Stochastic order parameter dynamics for phase coexistence in heat conduction

We propose a stochastic order parameter model for describing phase coexistence in steady heat conduction near equilibrium. By analyzing the stochastic dynamics with a nonequilibrium adiabatic boundary condition, where total energy is conserved over time, we derive a variational principle that determines thermodynamic properties in nonequilibrium steady states. The resulting variational principle indicates that the temperature of the interface between the ordered region and the disordered region becomes greater (less) than the equilibrium transition temperature in the linear response regime when the thermal conductivity in the ordered region is less (greater) than that in the disordered region. This means that a superheated ordered (supercooled disordered) state appears near the interface, which was predicted by an extended framework of thermodynamics proposed in Nakagawa and Sasa [Liquid-Gas Transitions in Steady Heat Conduction, Phys. Rev. Lett. 119, 260602 (2017).]

Figure 1

Schematic of setup. The configuration of a single interface is displayed, where the heat flux $J<0$.

Figure 2

Outline of key equations.

Figure 3

Schematic graph of $T_{\mathrm{eq}}$ as a function of $E$. The phase coexistence is observed at $T_{\mathrm{eq}}=T_c$ for $E_1\le E\le E_2$.

Figure 4

The statistical average of a single interface is represented by an effective interface whose width remains microscopic. By the spatial average over a region of length $\mathrm{\Lambda }$, the interface in the mesoscopic description is defined.

Figure 5

Schematic figure of a stationary interface in equilibrium. $\mathrm{\Lambda }=\sqrt{\eta }$ and $L=1$ in the dimensionless form.

Figure 6

Example of the graph ${\psi }_X^{\mathrm{qeq}}\left(x\right)$.

Figure 7

Space-time plot associated with interface motion whose timescale is $O\left({\eta }^{-1/2}\right)$. Its close-up at a timescale of $O\left(1\right)$ is also shown.

Figure 8

Coarse-grained description for determining the temperature gap at the interface. See (5.33) for $T_0^{\mathrm{int}}$.

Figure 9

Temperature configuration in the late stage of a relaxation process. Latent heat is generated at the moving interface and it diffuses into the bulk regions. See (5.23) and (5.24) for the expression of the profiles. A temperature gap appears in the interface region. See (5.54) for the expression of the temperature gap.

Figure 10

Temperature profile $T_X^{*}\left(x\right)$ that maximizes the modified entropy $\tilde{\mathcal{S}}({\alpha }_X;E,J)$ for a given $X$. ${\kappa }^{\mathrm{d}}>{\kappa }^{\mathrm{o}}$.

Figure 11

Schematic of the main result.

Figure 12

Free energy as a function of $m$ for $T$ fixed.

Figure 13

$T\left(m\right)$ as a function of $m$.

Figure 14

Schematic figure of nonlocal Onsager coefficient ${\mathcal{L}}^{ab}$.

Figure 15

Schematic figure of interface motion from the quasi-equilibrium state to the equilibrium state.