CORRECTED VERSION: this model was developed to study the dependence of thermophysical parameters under the assumption of quasi-static transients between regimes. The aim is to extend the applicability of Prigogine’s theorem to the entire continuous spectrum of frequencies $\omega$. However, the resulting equations are intrinsic to a functional $S^2$ and do not explicitly depend on the quasi-static transient hypothesis. This allows the model to be reduced to an algebraic system that enables the determination of the parameters $y(\omega_0)$ and $z(\omega_0)$ directly from discretized harmonic data, making the system applicable to real measurements (stabilized periodic regimes).
I have corrected a previous point that I had deleted when I revised the section on the CV treatment: Prigogine’s theorem remains applicable for an aperiodic regime provided that the system passes through conditions of local quasi-equilibrium and, therefore, the transition from $\Omega_0$ to $\Omega'$ can be considered quasi-static. I had deleted the part of the text that mentioned this fact; my apologies. I am trying to find some other strategy for exploring thermophysical parameters using the \textbf{Glansdorff-Prigogine} criterion, which generalizes minimization to systems far from equilibrium, but I don’t think I’ll be able to finish anytime soon (I’ll probably graduate before then). For now, I’ll stay focused on the assumptions of a quasi-static aperiodic system. I’d really appreciate advice and guidance from experts... This thesis is far from trivial for a simple bachelor’s degree.
N.N.B: It is of fundamental importance to note that, when considering the system evolving along the frequency axis $\omega$ , the Helmholtz structure acts as a topological invariant. Even though the thermal “wavelength” changes as the frequency varies, the “shape” of the function (its nature as an eigenfunction of the Laplace operator) cannot be broken. The constraint $f(y,\dot{y},z,\dot{z},\omega)=0$ is therefore not a simple numerical limit, but a differential connection. It requires that the variation of the structural parameters $y(\omega)$ and $z(\omega)$ not be random, but “rotate” or “bend” precisely to compensate for their kinetic nature introduced by $\omega$.
This statement holds true until singularities arise in the solution of the system. In such cases, to preserve the structure of the constraint, one could consider the scenario in which the multiplier depends on the frequency, e.g.: $\beta = \beta(\omega)$ for media exhibiting structural changes. In general, in these cases, the Helmholtz structure breaks down. This may suggest a possible transition in the nature of the phenomenon under study toward a new one in which different physical mechanisms come into play to govern its behavior: a new constraint $f(y,\dot{y},z,\dot{z},\omega)=0$ must then be determined that is physically consistent with the new nature of the phenomenon.
As a student, I’m open to suggestions and feedback from you experts; I’d really appreciate it.
N.B: I have corrected an oversight on my part when I introduced the functional $\bar{\sigma}_{\epsilon}^2$ without defining it as the dynamic component of the dissipations. I hope my final touches make the explanation clearer, thank you for your patience.
00) Hello everyone, I have updated the document regarding the discussion of the Cattaneo-Vernotte model. The method used to derive the constraint $f=0$ for Fourier diffusion does not apply as well to the hyperbolic case. In fact, that discussion was rather vague and not very rigorous. Upon further investigation, I have arrived at a new and more powerful formulation applicable to hyperbolic systems. I hope to receive some feedback and guidance from experts in the field who can suggest possible improvements to the document. Thank you for your attention.
0) I would also like to apologize for previous references to other fields such as electromagnetism and quantum mechanics. Those results stemmed from my own careless mistakes; I should have checked everything more carefully. Therefore, my model remains valid only for the topic discussed here and for cases that can be traced back to a treatment allowing the derivation of the constraint f as well as $f_n$. Alternatively, one would have to seek a different type of constraint not discussed or considered here. I apologize again; I am truly very sorry. I should not have published the thesis prematurely, despite my reasons: I am still too inexperienced.
1) I have corrected the Cattaneo-Vernotte case. I apologize for my oversight; I am truly sorry. There is a clear discrepancy between the two cases; however, the variational model can also be applied to the CV case in a manner entirely analogous to that used for the parabolic Fourier case.
NB: I developed this model on a purely theoretical basis in an attempt to solve the three-year exam question that was assigned to me. I am publishing this thesis in the hope of receiving feedback from experts who can guide me and offer suggestions on the matter.
The study of determining the thermophysical parameters of a wall based on its steady-state frequency is far from trivial; the use of the entropy generation function was the final idea I arrived at after observing that, in the frequency domain, the system clearly satisfied Prigogine’s conditions. From there came the idea of exploiting this theorem to determine the thermophysical parameters of the wall. Unfortunately, I have no way to conduct an experimental verification of the model, as I am unable to determine thermal signals at the H and W interfaces that satisfy the assumption of free evolution of the system. Furthermore, since this is a three-year thesis, I ultimately lack practical experience and do not have the necessary instrumentation to launch an experimental campaign for this variational model. I would kindly ask for advice and opinions from you experts on what you think of my current work. Thank you for the attention you are giving to my document; it exceeds my initial expectations to observe a v/d ratio of ~1:1 in just 5 days.
I apologize for having edited the file countless times before arriving at this version, which I hope will be the final one (at least as far as my bachelor’s degree is concerned). The fact is that studying this topic entirely on my own was no easy task, and I often found myself questioning my own work. My plan to post the document here in the hope of getting guidance from experts came to nothing. Don’t get me wrong, I’m not blaming anyone but myself: given the nature of my thesis, it’s normal that no one can offer immediate feedback. To be honest, I’m feeling a bit anxious right now… but on the other hand, I’m quite satisfied with my work. I really hope you’ll find my work to your liking as well.
I apologize for the Italian, but this document is my thesis itself, and I didn’t initially plan to publish it on this platform.
Il presente lavoro introduce un approccio teorico innovativo per l’identificazione delle proprietà termofisiche in sistemi stratificati, superando i limiti dei modelli diffusivi classici. Attraverso l'integrazione tra la termodinamica dei processi irreversibili e l’analisi spettrale, viene formalizzato un modello capace di operare in regime stabilizzato sotto sollecitazioni arbitrarie, garantendo la determinazione univoca della stratigrafia interna.
Il nucleo della scoperta risiede nell'individuazione di un'invarianza strutturale profonda: il formalismo matematico e il vincolo di coerenza fisica rimangono inalterati nel passaggio dalla fenomenologia della diffusione pura a quella della propagazione d'onda a velocità finita. Questa proprietà permette di isolare la complessità fisica della materia all'interno di parametri spettrali specifici, mantenendo un'architettura logica universale.
Un'estensione fondamentale del modello ne dimostra la validità geometrica generale: la teoria non è confinata alla semplice parete piana, ma risulta applicabile a qualunque sistema di coordinate (cilindriche, sferiche o curvilinee). Sfruttando la natura delle soluzioni armoniche nello spazio, il metodo si configura come una teoria di campo coerente, rendendo l'identificazione dei parametri materiali indipendente dalla forma dell'oggetto analizzato. Questo trasforma il modello in uno strumento di diagnostica non distruttiva estremamente versatile, con applicazioni che spaziano dall'efficienza energetica civile ai sistemi critici dell'industria aerospaziale.