Can a Lorentz Invariant Equation Describe Thermal Energy Propagation Problems?

Abstract

In the new technologies the development towards the small scales initiates and encourages the reformulation of those well-known transport equations, like heat and electric conduction, that were applied for bulk materials. The reason of it is that there are several physical evidences for the changes of the behavior of the signal propagation as the sample size is decreasing (Anderson & Tamma, 2006; Cahill et al., 2003; Chen, 2001; Liu & Asheghi, 2004; Schwab et al., 2000; Vazquez et al., 2009). The constructed different mathematical models clearly belong to the phenomena of the considered systems. However, presently, there is no a well-troddenway how to establish the required formulations in general. A great challenge is to establish and exploit the Lagrangian and Lorentz invariant formulation of the thermal energy propagation, since, on the one hand, the connection with other field theories including the interactions of fields can be done on this level, on the other hand, these provide the finite physical action and signal propagation. The results of the presented theory ensures a deeper insight into the phenomena, thus hopefully it will contribute to the technical progress in the near future. It is an old and toughish question how to introduce the finite speed propagation of action in such physical processes like the thermal energy propagation (Eckart, 1940; Joseph & Preziosi, 1989; Jou et al., 2010; Markus & Gambar, 2005; Sandoval-Villalbazo & Garcia-Colin, 2000; Sieniutycz, 1994; Sieniutycz & Berry, 2002). There is no doubt that the solution must exist somehow and the suitable description should be Lorentz invariant. Moreover, this Lorentz invariant formulation needs to involve anyway the Fourier heat conduction as the classical limit. The elaborated theory ensures that in the case of Lorentz invariant formulation both the speed of the signal and the action propagation is finite. Furthermore, for the Fourier heat conduction the temperature propagation is finite, however, the speed of action is infinite. This chapter treats the consequentmathematical formulation of a suitable relativistic invariant description of the above problem and its consequences, connections with other topics are also treated. As the author hopes it will be noticeable step-by-step that this synthesis theory may have a prominent role in the phenomena of nature. The construction of the Lorentz invariant thermal energy propagation, the Klein-Gordon equation with negative mass term providing the expected propagation modes, the limit to the classical heat conduction and the related dynamic phase transition between the dissipative – non-dissipative dynamic phase transitions are discussed in a coherent frame within Sec. 2. Two mechanical analogies are shown in Sec. 3 for the two kinds of Klein-Gordon type equations to see the distinct behavior due to 7