Second sound and characteristic temperature in solids

Ruggeri, Tommaso; Muracchini, Augusto; Seccia, Leonardo

doi:10.1103/physrevb.54.332

Cited by 26 publications

(47 citation statements)

References 10 publications

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“…It resolves the 'paradox' associated with instantaneous propagation of heat predicted in classic irreversible thermodynamics; see Jou & Casas-Vázquez (1988) for a brief discussion. It also leads to the presence of a second sound in solids, an effect that has been observed in laboratory experiments on dielectric crystals; see, for example, the discussion by Ruggeri et al (1996). As described by Jou et al (1993), the Cattaneo equation provided a key stimulus for the development of extended irreversible thermodynamics.…”

Section: The Extended Thermodynamics Viewmentioning

confidence: 99%

“…In other words, they do not represent the most general nonlinearities that one might envisage (in the dissipative problem). In the interest of clarity, we have chosen not to compare the nonlinear terms in our model to various attempts at constructing nonlinear heat-conducting models; see, for example, Morro & Ruggeri (1987), Ruggeri et al (1996), Jou et al (2004), Lebon et al (2008b) and Llebot et al (1983). Such a comparison would obviously be interesting given that nonlinearities are relevant for the development of both shocks and turbulence.…”

Section: The Extended Thermodynamics Viewmentioning

confidence: 99%

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Variational multi-fluid dynamics and causal heat conductivity

Andersson

Comer

2009

Proc. R. Soc. A.

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We discuss heat conductivity from the point of view of a variational multi-fluid model, treating entropy as a dynamical entity. We demonstrate that a two-fluid model with a massive fluid component and a massless entropy can reproduce a number of key results from extended irreversible thermodynamics. In particular, we show that the entropy entrainment is intimately linked to the thermal-relaxation time that is required to make heat propagation in solids causal. We also discuss non-local terms that arise naturally in a dissipative multi-fluid model, and relate these terms to those of phonon hydrodynamics. Finally, we formulate a complete heat-conducting two-component model and discuss briefly the new dissipative terms that arise.

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Section: The Extended Thermodynamics Viewmentioning

confidence: 99%

Section: The Extended Thermodynamics Viewmentioning

confidence: 99%

Variational multi-fluid dynamics and causal heat conductivity

Andersson

Comer

2009

Proc. R. Soc. A.

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“…Systems of conservation laws with characteristic fields that fulfill the assumption (A) physically arise in several contexts, for instance in studying elastodynamic (e.g., see [Dp2]) or rigid heat conductors at low temperature [RMS1,RMS2]. An example is given by the generalized Cattaneo model proposed by T. Ruggeri and co-workers [RMS1,RMS2] to describe the heat propagation in high-purity crystals (He, NaF, Bi):…”

Section: U(0 X)=ū(x)mentioning

confidence: 98%

“…An example is given by the generalized Cattaneo model proposed by T. Ruggeri and co-workers [RMS1,RMS2] to describe the heat propagation in high-purity crystals (He, NaF, Bi):…”

Section: U(0 X)=ū(x)mentioning

confidence: 99%

A Wavefront Tracking Algorithm for N×N Nongenuinely Nonlinear Conservation Laws

Ancona¹,

Marson²

2001

Journal of Differential Equations

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We introduce a wavefront tracking algorithm for N × N hyperbolic systems of conservation lawsthat admits characteristic fields that are neither genuinely nonlinear nor linearly degenerate in the sense of Lax. Instead we assume that, for any nongenuinely nonlinear ith characteristic family, the derivative of the ith eigenvalue l i (u) of DF (u) in the direction of the ith right eigenvector r i (u), vanishes on a single (N − 1)-dimensional hypersurface in the u-space, transversal to the field r i (u). Systems that fulfill this type of assumptions are of particular interest in studying elastodynamic or rigid heat conductors at low temperature. The first proof of the existence of weak solutions for nongenuinely nonlinear systems was given by T. P. Liu (Mem. Amer. Math. Soc. 30 (1981), no. 240), based on a Glimm scheme. Our construction here provides an alternative method for establishing the global existence of weak solutions for such systems. Moreover, relying on the stability analysis developed in Ancona and Marson, preprint S.I.S.S.A.-I.S. A.S. 27/99/11, 1999, and preprint, 2000, we show that these solutions are entropy admissible in the sense of Lax.

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“…Since the system (2.16), (2.17) is quasilinear and hyperbolic (cf. (2.3)-(2.5)), it is possible to account for this decay, but it also becomes possible for shocks to form in finite times [7], [8], [10] in temperatures below ϑ λ . The present analysis examines the situation under which solutions taking values at temperatures on both sides of ϑ λ should remain smooth.…”

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confidence: 99%

Phase Transitions and Change of Type in Low-Temperature Heat Propagation

Saxton¹,

Saxton

2006

SIAM J. Appl. Math.

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Abstract. Classical heat pulse experiments have shown heat to propagate in waves through crystalline materials at temperatures close to absolute zero. With increasing temperature, these waves slow down and finally disappear, to be replaced by diffusive heat propagation. Several features surrounding this phenomenon are examined in this work. The model used switches between an internal parameter (or extended thermodynamics) description and a classical (linear or nonlinear) Fourier law setting. This leads to a hyperbolic-parabolic change of type, which allows wavelike features to appear beneath the transition temperature and diffusion above. We examine the region around and immediately below the transition temperature, where dissipative effects are insignificant. Significantly, these features appear only at certain temperatures below which the materials reach their peak thermal conductivities (at approximately 18.5 K and 4.5 K for NaF and Bi, respectively). No wavelike behavior is found in NaF and Bi at higher temperatures, where only diffusive heat propagation is observed. Further, the speed, U E , at which small amplitude thermal waves propagate is a decreasing function of temperature in the region where the waves can be detected, after which the diffusion process dominates. This hyperbolic region appears separated from the diffusive region by a "critical" temperature, ϑ λ , at which U E = 0 [1]. The aim of this paper is to understand the dynamics of regular solutions having temperatures close to that of the phase transition. We begin, in section 2, by describing a phenomenological onedimensional model which uses an internal variable behaving as an order parameter. In section 3, we will examine properties of the phase transition, and in section 4, we obtain conditions under which this class of solutions remain smooth. Some explicit cases are, finally, examined in section 5.

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Second sound and characteristic temperature in solids

Cited by 26 publications

References 10 publications

Variational multi-fluid dynamics and causal heat conductivity

Variational multi-fluid dynamics and causal heat conductivity

A Wavefront Tracking Algorithm for N×N Nongenuinely Nonlinear Conservation Laws

Phase Transitions and Change of Type in Low-Temperature Heat Propagation

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