^{1}and Giovanni Volpe

^{2,a)}

### Abstract

An optically trapped Brownian particle is a sensitive probe of molecular and nanoscopic forces. An understanding of its motion, which is caused by the interplay of random and deterministic contributions, can lead to greater physical insight into the behavior of stochastic phenomena. The modeling of realistic stochastic processes typically requires advanced mathematical tools. We discuss a finite difference algorithm to compute the motion of an optically trapped particle and the numerical treatment of the white noise term. We then treat the transition from the ballistic to the diffusive regime due to the presence of inertial effects on short time scales and examine the effect of an optical trap on the motion of the particle. We also outline how to use simulations of optically trapped Brownian particles to gain understanding of nanoscale force and torque measurements, and of more complex phenomena, such as Kramers transitions, stochastic resonant damping, and stochastic resonance.

This work was partially supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under Grant Nos. 111T758 and 112T235, and Marie Curie Career Integration Grant (MC-CIG) under Grant PCIG11 GA-2012-321726.

I. INTRODUCTION

II. SIMULATION OF WHITE NOISE

III. FROM BALLISTIC MOTION TO BROWNIAN DIFFUSION

IV. OPTICAL TRAPS

V. FURTHER NUMERICAL EXPERIMENTS

### Key Topics

- Diffusion
- 11.0
- Random noise
- 10.0
- Stochastic processes
- 8.0
- Ballistics
- 7.0
- Difference equations
- 7.0

## Figures

As the time step decreases, we must employ larger values of the Gaussian white noise to approximate the solution of the free diffusion equation [Eq. (3) ] accurately. (a) , (b) 0.5, and (c) 0.1. The corresponding solutions of the finite difference free diffusion equation [Eq. (5) ] in (d)–(f) for (lines) behave similarly. Although these solutions differ because they are specific realizations of a random process, their statistical properties do not change, as can be seen by comparing the shaded areas, which show the regions within one standard deviation of the mean of 10,000 realizations.

As the time step decreases, we must employ larger values of the Gaussian white noise to approximate the solution of the free diffusion equation [Eq. (3) ] accurately. (a) , (b) 0.5, and (c) 0.1. The corresponding solutions of the finite difference free diffusion equation [Eq. (5) ] in (d)–(f) for (lines) behave similarly. Although these solutions differ because they are specific realizations of a random process, their statistical properties do not change, as can be seen by comparing the shaded areas, which show the regions within one standard deviation of the mean of 10,000 realizations.

(a) For times smaller or comparable to the inertial time the trajectory of a particle with inertia (solid line) appears smooth. In contrast, in the absence of inertia (dashed line) the trajectory is ragged and discontinuous. (b) For times significantly longer than both the trajectory with inertia (solid line) and without inertia (dashed line) are jagged, because the microscopic details are not resolvable. These trajectories are computed using Eqs. (8) and (10) with and the same realization of the white noise so that the two trajectories can be compared. (c) The velocity autocorrelation function [Eq. (12) ] for a particle with inertia (solid line) decays to zero with the time constant , while for a particle without inertia (dashed line) it drops immediately to zero demonstrating that its velocity is not correlated and does not have a characteristic time scale. (d) A log-log plot of the mean-square displacement [Eq. (14) ] for a particle with inertia (solid line) shows a transition from quadratic behavior at short times to linear behavior at long times, while for a particle without inertia (dashed line) it is always linear. The particle parameters are *R* = 1 *μ*m, *m* = 11 pg, , *T* = 300 K, and are used here and for the numerical solutions shown in the following figures.

(a) For times smaller or comparable to the inertial time the trajectory of a particle with inertia (solid line) appears smooth. In contrast, in the absence of inertia (dashed line) the trajectory is ragged and discontinuous. (b) For times significantly longer than both the trajectory with inertia (solid line) and without inertia (dashed line) are jagged, because the microscopic details are not resolvable. These trajectories are computed using Eqs. (8) and (10) with and the same realization of the white noise so that the two trajectories can be compared. (c) The velocity autocorrelation function [Eq. (12) ] for a particle with inertia (solid line) decays to zero with the time constant , while for a particle without inertia (dashed line) it drops immediately to zero demonstrating that its velocity is not correlated and does not have a characteristic time scale. (d) A log-log plot of the mean-square displacement [Eq. (14) ] for a particle with inertia (solid line) shows a transition from quadratic behavior at short times to linear behavior at long times, while for a particle without inertia (dashed line) it is always linear. The particle parameters are *R* = 1 *μ*m, *m* = 11 pg, , *T* = 300 K, and are used here and for the numerical solutions shown in the following figures.

(a) Trajectory of a Brownian particle in an optical trap ( and ). The particle explores an ellipsoidal volume around the center of the trap, as evidenced by the shaded area which represents an equiprobability surface. (b) and (c) The probability distributions of finding the particle in the *z*- and *y*-planes follow a two-dimensional Gaussian distribution around the trap center.

(a) Trajectory of a Brownian particle in an optical trap ( and ). The particle explores an ellipsoidal volume around the center of the trap, as evidenced by the shaded area which represents an equiprobability surface. (b) and (c) The probability distributions of finding the particle in the *z*- and *y*-planes follow a two-dimensional Gaussian distribution around the trap center.

(a) As the trap stiffness increases, the particles become more and more confined as shown by the theoretical (solid curve) and numerical (symbols) variance of the particle position around the trap center in the *y*-plane and, in particular, by the probability distributions corresponding to (b) , (c) , and (d) .

(a) As the trap stiffness increases, the particles become more and more confined as shown by the theoretical (solid curve) and numerical (symbols) variance of the particle position around the trap center in the *y*-plane and, in particular, by the probability distributions corresponding to (b) , (c) , and (d) .

(a) The position autocorrelation function of a trapped particle [Eq. (23) ] gives information about the effect of the trap restoring force on the particle motion. As the trap stiffness and, therefore, the restoring force are increased, the characteristic decay time of the position autocorrelation function decreases. (b) The mean square displacement, unlike for the free diffusion case, does not increase indefinitely but reaches a plateau, which also depends on the trap stiffness—the stronger the trap, the sooner the plateau is reached.

(a) The position autocorrelation function of a trapped particle [Eq. (23) ] gives information about the effect of the trap restoring force on the particle motion. As the trap stiffness and, therefore, the restoring force are increased, the characteristic decay time of the position autocorrelation function decreases. (b) The mean square displacement, unlike for the free diffusion case, does not increase indefinitely but reaches a plateau, which also depends on the trap stiffness—the stronger the trap, the sooner the plateau is reached.

(a) The probability distribution of an optically trapped particle shifts in response to an external force. The black histogram shows the initial distribution and the grey histogram represents the distribution after the application of a constant external force . (b) The position autocorrelation function of the trapped particle [black line, Eq. (23) ] and position cross-correlation function [grey line, Eq. (20) ], are modulated in the presence of a rotational force field such as the one in Eq. (19) with . This modulation demonstrates the presence of the rotational force field even though it is not clear from the trajectory (the inset shows a trajectory during a time interval equal to 0.1 s). (c) Dynamic transitions between the two equilibrium positions in a double-well potential (Kramers transitions) with and [Eq. (21) ].

(a) The probability distribution of an optically trapped particle shifts in response to an external force. The black histogram shows the initial distribution and the grey histogram represents the distribution after the application of a constant external force . (b) The position autocorrelation function of the trapped particle [black line, Eq. (23) ] and position cross-correlation function [grey line, Eq. (20) ], are modulated in the presence of a rotational force field such as the one in Eq. (19) with . This modulation demonstrates the presence of the rotational force field even though it is not clear from the trajectory (the inset shows a trajectory during a time interval equal to 0.1 s). (c) Dynamic transitions between the two equilibrium positions in a double-well potential (Kramers transitions) with and [Eq. (21) ].

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