brownian motion
The equation explained
In wordsOne part of the integral, measured against the whole integral, equals the other part measured against that same whole.OverviewEach part of the integral, divided by the whole of it, is the probability of finding the particles' centres in those regions at a moment picked at random. Where the particles move independently, the liquid is the same throughout and no forces act, equal regions must be equally probable, so the two are equal.Full explanationHere the molecular theory of heat enters. From the works cited in the footnote, the paper says, it follows easily that , and likewise , is the probability that at an arbitrarily chosen instant the particles' centres lie in the regions , respectively in the regions . Both are divided by the same , because is the whole integral over every position the particles may take, and it is the same whichever set of regions one asks about. Three conditions are then named: the motions of the individual particles are, with sufficient approximation, independent of one another; the liquid is homogeneous; and no forces act on the particles. Under them, regions of equal size must carry equal probabilities, since nothing distinguishes one place in from another. That is the display. It is the only place in the section where a physical assumption about the liquid is made, and the three conditions are exactly where it could fail: a force such as gravity, a liquid that differs from place to place, or particles close enough to feel one another would each single out positions.Every stepThe probability that at an arbitrarily chosen instant the particles' centres lie in the first set of regions.The probability for the second set. The denominator is the same , the integral over all positions available to the particles.The regions are of equal size, the liquid is homogeneous, no forces act, and the particles move independently of one another, so no set of positions in is more probable than another of the same size. The printed form.Also