Nonlinear energy dissipation and transfers in coarsening systems

During coarsening, small structures disappear, leaving behind only large ones. Here we study the spectral energy transfers in Model A, where the order parameter $\varphi$ evolves via nonconserved dynamics. We show that the nonlinear interactions dissipate fluctuations and facilitate energy transfers among the Fourier modes so that only $\varphi (k=0)$, where $k$ is the wave number, survives at the end and approaches the asymptotic value $+1$ or $-1$. We contrast the coarsening evolution for the initial conditions with $\langle \varphi (x,t=0)\rangle =0$ and with uniformly positive or negative $\varphi (x,t=0)$.

Figure 1

Schematic diagram illustrating the energy transfers (a) during the coarsening process in the TDGL equation and (b) in hydrodynamic turbulence (HDT).

Figure 2

Schematic evolution of the order parameter $\varphi (x,t)$: profile A (black curve) at $t=t_1$, profile B (red curve) at $t=t_2$, and the asymptotic profile C (green curve) at $t=t_3$, with $t_1<t_2<t_3$.

Figure 3

Plots of the energy spectra $E(k,t)$ for profiles A and B of Fig. 2. The solid line indicates Porod's law, $E\left(k\right)\sim k^{-2}$.

Figure 4

(a) Plots of $2E(k,t)$ (circles) and $T(k,t)$ (squares) for profiles A, B, C of Fig. 2 at small wave numbers. Here the contributions from wave numbers $k$ and $-k$ have been added. Note that $T(k,t)\approx -2E(k,t)$. (b) Plots of quantity $2E\left(k\right)+T\left(k\right)$ (squares) and dissipation spectrum $2k^{2}E\left(k\right)$ (circles) for profiles A and B.

Figure 5

Simulation of merger of a kink and an antikink discussed in Sec. 5a: (a) plots of $\varphi (x,t)$ at different times (see legend), (b) plots of $2E(k,t)$ (circles) and $T(k,t)$ (squares) at different times [same color convention as (a)], and (c) plots of quantity $2E(k,t)+T(k,t)$ (squares) and dissipation spectrum $2k^{2}E(k,t)$ (circles).

Figure 6

Simulation of merger of a kink and an antikink discussed in Sec. 5a, plot of $1/2-\tilde{E}\left(t\right)$ vs $t$. See Fig. 5.

Figure 7

Simulation of merger of a kink and an antikink discussed in Sec. 5a, plot of $E(k,t)$ vs $t$ for $k=1,2,3$ during the merger. The straight lines represent best-fit curves.

Figure 8

Simulation of biased $\varphi (x,t)$ discussed in Sec. 5b. (a) Plot of $\varphi (x,t)$ at different times (see legend). $\varphi (x,t)$ proceeds rapidly to 1 due to the bias in the initial condition. (b) Plots of $6E(k,t)$ and $-T(k,t)$ for small $k$'s for $\varphi$ of (a). We follow the same color convention. $T(k,t)\approx -6E(k,t)$, except $T(k=0,t)\approx -2E(k=0,t)\approx -1$ for $t>2$. (c) Plots of $6E(k,t)+T(k,t)$ and $2k^{2}E(k,t)$, which are small.

Figure 9

Simulation of biased $\varphi (x,t)$ discussed in Sec. 5b: (a) plot of $\langle \varphi (x,t)\rangle$ vs $t$ for various system sizes (see legend) and (b) plot of $E(k,t)$ vs $t$ for $k=1,3$. The dashed lines represent best-fit curves.

Figure 10

Simulation of Model A with disorder discussed in Sec. 5c: (a) plot of $\varphi (x,t)$ vs $x$ for intermediate time $t=50$, (b) plot of $2E(k,t)$ (circles) and $T(k,t)$ (squares) for the above profile, and (c) plot of $2E(k,t)+T(k,t)$ (squares) and dissipation spectrum $2k^{2}E(k,t)$ (circles).

Figure 11

Simulation of Model A with disorder discussed in Sec. 5c, averaging over 200 independent realizations. (a) Plots of $2\overline{E(k,t)}$ (circles) and $-\overline{T(k,t)}$ (squares) at different times (see legend) after a quench in initially disordered state. The dashed curve represents $E\left(k\right)\sim k^{-2}$ curve. (b) Plots of $2\overline{E(k,t)}+\overline{T(k,t)}$ (squares) and $2k^2\overline{E(k,t)}$ (circles) at different times.

Figure 12

Simulation of Model A with disorder discussed in Sec. 5c. Plots of $12\tilde{E}\left(t\right)\overline{E(k,t)}$ (circles) and energy transfer $\overline{T(k,t)}$ at different times. Empty and filled squares denote $\overline{T(k,t)}$ and $-\overline{T(k,t)}$, respectively (same convention as Fig. 4).

Figure 13

Plot of kink-antikink spacing $R\left(t\right)$ vs $t$ on a log-linear scale for data of Fig. 5. The inset shows $exp\left[R\right(t)/\xi ]$ vs $t$, where $\xi =1/\left(2\sqrt{2}\right)$. Equation (25) describes $R\left(t\right)$ until $t=t_c\simeq 5000$. In the inset we plot $exp(R/\xi )\times {10}^{-12}$ vs $t$ as the red-dashed curve. Following Eq. (25), $exp(R/\xi )=exp(R_0/\xi )-t/\xi$, which is illustrated as a black line in the inset (for $t<t_c$). The simulation box size is $L=100$.