Hyperbolic heat conduction, effective temperature, and third law for nonequilibrium systems with heat flux

Some analogies between different nonequilibrium heat conduction models, particularly random walk, the discrete variable model, and the Boltzmann transport equation with the single relaxation time approximation, have been discussed. We show that, under an assumption of a finite value of the heat carrier velocity, these models lead to the hyperbolic heat conduction equation and the modified Fourier law with relaxation term. Corresponding effective temperature and entropy have been introduced and analyzed. It has been demonstrated that the effective temperature, defined as a geometric mean of the kinetic temperatures of the heat carriers moving in opposite directions, acts as a criterion for thermalization and is a nonlinear function of the kinetic temperature and heat flux. It is shown that, under highly nonequilibrium conditions when the heat flux tends to its maximum possible value, the effective temperature, heat capacity, and local entropy go to zero even at a nonzero equilibrium temperature. This provides a possible generalization of the third law to nonequilibrium situations. Analogies and differences between the proposed effective temperature and some other definitions of a temperature in nonequilibrium state, particularly for active systems, disordered semiconductors under electric field, and adiabatic gas flow, have been shown and discussed. Illustrative examples of the behavior of the effective temperature and entropy during nonequilibrium heat conduction in a monatomic gas and a strong shockwave have been analyzed.

Figure 1

Nondimensional effective temperature $\theta /T$ as a function of the nondimensional heat flux $\widehat{q}$: solid line, the effective temperature from the present model, Eq. (34); dashed line, the effective temperature from EIT [5, 6].

Figure 2

Nonequilibrium entropy $S_{\mathrm{neq}}$, Eq. (45), scaled with $S_{\mathrm{eq}}$ (solid line) and the entropy production ${\sigma }_S$, Eq. (46) (dashed line), as functions of the nondimensional heat flux $\widehat{q}$. The nonequilibrium entropy obtained by Camacho [27] from a maximum entropy formalism is placed for comparison (dash-dotted line).

Figure 3

Parameter minus γ as a function of the nondimensional heat flux $\widehat{q}$: solid line, the present model; dashed line, the quantum limit by Camacho [27]; dash-dotted line, the classical limit by Camacho [27].

Figure 4

(a) Schematic representation of the nonequilibrium state with the maximum heat flux $q=q^{max}$ when all the heat carriers move in the same direction. In this case $T_1=2T$, $T_2=0$, and $\theta =0$. (b) Schematic representation of the equilibrium state with $q=0$. In this case $T_1=T_2=\theta =T$.

Figure 5

Nondimensional effective temperature $\theta /T$ (solid line) as a function of nondimensional time $t/\tau$ during relaxation from the nonequilibrium state [see Fig. 4] to the equilibrium state [see Fig. 4]. The temperatures $T_1/T$(upper dashed line) and $T_2/T$ (bottom dashed line) are also shown for comparison.

Figure 6

Nonequilibrium entropy $S_{\mathrm{neq}}$, Eq. (45), scaled with $S_{\mathrm{eq}}$ (solid line) and the entropy production ${\sigma }_S$, Eq. (46) (dashed line), as functions of nondimensional time $t/\tau$ during relaxation from the nonequilibrium state [see Fig. 4] to the equilibrium state [see Fig. 4].

Figure 7

Nondimensional temperatures and heat flux distributions as functions of coordinate $x$ for a strong shockwave ($x=0$ is the wave front). The solid line denotes the longitudinal component of the temperature in the direction of the shockwave T (or $T_{xx}$ in terms of Ref. [30]) calculated from Eq. (34); solid circles denote the nonequilibrium molecular dynamics simulation data for $T_{xx}$ [30]; the dashed line and dash-dotted line denote the average (or effective) temperature θ and the heat flux $q$, respectively, taken from Ref. [30].