Universal energy transport law for dissipative and diffusive phase transitions

We present a scaling law for the energy and speed of transition waves in dissipative and diffusive media. By considering uniform discrete lattices and continuous solids, we show that—for arbitrary highly nonlinear many-body interactions and multistable on-site potentials—the kinetic energy per density transported by a planar transition wave front always exhibits linear scaling with wave speed and the ratio of energy difference to interface mobility between the two phases. We confirm that the resulting linear superposition applies to highly nonlinear examples from particle to continuum mechanics.

Figure 1

Example of a moving transition wave: (a) bistable topology of the on-site potential $\psi$ with minima at $u=0$ and $u=2$; (b) resulting transition wave profile (displacement vs position); (c) evolution of the kinetic energy per density vs time (the kinetic energy stabilizes at a constant value once the kink assumes a steady wave form); (d) contour plot of the wave propagation in $x-t$ form. The phase boundary moves at a constant velocity once it assumes a steady kink wave form.

Figure 2

Plots of the kinetic energy $E$ of the traveling wave vs kink propagation speed $v$ for (a) various examples of interaction potentials and (b) varying numbers of interacting neighbors. All examples use the bistable energy of Fig. 1 with $m=1$ for dissipative and $m=0.0001$ for weakly inertial or diffusive cases, and $a=1$. All results lie almost perfectly on the predicted lines with slopes $E/v=\mathrm{\Delta }\psi /2\gamma =1$.

Figure 3

(a) Three different topologies of the on-site potential $\psi$ with different equilibrium distances but with the same energy jump $\mathrm{\Delta }\psi$; (b) resulting energy per density $E$ vs wave speed $v$ for the different topologies and interaction potentials (all other parameters as in Fig. 2). Again, all computed values fall onto the predicted line with slope $E/v=\mathrm{\Delta }\psi /2\gamma =1$.

Figure 4

(a) Three possible topologies of a triple-well potential that generate propagating kinks: (I) $\mathrm{\Delta }{\psi }_1,\mathrm{\Delta }{\psi }_2>0$, (II) $\mathrm{\Delta }{\psi }_1>0,\mathrm{\Delta }{\psi }_2<0$, and (III) $\mathrm{\Delta }{\psi }_1<0,\mathrm{\Delta }{\psi }_1+\mathrm{\Delta }{\psi }_2>0$. (b) Resulting wave forms for the three cases: (I) two transition waves travel with different velocities, (II) only one partial transition wave propagates (the other is stationary as $\mathrm{\Delta }{\psi }_2<0$), and (III) one complete transition wave propagates with a constant velocity (the second transition drags the first along).

Figure 5

(a) Displacement profile and (b) kinetic energy when discreteness effects dominate and the wave profile is not smooth ($m=1000$ with the hyperelastic interaction potential).