The sampling rate of (B) is 10 times of that of (A). Red dots are measured positions, and red curves are measured trajectories. The black curve is assumed to be a true trajectory of the particle. (online color at: A 2D trajectory of a Brownian particle. If the measured displacement of the particle is during time, then the velocity of the particle is approximately. Now the measured trajectory is very close to the true trajectory of the particle. It is impossible to obtain the velocity of the particle from the measured trajectory in Fig. The measured trajectory (red curve) is completely different from the true trajectory, and appears chaotic. 1 A, the sampling rate is too small to measure the velocity of the Brownian particle. Figure 1 shows a 2D trajectory of a Brownian particle. This is termed “ballistic Brownian motion” to be distinguished from the common “diffusive Brownian motion”. This diverges as t approaches 0, and therefore does not represent the real velocity of the particle 11, 12.Īt short time scales (, where is the momentum relaxation time of a particle with mass M), the dynamics of a Brownian particle is expected to be dominated by its inertia and its trajectory cannot be self-similar. 1, the mean velocity measured over an interval of time t is. Since its trajectory is not differentiable, the velocity of a Brownian particle is undefined. They are commonly assumed to be continuous everywhere but not differentiable anywhere 10. The trajectories of Brownian particles are classic examples of fractals 9. Persistence and randomness are generally accepted as two key characteristics of Brownian motion. 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. 1 was provided by the brilliant experiments of Jean Perrin 7, recognized by the Nobel Prize in Physics in 1926. Langevin's approach is more intuitive than Einstein's approach, and the resulting “Langevin equation” has found broad applications in stochastic physics 6. In 1908, Paul Langevin introduced a stochastic force and derived Eq. von Smoluchowski also derived the expression of MSD independently in 1906 3, with a result that differed from Eq. The diffusion constant can be calculated by, where is the Boltzmann constant, T is the temperature, and is the Stokes friction coefficient for a sphere with radius R. Where is the mean-square displacement (MSD) of a free Brownian particle in one dimension during time t, and D is the diffusion constant.
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