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Introduction to statistical methods. 4 Many particle systems 4 Microscopic laws of physics -F4 describe behavior of many particle systems(1026) (a) Is it possible? hopeless!! insufficiency of memory & speed, chaos (b) Are we interested in microscopic details? Usually not! Only interested in macroscopic parameters like volume, pressure, conductivity, ... (kind of averaged quantity over very many particles) (c) Then, is that possible to understand macroscopic behavior? 4 Equilibrium & nonequilibrium *equilibrium; macroscopic parameters do not change in times easier to handle than non-equilibrium situation. *non-equilibrium ; much more difficult 4statistical average so many particles(1026) F4 great to apply statistical arguments! Elementary probability theory F4 take a example of "random walks" (most important and simple problem in probability theory) 1-1 Random walks * Statistical treatment F4 probability theory. * need "an assembly of very large # N of similarly prepared systems" <ensemble> (eg. dice throw, public opinion survey, . . .) * Outcome depends crucially on the characteristics of the ensemble chosen. * Simple example : "random-walk problem" drunken walker (Brownian motion of a suspended particle in a fluid < diffusion>) - How far would he reach in time ? - What is the probability of his being at after time elapsed? : probability distribution function <large number os similar trials> a4then average over trials with equal weight. 1-2. One-dimensional random walks Why? easy and applications ( magnetism: each atom either spins up or down total magnetization of N atoms) (always walk) Question : What is the probability of his being at after N steps? (1) Probability distribution Fundamental assumptions: Steps are "statistically independent" - no history dependency - # of different possible ways. normalization : ^