page_title: Brownian motion
section_title: Einstein's theory
hierarchical_section_title: Brownian motion / Statistical mechanics theories / Einstein's theory
caption_attribution_description: English: "A sketch of Perrin's results of a microscopic examination of the equilibrium or barometric distribution of granules of gamboge, which is a sticky substance used as a bright yellow water color. The granules are approximately uniform in diameter (0.6 microns). The relative change in density observed in 10 microns of suspension is equivalent to that occurring in 6 km of air or a magnification of 600 000 000."
context_page_description: Brownian motion, or pedesis, is the random motion of particles suspended in a medium.
  This pattern of motion typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow. More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy.
  This motion is named after the botanist Robert Brown, who first described the phenomenon in 1827, while looking through a microscope at pollen of the plant Clarkia pulchella immersed in water. In 1905, almost eighty years later, theoretical physicist Albert Einstein published a paper where he modeled the motion of the pollen as being moved by individual water molecules, making one of his first major scientific contributions.
context_section_description: There are two parts to Einstein's theory: the first part consists in the formulation of a diffusion equation for Brownian particles, in which the diffusion coefficient is related to the mean squared displacement of a Brownian particle, while the second part consists in relating the diffusion coefficient to measurable physical quantities. In this way Einstein was able to determine the size of atoms, and how many atoms there are in a mole, or the molecular weight in grams, of a gas. In accordance to Avogadro's law this volume is the same for all ideal gases, which is 22.414 liters at standard temperature and pressure. The number of atoms contained in this volume is referred to as the Avogadro number, and the determination of this number is tantamount to the knowledge of the mass of an atom since the latter is obtained by dividing the mass of a mole of the gas by the Avogadro constant.
  The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval. Classical mechanics is unable to determine this distance because of the enormous number of bombardments a Brownian particle will undergo, roughly of the order of 10¹⁴ collisions per second. Thus Einstein was led to consider the collective motion of Brownian particles.
  He regarded the increment of particle positions in time  in a one dimensional (x) space (with the coordinates chosen so that the origin lies at the initial position of the particle) as a random variable () with some probability density function . Further, assuming conservation of particle number, he expanded the density (number of particles per unit volume) at time  in a Taylor series,
  where the second equality in the first line is by definition of . The integral in the first term is equal to one by the definition of probability, and the second and other even terms (i.e. first and other odd moments) vanish because of space symmetry. What is left gives rise to the following relation:
  Where the coefficient after the Laplacian, the second moment of probability of displacement , is interpreted as mass diffusivity D:
  Then the density of Brownian particles ρ at point x at time t satisfies the diffusion equation:
  Assuming that N particles start from the origin at the initial time t = 0, the diffusion equation has the solution
  This expression (which is a normal distribution with the mean  and variance  usually called Brownian motion ) allowed Einstein to calculate the moments directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to the right. The second moment is, however, non-vanishing, being given by
  This equation expresses the mean squared displacement in terms of the time elapsed and the diffusivity. From this expression Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its square root. His argument is based on a conceptual switch from the "ensemble" of Brownian particles to the "single" Brownian particle: we can speak of the relative number of particles at a single instant just as well as of the time it takes a Brownian particle to reach a given point.
  The second part of Einstein's theory relates the diffusion constant to physically measurable quantities, such as the mean squared displacement of a particle in a given time interval. This result enables the experimental determination of Avogadro's number and therefore the size of molecules. Einstein analyzed a dynamic equilibrium being established between opposing forces. The beauty of his argument is that the final result does not depend upon which forces are involved in setting up the dynamic equilibrium.
  In his original treatment, Einstein considered an osmotic pressure experiment, but the same conclusion can be reached in other ways.
  Consider, for instance, particles suspended in a viscous fluid in a gravitational field. Gravity tends to make the particles settle, whereas diffusion acts to homogenize them, driving them into regions of smaller concentration. Under the ac
caption: The equilibrium distribution for particles of gamboge shows the tendency for granules to move to regions of lower concentration when affected by gravity.
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