Anomalous Heat Diffusion

Consider anomalous energy spread in solid phases, i.e., ${⟨\mathrm{\Delta }{x}^2(t)⟩}_E\equiv \int {(x-{⟨x⟩}_E)}^2{\rho }_E(x,t)dx\propto t^{\beta }$, as induced by a small initial excess energy perturbation distribution ${\rho }_E(x,t=0)$ away from equilibrium. The second derivative of this variance of the nonequilibrium excess energy distribution is shown to rigorously obey the intriguing relation $d^2{⟨\mathrm{\Delta }{x}^2(t)⟩}_E/d{t}^2=2{\mathcal{C}}_{JJ}(t)/(k_B{T}^{2}c)$, where ${\mathcal{C}}_{JJ}(t)$ equals the thermal equilibrium total heat flux autocorrelation function and $c$ is the specific volumetric heat capacity. Its integral assumes a time-local Helfand-like relation. Given that the averaged nonequilibrium heat flux is governed by an anomalous heat conductivity, the energy diffusion scaling determines a corresponding anomalous thermal conductivity scaling behavior.

Figure 1

Numerical validation of the main result in (9) for a FPU chain with a length $N=401$, specific heat $c=0.828$ at a dimensionless temperature $T=1$ [33]. The red circles and blue squares are the second derivative ${d^2⟨\mathrm{\Delta }{x}^2(t)⟩}_E/d{t}^2$ as obtained from the insets (a) and (b), respectively. The black solid line depicts the result for the total heat flux autocorrelation ${\mathcal{C}}_{JJ}(t)$, i.e., the right-hand side of Eq. (9). Insets: (a) energy diffusion along the FPU chain using the linear response result (6) with an initial small Hamiltonian perturbation $\eta (x)$ that is composed of two Gaussians of opposite weights; (b) nonequilibrium energy diffusion as obtained from an initial near-equilibrium steady state. For further details, see Ref. [35].