Thermal motion of a nonlinear localized pattern in a quasi-one-dimensional system

We study the dynamics of localized nonlinear patterns in a quasi-one-dimensional many-particle system near a subcritical pitchfork bifurcation. The normal form at the bifurcation is given and we show that these patterns can be described as solitary-wave envelopes. They are stable in a large temperature range and can diffuse along the chain of interacting particles. During their displacements the particles are continually redistributed on the envelope. This change of particle location induces a small modulation of the potential energy of the system, with an amplitude that depends on the transverse confinement. At high temperature, this modulation is irrelevant and the thermal motion of the localized patterns displays all the characteristics of a free quasiparticle diffusion with a diffusion coefficient that may be deduced from the normal form. At low temperature, significant physical effects are induced by the modulated potential. In particular, the localized pattern may be trapped at very low temperature. We also exhibit a series of confinement values for which the modulation amplitudes vanishes. For these peculiar confinements, the mean-square displacement of the localized patterns also evidences free-diffusion behavior at low temperature.

Figure 1

Three snapshots of a bubble in a system of $N=32$ particles in a cell of length $L=60$ mm, for a temperature $T=5\times {10}^9$ K and $\epsilon =0.08$. The blue circles are the coordinates $(x_i,y_i)$ of the point particles at time (a) 100 s, (b) 500 s, and (c) 700 s. The solid cyan line is the analytical bubble shape of Eq. (5). This shape is the same in every plot. The red straight line indicates the bubble position given by Eq. (14).

Figure 2

Discrete configurations of the particles in a bubble, zoomed in on the center particles. The number of particles is $N=64$ and the cell length is $L=120$ mm. The abscissa is the longitudinal position in units of $d$ and the ordinate is the transverse positions in mm. The dashed cyan line is the solitary-wave envelope (5) centered at $x=0$. The open circles indicate the unperturbed chain of particles and the crosses are the actual positions of the particles in the bubble. The arrows show the displacements of the particles. (a) The center particle is at the bubble apex $\mathrm{\Delta }=0$ [see Eq. (11)]. The solid black line indicates the plane of mirror symmetry. (b) Configuration shifted by half a period $\mathrm{\Delta }=1/2$ [see Eq. (12)]. The thick black plus sign indicates the symmetry center.

Figure 3

(a) and (b) The thick solid cyan line plots the potential energy for a bubble configuration (${10}^{-10}$ nJ) as a function of the dimensionless shift $\mathrm{\Delta }$, for $N=64$ particles in a cell of length $L=120$ mm and for confinement (a) $\epsilon =0.077$ and (b) $\epsilon =0.22$. The thin black solid line is a sinusoidal fit. Below we plot the particle configurations that correspond to the indices 1, 2, and 3 in (a) and (b). The dashed red line indicates the mirror symmetry plane and the red plus sign indicates the center of symmetry.

Figure 4

(a) Plot of the energetic barrier $\mathrm{\Delta }E\left(\epsilon \right)$ (in ${10}^{-10}$ nJ) as a function of the confinement $\epsilon$, for $N=64$ particles in a cell of length $L=120$ mm. The dashed blue line corresponds to confinements for which the configurations of minimal energy exhibit a mirror symmetry and the solid red line corresponds to confinements for which the configurations of minimal energy exhibit a center symmetry. The peculiar confinements ${\epsilon }_1,{\epsilon }_2$, and ${\epsilon }_3$ are discussed in Sec. 5. The blue square indicates the confinement of Fig. 3 and the red dot indicates the confinement of Fig. 3. (b) Plot of the difference between two successive zeros of $\mathrm{\Delta }E\left(\epsilon \right)$ as a function of the confinement $\epsilon$.

Figure 5

Plot of the longitudinal displacements (in mm) as a function of time (in s), for a system of $N=32$ particles, $L=60$ mm, at $T=5.\times {10}^9$ K, $\gamma =1{\mathrm{s}}^{-1}$, and $\epsilon =0.08$. (a) Typical trajectory of a bubble, measured by $x_{\mathrm{B}}\left(t\right)$ [red (dark gray) solid line], and longitudinal motion of a typical particle [cyan (light gray) solid line]. (b) Zoom in on the bubble trajectory (red solid line) [note the scale difference from (a)] and instantaneous position of the particle with maximum transverse displacement (green dots).

Figure 6

(a) and (c) Plots of the bubble MSD (logarithmic scale, in ${\mathrm{mm}}^2$) as a function of time (logarithmic scale, in s) for $N=32$ particles in a cell of length $L=60$ mm and $\epsilon =0.08$. The temperatures are $T={10}^9$ K (blue squares), $T=5\times {10}^9$ K (red triangles), and $T={10}^{10}$ K (green circles). The damping constant is (a) $\gamma =1{\mathrm{s}}^{-1}$ and (c) $\gamma =10{\mathrm{s}}^{-1}$. For the sake of comparison, the longitudinal MSDs of a typical particle in the system with temperature $T={10}^9$ K are shown as cyan diamonds. The dashed black lines correspond to the ballistic regime and the solid black lines to the diffusive regimes for a single particle in the chain [30]. (b) and (d) Plots of $\langle {\mathrm{\Delta }{x}_{\mathrm{B}}}^2\left(t\right)\rangle /k_{B}T$ (logarithmic scale, in ${\mathrm{s}}^2{\mathrm{kg}}^{-1}$) as a function of time (logarithmic scale, in s) for (b) $\gamma =1{\mathrm{s}}^{-1}$ and (d) $\gamma =10{\mathrm{s}}^{-1}$. The solid black line is the free diffusion of a bubble of mass $M_{\mathrm{B}}$ defined in Eq. (10) and the dashed black line is the ballistic regime of Eq. (15), without any fitting parameter.

Figure 7

Plot of the bubble MSD (logarithmic scale, in ${\mathrm{mm}}^2$) as a function of time (logarithmic scale, in s) for $\epsilon =0.06$ (green circles), $\epsilon =0.07$ (cyan triangles), $\epsilon =0.10$ (purple squares), and $\epsilon =0.18$ (red diamonds); $N=32$ particles; $L=60$ mm; temperature $T=5\times {10}^9$ K; and $\gamma =1{\mathrm{s}}^{-1}$.

Figure 8

Plot of the diffusion coefficient $D_{\mathrm{B}}$ extracted from the long-time behavior of the bubble MSD, normalized by the diffusion coefficient of a free single particle $D_0=k_{B}T/m\gamma$, as a function of the confinement $\epsilon$. We show data for $N=16$ and $L=30$ mm (green diamonds), $N=32$ and $L=60$ mm (red triangles), and $N=64$ and $L=120$ mm (blue circles).

Figure 9

(a) Three identical bubbles obtained for three sets of parameters $(N,\epsilon )$: $(16,0.25)$ and $L=30$ mm, green triangles; $(32,0.1)$ and $L=60$ mm, red crosses; and $(64,0.03)$ and $L=120$ mm, blue circles. We display the configurations $x_i$ and $y_i$ (both with length in mm), with the bubble apex in the middle of the cell. (b) Plot of the bubble MSD (logarithmic scale, in ${\mathrm{mm}}^2$) as a function of time (logarithmic scale, in s) for the three configurations of (a), using the same symbols. The temperature is $T=5\times {10}^9$ K and the dissipation is $\gamma =1{\mathrm{s}}^{-1}$ in every simulation. The solid black line is the free diffusion of a bubble of mass $M_{\mathrm{B}}$ defined in Eq. (10) and the dashed black line is the ballistic regime of Eq. (15), without any fitting parameter.

Figure 10

(a) Plot of the diffusion coefficient $D_{\mathrm{B}}$, normalized by the free particle diffusion coefficient $D_0=k_{B}T/m\gamma$, as a function of the confinement $\epsilon$, for $N=32$ and $L=60$ mm (red triangles) and $N=64$ and $L=120$ mm (blue circles). The closed symbols correspond to the diffusion coefficients deduced from the long-time MSD and the open symbols correspond to the calculated diffusion coefficients $k_{B}T/M_{\mathrm{B}}\gamma$, where the mass $M_B$ is given by (10). (b) A log-log plot of $D_{\mathrm{B}}/D_0$ as a function of the dimensionless mass $M_{\mathrm{B}}/m$ of the quasiparticle. The dashed line is the theoretical expectation of slope $-1$.

Figure 11

(a) and (c) Plots of the longitudinal bubble position $x_{\mathrm{B}}$ (in mm) as a function of time (in s), for a system of $N=64$ particles in a cell of $L=120$ mm with $\gamma =1{\mathrm{s}}^{-1}$. (a) and (b) The confinement is ${\epsilon }_t=0.085$ and $\mathrm{\Delta }E\left({\epsilon }_t\right)=7.9\times {10}^{-10}$ nJ. (c) and (d) The confinement is ${\epsilon }_b=0.093$ and $\mathrm{\Delta }E\left({\epsilon }_b\right)=1.0{10}^{-10}$ nJ. The temperatures are $T={10}^6$ K (cyan circles), $T=5\times {10}^6$ K (green squares), $T=7\times {10}^6$ K (red triangles), and $2\times {10}^7$ K (blue diamonds). (b) and (d) Plots of the corresponding MSD $\langle \mathrm{\Delta }{x}_{\mathrm{B}}^2\left(t\right)\rangle /k_{B}T$ (logarithmic scale, in ${\mathrm{s}}^2{\mathrm{kg}}^{-1}$) as a function of time (logarithmic scale, in s).

Figure 12

Plots of the bubble MSD $\langle \mathrm{\Delta }{x}_{\mathrm{B}}^2\left(t\right)\rangle$ (logarithmic scale, in ${\mathrm{mm}}^2$) as a function of time (logarithmic scale, in s) for a system of $N=64$ particles in a cell of length $L=120$ mm in a thermal bath at $T={10}^6$ K and $\gamma =1{\mathrm{s}}^{-1}$ for the confinements (a) ${\epsilon }_1=0.12$ (green circles), $\epsilon =0.13$ (cyan diamonds), and ${\epsilon }_2=0.134$ (blue triangles) and (b) ${\epsilon }_2=0.134$ (blue triangles), $\epsilon =0.138$ (purple diamonds), and ${\epsilon }_3=0.145$ (red circles). The solid black line is the long-time diffusive behavior with $D_{\mathrm{B}}=k_{B}T/M_{\mathrm{B}}\gamma$. Below we display the minimum energy configurations associated with each confinement, in increasing order of $\epsilon$ from left to right. The dashed lines indicate the mirror symmetry plane and the plus signs indicate the centers of symmetry.

Figure 13

(a) Plot of the saturation values of the MSD $\langle \mathrm{\Delta }{x}_{\mathrm{B}}^2\left(t\right)\rangle$ (in ${\mathrm{mm}}^2$) as a function of the confinement $\epsilon$, for $N=64$ particles in a cell of $L=120$ mm. (b) Plot of the amplitude $\mathrm{\Delta }E\left(\epsilon \right)$ (in ${10}^{-7}$ nJ) as a function of the confinement $\epsilon$ (see the text for the calculation method). In both plots the symmetries of the minimal energy configuration are, respectively, indicated in blue for the vertical plan symmetry (VP) and in red for the inversion point symmetry (IP).