Brownian motion

Principle

Brownian motion is the random movement of particles suspended in a fluid. However, it is also a mathematical model, often called a Wiener process (continuous-time stochastic processes such as white noise generation) with slightly different properties than the physical process. The mathematical model has no direct relevance for medical physics and biophysics, and therefore will be discussed briefly.

Fig. 1 Intuitive metaphor for Brownian motion

It is believed that Brown was studying pollen particles floating in water under the microscope. The pollen, as holds for all other kind of small particle are swimming randomly in water. One molecule of water is about 0.1 to 0.2 nm, (a hydrogen-bonded cluster of 300 atoms has a diameter of approximately 3 nm) where the pollen particle is roughly 1 µm in diameter, roughly 10,000 times larger in diameter than a water molecule. So, the pollen particle can be considered as a very large balloon constantly being pushed by water molecules. The Brownian motion of particles in a liquid is due to the instantaneous imbalance in the force exerted by the small liquid molecules on the particle.

Particles of a liquid (also a gas, so actually in a fluid) will show random movements due to the collisions with other liquid particles. Therefore, their displacement in a certain time t is less then linear with t. This also holds for the large molecules diluted or suspended in liquid. They make some 10¹³-10¹⁵ collisions per second with liquid molecules. It appeared that the mean square displacement in the x-direction is proportional with t:

. (1)

C is a constant to be solved. A macromolecule with velocity v is subjected to a friction force F_f :

F_f = ─fv, (2)

where f is the friction coefficient of the particle.

According to Stokes’ law (see Stokes’ law and hematocrit), applied to spherical particles f equals:

f = 6πηr, (3)

where η the dynamic viscosity of the medium and r the radius of the Brownian particle. This finally yields:

, (4)

where k is Boltzmann’s constant and T absolute temperature. t should be > 1 μs.

Example

Equation (4) applied to a blood platelet, supposing that it is by approximation spherical of shape (whereas in reality it is rather disk-like) with a radius of 0.001 mm, T = 310 K, and η of plasma is 2.0 mPa∙s (20 ⁰C) and t = 1, we get μm per second, which should be visible under a light microscope. (Often reality is more complicated: plasma behaves as a so called non-Newtonian fluid. Applying a high stress means a high viscosity and a low stress a low viscosity. Consequently, the plasma under high pressure should show smaller Brownian motion.)

Application

Brownian motion occurs in cells, alive or dead. Experimental biomedical applications are found in nanomedicine and special microscopic techniques (e.g. size estimation, mobility of fluorescent macromolecules, estimating diffusion coefficients).

Mathematical models (Wiener models) have several real-world applications, for example stock market fluctuations.

More Info

The diffusion equation (see Diffusion: general) yields an approximation of the time evolution of the probability density function associated to the position of the particle undergoing a Brownian movement under the physical definition. The approximation is valid on short timescales and can be solved with the Langevin equation, a stochastic differential equation describing Brownian motion in a potential involving a random force field representing the effect of the thermal fluctuations of the solvent on the particle. The displacement of a particle undergoing Brownian motion is obtained by solving the diffusion equation under appropriate boundary conditions. This shows that the displacement indeed varies as the square root of the time, not linearly.

Brownian motion always goes on. This is not in conflict with the first principal law of thermodynamics, since the kinetic energy of the medium particles provide continuously energy. This will result in local cooling, but this is balanced by heat transport from areas with a higher temperature. In this way the 2^nd principal law of thermodynamics is not violated. In other words, only locally and temporally the entropy is diminished (but can also be enlarged), but the entropy of the total system will not change.

Mathematical models of Brownian motion

One-dimensional random walk takes place on a line. So, one starts at zero, and at each step one moves by a fixed amount along one of the two directions from the current point, with the direction being chosen randomly. The average straight-line distance between start and finish points of a one-dimensional random walk of n steps is π^-1√(2n) ≈ 0.8√n. If "average" is understood in the sense of root-mean-square, then the average distance after n steps is √n times the step length exactly, just as the time t of the physical Brownian movement.

Brownian motion is among the simplest continuous-time stochastic processes, and it is a limit of both simpler and more complicated stochastic processes as 3D random walk.