, but with a zero amplitude of the perturbation reaching an infinite point. What 'occur' in elliptic models You can see in Chapter 4 of our article 'Vapour explosion under hot water depressurization' in Research Gate.

All parabolic models do not agree with the theory of Special theory of Relativity. They predict an infinite velocity for the perturbations propagation, while according the Special theory of relativity the velocity cannot exceed the velocity of light. Already Antique Philosophiers found out that a hyperbolic model is the most 'complete' and 'correct'. Let us show the origination and the area of applicability for parabolic models on the example.

A simplest hyperbolic model consists of a single Wave Equation. This model describes the propagation of small (acoustic) perturbations through homogeneous media. A particular solution of this equation is called a 'wave'. It describes the perturbation traveling with no changing its form with a constant velocity of sound (or light).

Now consider a more complex hyperbolic model describing transient processes in a bubbly medium where perturbation change their form. The deformation of a perturbation is linked with the compression of bubbles by the perturbation arrival. The bubbles absorbs and then return some part of the perturbation energy. The velocity of the perturbation spread will change as a result. The only fore-part of the perturbation, precursor, propagates at a constant velocity. The velocity of the precursor movement corresponds to the speed of extremely high frequency perturbations which do not deform bubbles. Therefore, the precursor's velocity is called a 'frozen' speed of sound. Therefore in complex media a perturbation is decomposed into two parts, a fore-part traveling with a great, 'frozen', velocity and a tail one which propagate with much less velocity.

If the bubble content is great, the amplitude of the first part of perturbation will be very small. The precursor may be leaved out of account by assuming the velocity of the sound in a carrying phase (in liquid) to be infinity. Then, for modeling the perturbations propagation in a bubbly medium, we will obtain a parabolic equation.

Under transition to the parabolic model the front part of the perturbation "disappears". A traveling step will be substituted by a gently sloping profile with a zero amplitude at an infinite point. A main part of the perturbation will be absolutely similar in the calculations with the two models. (This perturbation is not a wave, in mathematical terms, since it is not a solution of a wave equation.)

The main part of the perturbation will spread with an effective velocity close to the velocity of the speed of sound in an equilibrium bubbly medium, that is with the velocity at which the perturbations of an extremely low frequency travel.

Just "alpha" in the Fourier Equation, You have given, that is an effective velocity of the heat spread (diffusion).

For describing the heat conduction in different media, there are applied different models. For solid bodies (soils), the equation of thermal heat conductivity is enough. In gases, besides heat diffusion, the heat convection is essential. Therefore the process is described by three equations of gas dynamics, the heat diffusion should be added to the equation of energy conservation. To describe the heat conduction in cosmos, where heating due to electromagnetic radiation is essential, Maxwell's equation should be added to the three equations of gas dynamics. The more 'convenient' is the model which describes the physics of the medium..

All the parameters of all the models consisting of equation in partial derivatives are fields. The heat, which is the gradient of the temperature field, is a field, too.

Dear Marina:

Thank you for sharing on the more mathematical side of this ,

Just one thing ...

The Gradient of the temperture (which is a scalar field) is a conservative vector field ... am I right ? in this case the Heat Flow, which we can see as a conservative vector field in every point in space (3D space) ... So, could we apply the framework of harmonic oscillators and wave description to this field(Heat)?

Best Regards ! :)

Dear Franclin,

1. You are right. The temperature field is a scalar one. Whereas its Gradient, the Heat Flux, is a vector field. The heat flux that moves from a warm body to a colder one is directed.

2. Since the Heat Flux field has a potential, which is the temperature, the field is conservative.

3. A General solution of any linear equation in Partial Derivatives is sought as a sum of its particular solutions, the harmonics. But only a partial solutions of a Wave Equation is called a (simple, running) wave. The particular solutions of the heat equation are not named 'the waves' but one can sought the solution of this equation in the form of harmonics (Fourier series).

Marina Ivashneva