Nucleation rate of critical droplets on an elastic string in a ${\varphi }^6$ potential

We obtain the nucleation rate of critical droplets for an elastic string moving in a ${\varphi }^6$ local potential and subject to noise and damping forces. The critical droplet is a bound soliton-antisoliton pair that carries a section of the string out of the metastable central minimum into one of the stable side minima. The frequencies of small oscillations about the critical droplet are obtained from a Heun equation. We solve the Fokker-Planck equation for the phase-space probability density by projecting it onto the eigenfunction basis obtained from the Heun equation. We employ Farkas’ “flux-overpopulation” method to obtain boundary conditions for solving the Fokker-Planck equation; these restrict the validity of our solution to the moderate to heavy damping regime. We present results for the rate as a function of temperature, well depth, and damping.

Figure 1

The local potential $V_1\left(u\right)$. From bottom to top the values of the shape parameter $a$ are 0.01, 0.1, 0.18, 0.22222.

Figure 2

From bottom to top along the ordinate, the bounce solution for $a=0.01$, 0.1, 0.18, 0.2222. The independent variable is $y=\sqrt{a}x∕c_0$.

Figure 3

From bottom to top along the ordinate, the “potential energy” function in Eq. (3.8) for $a=0.01$, 0.14223, 0.21, 0.222222.

Figure 4

The values of the squared frequencies for small oscillations for the two discrete eigenvalues for Eq. (2.22): the unstable mode with ${\omega }_0^2<0$ and the translation mode with ${\omega }_1^2=0$. The dashed line is the lower limit of the continuum, which begins at ${\omega }^2=a$.

Figure 5

Temperature dependence of the scaled nucleation rate for $\gamma =0.3$ and for different $a$ values. From right to left, $a=0.030$, 0.040, 0.050, 0.060, 0.070, 0.0878.

Figure 6

Temperature dependence of the scaled nucleation rate for $\gamma =0.3$ and for different $a$ values. From right to left, $a=0.0972$, 0.109, 0.124, 0.142, 0.167, 0.191, 0.210.

Figure 7

Scaled nucleation rate per unit length, $c_{0}I∕L$ as a function of $c_0∕T$ for fixed $a$ and for different $\gamma$ values.