Quantitative approach to small-scale nonequilibrium systems

In a nanoscale system out of thermodynamic equilibrium, it is important to account for thermal fluctuations. Typically, the thermal noise contributes fluctuations, e.g., of distances that are substantial in comparison to the size of the system and typical distances measured. If the thermal fluctuations are ignored, misinterpretation of measured quantities such as interaction forces, potentials, and constants may result. Here, we consider a particle moving in a time-dependent landscape, as, e.g., in an optical tweezers or atomic force nanoscopic measurement. Based on the Kramers equation [H. A. Kramers, Physica 7, 284 (1940)], we propose an approximate but quantitative way of dealing with such an out-of-equilibrium system. The limits of this approximate description of the escape process are determined through optical tweezers experiments and comparison to simulations. Also, this serves as a recipe for how to use the proposed method to obtain knowledge about the underlying energy landscape from a set of experimental measurements. Finally, we perform estimates of the error made if thermal fluctuations are ignored.

Figure 1

(a) An optically trapped colloid approaching a surface. The measured height of the colloid over the surface is shown as a function of time. A $157\mathrm{nm}$ jump from the equilibrium position of the trap to the surface is observed at $t\simeq 9\mathrm{s}$. (b) Average jump length as function of approach velocity.

Figure 2

The interaction potentials used in Eq. (1), with $\kappa =0.01\mathrm{pN}∕\mathrm{nm}$, and $AR∕6=1440\mathrm{pN}{\mathrm{nm}}^2$. Dotted line: Harmonic potential of the measurement apparatus. Dashed line: van der Waals surface interaction energy between a colloid and a planar surface. Solid line: Total interaction energy from Eq. (1), sum of dotted and dashed lines.

Figure 3

Predictions of the jump distances using $\kappa =0.01\mathrm{pN}∕\mathrm{nm}$, and $AR∕6=1440\mathrm{pN}{\mathrm{nm}}^2$ (as in Fig. 2). Circles show a histogram of jump lengths from a Langevin simulation with $v=20\mathrm{nm}∕\mathrm{s}$. Solid line: Numerical integration of Kramers equation Eq. (9). Dashed line: Fit of Eq. (22) to the simulation data. Inset: Most likely jump length $z^{*}$ as function of approach velocity. Squares indicate numerical integration of Kramers equation and circles Langevin simulation.

Figure 4

Histogram (bin $\text{width}=20\mathrm{nm}$) of jumps at approach speed $20\mathrm{nm}∕\mathrm{s}$. The solid line shows a fit of Eq. (22). For comparison, a Gaussian fit (dashed line) is also shown. Inset: $z^{*}$ as a function of approach velocity. Circles (with error bars) denote experimentally obtained data. Asterisks indicate Langevin simulation data. The dashed line is a linear fit to the experimental data.