Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

3 TheoryHistorically, it is not so easy to make out a definite time when the problem of diffusion enteredthe world of natural sciences. One reason is that the problem of diffusion is really diverse, andthere are many phenomenons that can be seen and treated as diffusional movement, equal if it istaking place in a molecular environment like in cells, or if it is taking place in a more macroscopicarea. The most of the literature on the topic is familiar in the opinion that it was the date, as theEnglish botanist Robert Brown made his famous research on pollen grains, suspended in water.But it is more so that Brown rediscovered or described a phenomenon that was already knownor described in the history of human science. Going back further this science history, namely tothe first century A.D. when the roman philosopher Lucretius lived and wrote a scientific poem,called ”De Rerum Natura” (On the nature of things), one is impressed by the following paragraph,describing the irregular movement of dust particles in the air that one can make out as a kind ofBrownian motion:”Observe what happens when sunbeams are admitted into a building and shed light onits shadowy places. You will see a multitude of tiny particles, mingling in a multitudeof ways. [...] Their dancing is an actual indication of underlying movements ofmatter that are hidden from our sight. [...] It originates with the atoms which moveof themselves. Then those small compound bodies that are least removed from theimpetus of the atoms, are set in motion by the impact of their invisible blows, andin turn cannon against slightly larger bodies. So the movement mounts up from theatoms, and gradually emerges to the level of our senses so that those bodies are inmotion that we see in sunbeams, moved by blows that remain invisible.”Lucretius who stood in the tradition of the Greek philosopher Epicure, can be seen as an atomistwho believed in the existence of atoms as the primary and indivisible elements of nature, in orderto explain the whole existence and behavior in the universe.Much later, in the year 1811, the brilliant physicist and mathematician Baron Jean Joseph Fourier,who at this time was the prefect of the department of Isère, France, received the price of the frenchacademy of science for his tremendous work on the spreading of heat in solid bodies [28, p. 112].In fact Fourier can be seen as the first who wrote down a so called heat conductivity equation, andtoday we know that thermal conductivity within an arbitrary material is based on the molecularmovement in it, and on the other hand, as was first proved by Einstein and Smoluchowski, the socalled Brownian motion or Diffusion is arising due to the thermal movement (see below). By thiswork Fourier can be seen as the one, who constituted the area of science, dealing with heat and itsproperties, today known as thermodynamics.In 1807 Fourier gave ”a careful justification of the assumption that the heat flow per unit of areais proportional to the gradient of temperature T , the constant of proportionality k (the internalconductivity) depending on the substance in question” [14, p. 165].In this and other later works Fourier derived a differential equation ”for the propagation of heat inthe interior of a continuous solid,( ) ∂Tcd = k∇ 2 T, (3.34)∂twhere d was the density, c the specific heat per unit and k the proportionality constant” [14, p.30