Pulling a harmonically bound particle subjected to Coulombic friction: A nonequilibrium analysis

We address the effects of dry friction, which has emerged only recently to play an important role in some biological systems. In particular, we investigate the nonequilibrium dynamics of a mesoscopic particle, bound to a spring being pulled at a definite speed, moving on a surface with dry friction in a noisy environment. We model the dry friction phenomenologically with a term that is proportional to the sign of the velocity, and by means of numerical simulations of a Langevin equation we show that (a) the frictional force scales with the logarithm of the pulling velocity, (b) the probability distribution function of the spatial displacement away from the potential minimum is non-Gaussian, (c) the fluctuation-dissipation theorem is violated as expected, but (d) the work function obeys the stationary fluctuation theorem, with an effective temperature related to the noise of the system.

Figure 1

A mesoscopic particle of mass $m$ bound to harmonic potential with spring constant $K_s$, pulled with a speed $v_s$ on a surface with Coulombic friction, and subjected to a Gaussian noise from the environment.

Figure 2

The stationary state phase space distribution $P(\tilde{v},\tilde{x})$ as obtained from numerical simulations of Eq. (1) for $\tilde{g}=1$ where $\tilde{x}=x^{\prime }/\ell , \tilde{v}=(\tau /\ell )v^{\prime }$, and $\tilde{g}=g/\left(\mathrm{\Delta }\sqrt{\tau }\right)$. (a) The normalized velocity distribution in the laboratory frame $P\left(\tilde{v}\right)/P_M$ [where $P_M$ is the maximum of $P\left(\tilde{v}\right)$ and $P\left(\tilde{v}\right)\equiv \int d\tilde{x}P(\tilde{v},\tilde{x})]$ for different values of ${\tilde{v}}_s$. From the left, ${\tilde{v}}_s=0(\square ), 1(\circ ), 2\left(▵\right)$, and $10\left(★\right)$. The sharp peak at $\tilde{v}\approx 0$ for all values of ${\tilde{v}}_s$ shown is a signature of dry friction. However, as ${\tilde{v}}_s$ increases, there appears a secondary maximum, roughly at $\tilde{v}\sim {\tilde{v}}_s$, corresponding to the object sliding due to the force from the spring. Increasing ${\tilde{v}}_s$ increases the magnitude of the peak and ${\tilde{v}}_s$ becomes the most probable speed. (b) The normalized spatial distribution in the comoving frame $P\left(\tilde{x}\right)\equiv \int d\tilde{v}P(\tilde{v},\tilde{x})$ for different values of ${\tilde{v}}_s$ [same legends as in (a)]. Note that the distribution is not Gaussian in $\tilde{x}$; this can be seen from the inset of (b) which shows the kurtosis of $P\left(\tilde{x}\right)$ vs ${\tilde{v}}_s$ for $\tilde{g}=1$. While the calculated skewness is small, the kurtosis starts out at roughly 5 for ${\tilde{v}}_s=0$, and goes asymptotically to 2 as ${\tilde{v}}_s$ increases. For comparison, a purely Gaussian distribution has a kurtosis of 3 (solid line) and uniform distribution has a kurtosis of 1.8.

Figure 3

Frictional force, as obtained from $F=K_s\langle x^{\prime }\rangle =\mathrm{\Delta }\langle \tilde{x}\rangle$ as a function of the pulling velocity, ${\tilde{v}}_s$, for increasing values of $\tilde{g}=0.1(+),1(\circ ),5\left(▵\right),10(\square )$. When fluctuations are weak, the force curve resembles a step function, but as $\tilde{g}$ increases, the force scales sublinearly for small ${\tilde{v}}_s$ and asymptotically goes to a constant for large ${\tilde{v}}_s$ given by $\mathrm{\Delta }$. (b) The force vs pulling velocity curve as in (a) but plotted in a linear-log scale [same legends as in (a)], showing that the force scales with ${\tilde{v}}_s$ as $\text{ln}{\tilde{v}}_s$ for sufficiently large $\tilde{g}$. This scaling is a signature of dry friction. The error bars represent the standard deviation of simulations [15].

Figure 4

The mean-squared displacement as a function of time for $\tilde{g}=0.2$ (a) and $\tilde{g}=1$ (b) for ${\tilde{v}}_s=0(\circ )$ and ${\tilde{v}}_s=2(▿)$. At low $\tilde{g}$, the MSD resembles the motion of the particle in the deterministic limit: at low ${\tilde{v}}_s$, the particle is in the diffusive regime, but at high ${\tilde{v}}_s$, the particle exhibits oscillatory motion. At high $\tilde{g}$, MSD resembles that of a damped harmonic oscillator in the underdamped limit. Note that in both cases, increasing ${\tilde{v}}_s$ increases the amplitude of the oscillation. Inset of (b): A phase diagram showing the region in the parameter space in which the particle exhibits diffusive (D) and underdamped (U) behaviors. The regions are obtained by visually inspecting the behavior of MSD at a particular $\tilde{g}$ and ${\tilde{v}}_s$. If the MSD monotonically increases to a constant value, we consider it as diffusive (D). On the other hand, if the MSD exhibits oscillations we consider it as underdamped (U). We note that only for a small range of $\tilde{g}$ and ${\tilde{v}}_s$ we have diffusive motion. The system crosses over the underdamped regime by increasing $\tilde{g}$ and ${\tilde{v}}_s$. Red points represent roughly the boundary of the diffusive and underdamped regimes.

Figure 5

Plots of the Fourier transform of the linear response function $k_B{T}_{\mathrm{eff}}\text{Im}\chi \left(\omega \right)$ ($\times$) and correlation function $\omega C\left(\omega \right)$ ($⬡$) for different values of $\tilde{g}$ and ${\tilde{v}}_s$, where $T_{\mathrm{eff}}$ represents the effective temperature. Here, case a corresponds to ($\tilde{g}=1,{\tilde{v}}_s=2)$, case b corresponds to $(\tilde{g}=1,{\tilde{v}}_s=5)$, case c corresponds to ($\tilde{g}=0.1,{\tilde{v}}_s=0$), and case d corresponds to ($\tilde{g}=1,{\tilde{v}}_s=0$). For a fluctuation-dissipation theorem to hold, $\omega C\left(\omega \right)$ and $2k_B{T}_{\mathrm{eff}}\text{Im}\chi \left(\omega \right)$ must overlap. For case a, at this particular $\tilde{g}$ and $\widetilde{v_s}$, the FDT is obeyed. For cases b–d, the FDT is violated. However, for case c, the FDT can be restored with a redefinition of the effective temperature, about $1.4\times T_{\mathrm{eff}}$. (e) A plot of the response function $2k_B{T}_{\mathrm{eff}}\text{Im}\chi \left(\omega \right)$ vs the correlation functions $\omega C\left(\omega \right)$ for cases a ($◼$), b ($⬢$), c ($•$), and d ($▼$), as indicated. Note that the response function and the correlation functions have been normalized by the maximum values of $\omega C\left(w\right)$ for each cases. For a fluctuation-dissipation-like theorem to hold, the slope would be 1, as indicated by the solid black line. We also observe that cases b and d are not single valued, implying a frequency dependent temperature.

Figure 6

Fluctuation theorem for the time-averaged work done (over a time period $\mathcal{T}$) on the particle: $W_{\mathcal{T}}\equiv -(K_s/\mathcal{T}){\int }_t^{t+\mathcal{T}}d{t}^{\prime }(x-v_s{t}^{\prime })v_s$. We plot from our numerical simulations, ${\mathcal{T}}^{-1}ln[P(W_{\mathcal{T}}=w)/P(W_{\mathcal{T}}=-w)]$ vs $W_{\mathcal{T}}/\left(k_B{T}_{\mathrm{eff}}\right)$ for (a) ($\tilde{g}=1,{\tilde{v}}_s=2$), and (b) ($\tilde{g}=1,{\tilde{v}}_s=5$), at three different values of $\mathcal{T}/\tau =5(\circ ), 15\left(▵\right)$, and $20\left(★\right)$, where $\tau \equiv \sqrt{m/K_s}$. The error bars indicate the standard deviation of simulations and for most of the points it is smaller than the symbols used [15]. We should note that the parameters for (a) and (b) correspond to the cases in which the FDT holds or does not, hold, respectively (see Fig. 5). To verify the stationary state FT, the value of $\mathcal{T}$ should be large enough. When increasing the value of $\mathcal{T}$, the slope of (a) approaches 1, while for (b) it is 0.84. Thus, the FT holds for (a) but for (b) it holds only if we redefine $T_{\mathrm{eff}}$, as 1.188 times the average kinetic energy of the particle. Insets: The probability distribution of work $W_{\mathcal{T}}$ for different values of $\mathcal{T}$.