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La dinamica del moto browniano
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  1. La dinamica del moto browniano Cosa hanno in comune ubriachi, luce, virus e mercati finanziari ?

  2. Le ricerche di Robert Brown(1773-1858)

  3. Cosa e’ il moto browniano

  4. Cosa non e’ il moto browniano

  5. Cosa e’ il moto browniano Remarks on Active Molecules (1827) “… extremely minute particles of solid matter, whether obtained from organic or inorganic substances, when suspended in pure water, or in some other aqueous fluids, exhibit motions for which I am unable to account, and which from their irregularity and seeming independence resemble in a remarkable degree the less rapid motions of some of the simplest animalcules of infusions.” … “These causes of motion (evaporation, currents), however, either singly or combined with others, -- as, the attractions and repulsions among the particles themselves, their unstable equilibrium in the fluid in which they are suspended, their hygrometrical or capillary action, and in some cases the disengagement of volatile matter, or of minute air bubbles, -- have been considered by several writers as sufficiently accounting for the appearances. … the insufficiencyof the most important of those enumerated, may, I think, be satisfactorily shown by means of a very simple experiment.

  6. Cosa e’ il moto browniano

  7. La teoria atomica e i metodi statistici: la teoria cinetica dei gas J.C. Maxwell (1831-1879) L. Boltzmann (1844-1906)

  8. La teoria cinetica dei gas • N molecole, ognuna di massa m, in un contenitore cubico di volume V (e lato L). • Variazione di velocita’ alla parete: 2 vx • Tempo tra collisioni: 2 L / vx • Forza sulla parete: m vx2 / L • In media < vx2 >= < vy2 >= < vz2 >= < v2 >/3 • Forza media sulla parete = N m v2 /3L • Forza media per unita’ di superficie (pressione): P = N m v2 /3V

  9. La teoria cinetica dei gas P V = N m v2 /3 = n R T K = m v2 /2 = 3 R T / 2 NA

  10. Effetti statistici di moti casuali(random walks) • Rayleigh (sovrapposizione di luce da sorgenti scorrelate) • Bachelier “Theorie de la speculation - 1900 (tesi di dottorato con Poincare’) • Einstein, Smoluchowsky, Langevin, Ornstein-Uhlenbeck,… (moto browniano)

  11. Meccanica statistica Combinazione di meccanica e descrizione probabilistica per arrivare a conclusioni sulle conseguenze mediee sulle fluttuazioni di proprieta’ macroscopiche. Concentrarsi sulle proprieta’ statistiche permette di utilizzare tecniche alternative alla soluzione delle equazioni del moto mcroscopiche per ottenere le stesse distribuzioni statistiche (metodi stocastici ovvero basati su probabilita’ e numeri casuali).