|Scottish botanist Robert Brown's 1828 A Brief Account of Microscopical Observations, wherein he reports on the ransom movement of active molecules. |
The term “Brownian motion” is named after Scottish botanist Robert Brown who studied the movement of pollen grains suspended in water in 1827.
In 1905, German-born physicist Albert Einstein explained Brownian motion mathematically. 
In 1909, French chemist Jean Perrin, in his “Brownian Motion and Molecular Reality”, building on Einstein's 1905 Brownian motion work, stated that Avogadro's hypothesis is equivalent to saying that 'any two gram-molecules contain the same number of molecules', and commented further by stated:
“The invariable number N is a universal constant, which may appropriately be designated Avogadro’s constant.”
Perrin gave the following value: 
which, it seems, to the number of molecules in 32-grams of oxygen.
In 1912, Austrian-born Polish statistical physicist Marian Smoluchowski posited a conceptualized type of Maxwell demon trap door (Smoluchowski’s demon) that "vibrates" owing to Brownian motion energy impacts. 
in 1926, Perrin received the Nobel Prize in physics for proving, conclusively, the existence of molecules, by calculating Avogadro's number using three different methods, all involving liquid phase systems. First, he used a gamboge soap-like emulsion, second by doing experimental work on Brownian motion, and third by confirming Einstein’s theory of particle rotation in the liquid phase. 
In 1975, Austrian-born American astrophysicist Erich Jantsch, in his Design for Evolution: Self-organization and planning in the Life of Human Systems, discusses the second law of thermodynamics in the context of the Brownian motion of human molecules; an excerpt of which is as follows: 
“… dream of a pressure-free end state of maximum entropy in physical and social energy, in which action is neither possible nor necessary in any direction, only a kind of thermal (so-called Brownian) motion of the human "molecules" ...”
In 1978, American physicist Arthur Iberall applied the model of the thermodynamics of Brownian motion to interactions in chnopsological (biological) systems. 
In economics, and especially, econophysics, Brownian motion is often found convoluted together with French mathematician Louis Bachelier’s 1900 “random walk” hypothesis, wherein he treated securities prices similar to gas molecules, moving independently of each other, with future movements being independent of past movements. 
The following are related quotes:
“This suicide must be ranked as one of the great tragedies in the history of science, made all the more ironic by the fact that the scientific world made a complete turnabout in the next few years and accepted the existence of atoms, following Perrin’s experiments on Brownian motion.”— Stephen Brush (1964) on Boltzmann’s ironic death 
“Human beings mimicking Brownian motion seems not by itself much socially enlightening.”— Aaron Agassi (2009), comment #19 in Moriarty-Thims debate
1. Brown, Robert. (1828). “A Brief Account of Microscopical Observations, made in the months of June, July and August, 1827, On the Particles Contained in the Pollen of Plants; and On the General Existence of Active Molecules in Organic and Inorganic Bodies” (pdf), Philosophical Magazine, N.S. 4, 161-73.
2. Weatherall, James O. (2013). The Physics of Wall Street: A Brief History of Predicting the Unpredictable (pgs. 12-13). Houghton Mifflin Harcourt.
3. Perrin, Jean, B. (1926). Discontinuous Structure of Matter, Nobel Lecture, December 11.
4. Iberall, Arthur S. (1978). “A field and Circuit Thermodynamics for Integrative Physiology. III. Keeping the Books-a General Experimental Method.” American Journal of Physiology Regulatory Integrative and Comparative Physiology, 234 : 85-97.
5. Brush, Stephen. (1964). “Translator’s Introduction”, in: Lectures on Gas Theory (by Ludwig Boltzmann). University of California Press, Berkeley.
7. Jantsch, Erich. (1975). Design for Evolution: Self-organization and planning in the Life of Human Systems (human molecules, pg. 35). G. Braziller.
8. (a) Perrin, Jean. (1909). “Brownian Motion and Molecular Reality” ("Mouvement brownien et réalité moléculaire"), Annales de Chimie et de Physique, 18: 1–114.
(b) Engl. Trans. by Frederick Soddy (London: Taylor and Francis, 1910) [Excerpt: sections 1-6 complete (from: David M. Knight, ed., Classical Scientific Papers: Chemistry (New York: American Elsevier, 1968) and the abridgment reprinted in Henry A. Boorse & Lloyd Motz, The World of the Atom, Vol. 1 (New York: Basic Books, 1966)].
9. Norton, John D. (2010). “When a Good Theory meets a Bad Idealization: the Failure of the Thermodynamics of Computation”, Pitt University.
● Brownian motion – Wikipedia.