### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 Brownian motion in a gas
- 3 Brownian motion in a liquid
- 4 Effects of detection noise on studying Brownian motion
- 5 Future
- Acknowledgements
- References
- Biographies

Brownian motion has played important roles in many different fields of science since its origin was first explained by Albert Einstein in 1905. Einstein's theory of Brownian motion, however, is only applicable at long time scales. At short time scales, Brownian motion of a suspended particle is not completely random, due to the inertia of the particle and the surrounding fluid. Moreover, the thermal force exerted on a particle suspended in a liquid is not a white noise, but is colored. Recent experimental developments in optical trapping and detection have made this new regime of Brownian motion accessible. This review summarizes related theories and recent experiments on Brownian motion at short time scales, with a focus on the measurement of the instantaneous velocity of a Brownian particle in a gas and the observation of the transition from ballistic to diffusive Brownian motion in a liquid.

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Brownian motion in a gas
- 3 Brownian motion in a liquid
- 4 Effects of detection noise on studying Brownian motion
- 5 Future
- Acknowledgements
- References
- Biographies

Brownian motion is the apparently perpetual and random movement of particles suspended in a fluid (liquid or gas), which was first observed systematically by Robert Brown in 1827 [1]. When Brown used a simple microscope to study the action of particles from pollen immersed in water [1], he “observed many of them very evidently in motion”. The size of those particles was about 5 m. He also observed the same kind of motion with powders of many other materials, such as wood and nickel, suspended in water.

As first explained by Einstein in 1905 [2], the Brownian motion of a suspended particle is a consequence of the thermal motion of surrounding fluid molecules. Einstein's theory of Brownian motion predicts that

- (1)

M. von Smoluchowski also derived the expression of MSD independently in 1906 [3], with a result that differed from Eq. (1) by a factor of about 2. In 1908, Paul Langevin introduced a stochastic force and derived Eq. (1) from Newton's second law [4, 5]. Langevin's approach is more intuitive than Einstein's approach, and the resulting “Langevin equation” has found broad applications in stochastic physics [6]. Experimental confirmation of Eq. (1) was provided by the brilliant experiments of Jean Perrin [7], recognized by the Nobel Prize in Physics in 1926. Theodor Svedberg also verified the Einstein-Smoluchowski theory of Brownian motion and won the Nobel Prize in Chemistry in 1926 for related work on colloidal systems [8].

Persistence and randomness are generally accepted as two key characteristics of Brownian motion. The trajectories of Brownian particles are classic examples of fractals [9]. They are commonly assumed to be continuous everywhere but not differentiable anywhere [10]. Since its trajectory is not differentiable, the velocity of a Brownian particle is undefined. According to Eq. (1), the mean velocity measured over an interval of time *t* is . This diverges as *t* approaches 0, and therefore does not represent the real velocity of the particle [11, 12].

In 1900, F. M. Exner made the first quantitative study of Brownian motion by measuring the velocity of Brownian particles suspended in water [13, 14]. He found that the measured velocity decreased with increasing particle size and increased with increasing water temperature. However, his measured velocities were almost 1000-fold smaller than those predicted by the energy equipartition theorem [13]. The reason of this discrepancy was not understood until A. Einstein developed his kinetic theory about Brownian motion [2].

In 1907, Einstein published a paper entitled “Theoretical observations on the Brownian motion” in which he considered the instantaneous velocity of a Brownian particle [11, 12]. Einstein showed that by measuring this quantity, one could prove that “the kinetic energy of the motion of the centre of gravity of a particle is independent of the size and nature of the particle and independent of the nature of its environment”. This is one of the basic tenets of statistical mechanics, known as the equipartition theorem. However, Einstein concluded that due to the very rapid randomization of the motion, the instantaneous velocity of a Brownian particle would be impossible to measure in practice [11, 12]:

“We must conclude that the velocity and direction of motion of the particle will be already very greatly altered in the extraordinary short time θ, and, indeed, in a totally irregular manner. It is therefore impossible – at least for ultramicroscopic particles – to ascertain by observation.”

Nondiffusive Brownian motion of colloidal suspensions with high concentrations at short time scales have been studied by measuring the autocorrelation functions of multiply scattered, transmitted light [15-17]. Recent developments in optical tweezers and detection systems with unprecedented resolution now prove to be an indispensable tool for studying the Brownian motion of a single particle at short time scales [18-26]. For example, we have observed the ballistic Brownian motion and measured the instantaneous velocity of a Brownian particle for the first time with an optically trapped bead in air [21]. Huang et al. have observed the transition from ballistic Brownian motion to diffusive Brownian motion in a liquid [24]. Franosch et al. have observed resonances arising from hydrodynamic memory in Brownian motion in a liquid and the long-sought colored spectrum of the thermal force [25, 27].

Besides Brownian motion at short time scales, theoretical and experimental studies of anisotropic Brownian motion and Brownian motion in nonequilibrium systems are currently pursued by many groups. For example, several groups reported anisotropic Brownian motion of particles near interfaces [28-32], and Brownian motion of anisotropic particles such as ellipsoids [33, 34], nanotubes [35, 36] and helical bacteria [37]. Brownian motion in nonequilibrium systems is of particular interest because it is directly related to the transport of molecules and cells in biological systems. Important examples include Brownian motors [38, 39], active Brownian motion of self-propelled particles [40-46], hot Brownian motion [47], and Brownian motion in shear flows [48]. Recent theoretical studies also found that the inertias of particles and surrounding fluids can significantly affect the Brownian motion in nonequilibrium systems [49-54].

In this review, Section 'Brownian motion in a gas' introduces the theories of Brownian motion of particles in a gas, and the recent measurement of the instantaneous velocity of a Brownian particle in air. Section 'Brownian motion in a liquid' introduces the theories of Brownian motion of particles in a liquid at short time scales, the experimental observation of the colored thermal force, and the transition from ballistic to diffusive Brownian motion in a liquid. Section 'Effects of detection noise on studying Brownian motion' discusses the effects of detection noise on the measurement of different quantities of Brownian motion. Finally, in Section 'Future', we discuss future experiments on Brownian motion at short time scales.

### 3 Brownian motion in a liquid

- Top of page
- Abstract
- 1 Introduction
- 2 Brownian motion in a gas
- 3 Brownian motion in a liquid
- 4 Effects of detection noise on studying Brownian motion
- 5 Future
- Acknowledgements
- References
- Biographies

The main difference between the Brownain motion in a liquid and that in a gas is the hydrodynamic effects of the liquid [46]. The Brownian motion of colloidal particles in a liquid at high concentrations have been studied with diffusing wave spectroscopy, which requires each photon to be scattered many times before reaching the detector [15-17]. The recent developments in optical tweezers provide a new tool for studying the Brownian motion of single particles with unprecedented precision [18, 20, 21, 24, 25]. Meanwhile, precise calibrations (force, position, etc.) of optical tweezers demand better understanding of the Brownian motion of trapped particles [60, 69, 68, 70].

Table 1. Characteristic time scales of an optically trapped silica microsphere in water at 20 °C. Some examples of the spring constant of the optical trap (*k*) are shown in the 5th column. It is assumed to be inversely proportional to the diameter of the microsphere when the laser power is constantDiameter | | | | *k* | |
---|

(m) | (s) | (s) | (ns) | (N/m) | (s) |
---|

1.0 | 0.11 | 0.25 | 0.34 | 100 | 94 |

3.0 | 1.0 | 2.2 | 1.01 | 33.3 | 851 |

4.7 | 2.45 | 5.51 | 1.58 | 21.3 | 2083 |

10 | 11.1 | 25.0 | 3.4 | 10 | 9443 |

#### 3.1 Theory

Besides the inertia of the particle itself, the inertia of the surrounding liquid is also important for Brownian motion of particles in a liquid. The motion of a particle will cause long-lived vortices in the liquid that will affect the motion of the particle itself. This is the hydrodynamic memory effect of the liquid, which dominates the dynamics of the particle at short time scales. These hydrodynamic memory effects were first studied by Vladimirsky in 1945 [71]. In 1960s, several authors found in computer simulations that the velocity autocorrelation function (VACF) of fluid molecules had a power-law tail in the form of [72, 73], in contrast to the exponential decay in a dilute gas. Hinch obtained an analytical solution of the VACF for free particles from the original Langevin analysis [74]. Clercx and Schram calculated the MSD and VACF of a Brownian particle in a harmonic potential in an incompressible liquid [75], which can be used to describe the Brownian motion of an optically trapped microsphere in a liquid directly [30, 24, 76].

##### 3.1.1 A free particle in a liquid

The effective mass of the microsphere in an incompressible liquid is the sum of the mass of the microsphere and half of the mass of the displaced liquid [77, 78]:

- (16)

where

- (19)

is the momentum relaxation time of the particle due to its own inertia, characterizes the effect of liquid. Here η is the viscosity of liquid and *R* is the radius of the microsphere.

At long time scales, Eq. (18) approaches

- (20)

At short time scales, Eq. (18) approaches

- (21)

where

For a silica microsphere in water, . The normalized VACF approaches 1 at short time scales as

, rather than . Thus the dynamics of the particle is dominated by the hydrodynamic effects of the liquid. This is very different from the case in air.

##### 3.1.2 An optically trapped microsphere in a liquid

The optical trap provides a harmonic force on the microsphere when the displacement of the microsphere is small. where Ω is the natural angular frequency of the trap. Clercx and Schram [75] gave analytical solutions for the MSD and VACF of a trapped Brownian particle in a liquid, and Berg-Sørensen and Flyvbjerg [60] gave a solution for the power spectrum density (PSD) of a trapped Brownian particle in a liquid. This section introduces their analytical solutions and provides some numerical results to visualize those solutions.

Because the velocity of the Brownian motion of a microsphere in liquid is much smaller than the speed of sound in the liquid, the fluid motion can be described by the linearized incompressible time-dependent Navier-Stokes equation. The Langevin equation of the motion of a trapped microsphere in an incompressible liquid is [75]:

- (22)

The first term after the equal sign of Eq. (22) is the harmonic force, the second term is the ordinary Stokes's friction, the third term is a memory term associated with the hydrodynamic retardation effects of the liquid, and the last term is the Brownian stochastic force.

By the famous fluctuation-dissipation theorem, the thermal force is directly related to the frictional force. So the hydrodynamic memory of the liquid will affect both the thermal force and the frictional force. The thermal force is not a white noise, but becomes colored. The correlation in the thermal force is [25, 46, 75, 27]

- (23)

which is very different from a delta function (Eq. (4)).

The mean-square displacement of a trapped microsphere in a liquid is [75, 80]

- (24)

The coefficients *z*_{1}, *z*_{2}, *z*_{3}, and *z*_{4} are the four roots of the equation [80]

- (25)

where . For , Eq. (24) approaches

The normalized VACF of a trapped microsphere in a liquid is [75, 80]

- (26)

The power spectral density is [60, 80]:

- (27)

where

- (29)

- (30)

#### 3.2 Experimental observation of the color of thermal force in a liquid

Figure 11 shows the results of Franosch et al.’s experiment on the color of thermal force [25]. The data in Fig. 11(a) clearly shows the departure of thermal force from a white noise whose power spectrum is a horizontal line. The measured spectrum of the thermal force increases at higher frequencies. Franosch et al. also observed resonances in Brownian motion in liquid where overdamped motion is often assumed. In order to observe the resonance, they used optical tweezers with very large stiffness, and acetone as a liquid rather than water. The viscosity of acetone is about 3 times smaller than that of water. So the motion of a trapped bead is less damped in acetone than in water. The resonance can be clearly seen in Fig. 11(b). Remarkably, this resonance is mainly due to the inertia of the liquid, rather than the inertia of the particle itself.

#### 3.3 Experimental observation of the transition from ballistic to diffusive Brownian motion in a liquid

Recently, Huang et al. studied the Brownian motion of a single particle in an optical trap in water with sub-Angstrom resolution and measured the velocity autocorrelation function of the Brownian motion [24]. In their experiments, due to the confinement of the optical tweezers was typically two orders of magnitude larger than . So the role of the optical confinement can be neglected during the transition from ballistic to diffusive Brownian motion.

Although the spatial resolution of Huang et al.’s experiment is sufficient to observe the transition from ballistic to diffusive Brownian motion in MSD and even compute a VACF, it is not enough to measure the instantaneous velocity of Brownian motion in a liquid. The reason is that both MSD and VACF are insensitive to white noise of the detection system since they are averages over large data sets. On the other hand, the measurement of the instantaneous velocity is very sensitive to the detection noise. We will discuss about this issue in the following section.

### 4 Effects of detection noise on studying Brownian motion

- Top of page
- Abstract
- 1 Introduction
- 2 Brownian motion in a gas
- 3 Brownian motion in a liquid
- 4 Effects of detection noise on studying Brownian motion
- 5 Future
- Acknowledgements
- References
- Biographies

where the average is taken over all possible *t*_{0}. This derivation assumes no correlation between the real position of the microsphere and the detection noise. In this case, the real MSD of the microsphere can be obtained by , as in Ref. [18, 24]. is usually independent of time, as shown in Fig. 4.

The measured velocity of the microsphere is

- (33)

Thus is equivalent to .

The measured velocity autocorrelation function is

- (35)

If the noise of the detection system has almost no correlation (white noise), the last term of this equation can be neglected. Thus

- (36)

So the measurement of the velocity autocorrelation function is not sensitive to the noise of the detection system [24]. On the other hand, the measurement of the instantaneous velocity is very sensitive to the noise of the detection system [21].

The measured velocity becomes

- (38)

Then the velocity noise is

- (39)

### 5 Future

- Top of page
- Abstract
- 1 Introduction
- 2 Brownian motion in a gas
- 3 Brownian motion in a liquid
- 4 Effects of detection noise on studying Brownian motion
- 5 Future
- Acknowledgements
- References
- Biographies

As shown in this review, optical tweezers have become an indispensable tool for studying Brownian motion at short time scales. The instantaneous velocity of a Brownian particle trapped in a gas has been measured [21]. Recently, the velocity autocorrelation function of a Brownian particle in water was measured successfully for [30, 24, 79].

The instantaneous velocity of a Brownian particle in liquid is much more difficult to measure, and has not been measured to date. A successful measurement of the instantaneous velocity of a Brownian particle in a liquid will complete the task that was considered by Einstein more than 100 years ago [11, 12] and open a new door for studying Brownian motion. For example, the results can be used to test the modified energy equipartition theorem = and the Maxwell-Boltzmann velocity distribution. A measurement of the VACF > 0.35 will deepen our understanding of the hydrodynamic effects and compressibility effects of a liquid on Brownian motion [81-84]. The ability to measure the instantaneous velocity of a Brownian particle will be invaluable in studying nonequilibrium statistical mechanics [85, 86]. The Brownian motion of a suspended particle can be used for microrheology to probe the properties of fluids, such as viscoelastic fluids [87-89], and surrounding environments [90-92].