Solitons and thermal fluctuations in strongly nonlinear solids

We study a chain of anharmonic springs with tunable power law interactions as a minimal model to explore the propagation of strongly nonlinear solitary wave excitations in a background of thermal fluctuations. By treating the solitary waves as quasiparticles, we derive an effective Langevin equation and obtain their damping rate and thermal diffusion. These analytical findings compare favorably against numerical results from a Langevin dynamic simulation. In our chains composed of two-sided nonlinear springs, we report the existence of an expansion solitary wave (antisoliton) in addition to the compressive solitary waves observed for noncohesive macroscopic particles.

Figure 1

Top: (a) The velocity profile of the compressive solitary wave (SW) generated through simulation in an athermal chain of beads with one sided interaction. Here the circles represent beads and we see that the solitary wave is around 5-bead diameters. (b)–(c) The velocity profiles showing the formation of a train of SW-ASW pair for two-sided nonlinear springs. A single particle is initially given an impulse to the right, generating a train led by a SW moving in the direction of the impulse (b), while simultaneously generating a symmetric train led by an ASW that moves in the opposite direction (c). Here, velocity is measured in units of $\omega a$, where $a=2R$ is the lattice spacing and $R$ is the radius of the beads and $\omega =\sqrt{\frac{k}{m}{a}^{\alpha -2}}$.

Figure 2

The energy momentum relation for the leading SW (black)-ASW (red) in the three cases shown in Fig. 1 following the energy ($E$) momentum ($P$) relation $E=\frac{P^2}{2m_{\text{eff}}}$. Energy is measured in units of $k{a}^{\alpha }$, where $k$ is the bare spring constant, $a$ is the lattice spacing, and $\alpha$ is the exponent of the nonlinear potential. Velocity is measured in units of $\omega a$, where $\omega =\sqrt{\frac{k}{m}{a}^{\alpha -2}}$. (Inset) Zoom-in of the leading SW-ASW pair from Fig. 1. We see that the SW and ASW are both around 5-bead diameters.

Figure 3

The sound speed computed from the dispersion curves for a range of $\Gamma$ (inverse temperature) for $\alpha =\frac{5}{2}$. The circles are from numerical simulation while the dashed blue line is a linear fit, giving a slope of 0.11 for $\alpha =\frac{5}{2}$, close to the expected value $\frac{\alpha -2}{2\alpha }=\frac{1}{10}$. The inset shows the kinetic (KE) and potential energy (PE) approaching thermal equilibrium, where their ratio satisfies the Virial relation $\frac{\alpha }{2}$ which is 1.25 for $\alpha =\frac{5}{2}$. Here, energy is measured in units of $\frac{k{a}^{\alpha }}{{10}^3}$ and time in units of $\omega a$, where $k$ is the bare spring constant, $a$ is the lattice spacing, and $\omega =\sqrt{\frac{k}{m}{a}^{\alpha -2}}$.

Figure 4

(Left) Snapshot of two leading solitary waves in a background of thermal fluctuations for $\alpha =2.5$ (red) (narrow profile) and $\alpha =2.2$ (black) (broader profile). The solitary waves are obtained from the velocity field $v\left(x\right)$ measured in units of $\omega a$, where $a$ is the lattice spacing and $\omega =\sqrt{\frac{k}{m}{a}^{\alpha -2}}$, while the $x$ axis is measured in units of $a$. (Right) The attenuation of the solitary wave as a function of time plotted on a log-natural scale, for various values of $\gamma$, $\Gamma$, and $\alpha$. The light blue circles correspond to $\alpha =2.2,\Gamma ={10}^4,\gamma =5\times {10}^{-5}$; red circles correspond to $\alpha =2.5,\Gamma ={10}^4,\gamma =7\times {10}^{-5}$; green circles represent $\alpha =2.5,\Gamma ={10}^4,\gamma =5\times {10}^{-5}$; and the brown squared correspond to $\alpha =2.5,\Gamma ={16}^4,\gamma =5\times {10}^{-5}$. The analytic estimates for these values based on Eq. (17) are shown by solid black lines. In these plots, the initial SW $E_{\text{SW}}=0.5$ measured in units of $k{a}^{\alpha }$, where $k$ is the bare spring constant and $a$ is the lattice spacing and time is measured in units of $\omega =\sqrt{\frac{k}{m}{a}^{\alpha -2}}$.

Figure 5

(Left) The numerically obtained mean solitary wave energy for $\alpha =2.5,\gamma =0.01$ (purple circles), $\alpha =2.2,\gamma =0.02$ (blue squares), $\alpha =2.5,\gamma =0.02$ (black circles), $\alpha =3.0,\gamma =0.02$ (gray squares) decaying exponentially compared against the analytical expression (solid curves). The mean decay rate is independent of the temperature ${\Gamma }^{-1}$. (Right) The numerically obtained variance of the square root of the solitary wave energy for $\alpha =2.2,\gamma =0.02,\Gamma =6000$ (blue squares), $\alpha =2.5,\gamma =0.02,\Gamma =6000$ (black circles), $\alpha =3.0,\gamma =0.02,\Gamma =6000$ (gray squares), $\alpha =2.5,\gamma =0.01,\Gamma =6000$ (purple circles), and $\alpha =2.5,\gamma =0.02,\Gamma =10000$ (red squares) compared against the analytical expression Eq. (25) (solid curves). The variance has dimensions of energy expressed in units of $\frac{k{a}^{\alpha }}{{10}^5}$ where $k$ is the bare spring constant and $a$ is the lattice spacing.