Lecture2Jan28

Lecture2Jan28 - Section Times Monday 5pm-6pm Cabot Division...

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• Section Times: • Monday 5pm-6pm Cabot Division Rm Mallincrodt 102 • Tue 2pm-3pm M217 Chemistry Dept • All course materials and online discussions are at www.orwik.com (available to download before lecture) • (you should have received an invitation to join). Also an official website of the course will have slides of lectures available
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Previous Lecture:Main Points Concept of Statistical Ensemble: In case of uncertainty we have to make many experiments (like throwing dice) or observations (like sequence alignment). Set of these experiments forms Statistical Ensemble - an extremely important concept for our course. Having defined statistical ensemble we can now define probabilities of various outcomes for this Statistical Ensemble - Probabilities can be defined and measured only in the context of statistical ensemble whose definition MUST come first. Different outcomes in complex cases may be statistically independent - in this case their probabilities multiply - or statistically dependent reflecting deep and important properties of the system in question. In dealing with very complex systems such as materials or biological systems consisting of many particles or human society we often take statistical approaches describing these system in terms of probabilities of different outcomes. Nevertheless as we will see later this approach may be remarkably predictive for reasons which will become clear very soon.
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Key Concepts and Lessons: a) Coarse-grained statistics b) Self-Averaging c) Universality of Gaussian Distribution Reading: Reif, Ch1
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Example: Statistics of Random walks A group of N exuberant drunks (now in our wisdom we would say an ensemble of drunks ) leaves the pub and starts walking along the road. The probability to step right is p and probability to step left is q for each drunk (p may be not equal to q - the road may be sloped or the guys are just Republicans celebrating last election in Mass and making steps right more often than steps left). Each step advances them by one unit either right or left. Because the gentlemen are each of their steps are statistically independent of previous ones. Now the probability of a particular sequence of steps, say rrrllrlllrrrrlrrrrr equals to p s 1 q s 2 p s 3 q s 4 ... = p N r q N l where s 1 , s 2 , s 3 .... { } is a particular sequence of steps taken and N r and N l are total numbers of right and left steps taken. So - where would we find our friends after they have made N>>1 steps? At first glance it may appear that everywhere - each particular walk is not better or worse than any other walk Why then should there be any preferred ‘’location’’ to collect our drunks?
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Not all sequences of steps are created equal… In fact we are missing another important factor. Indeed at given Nr and N l we have assess how many sequences of steps correspond to given N r and N l (of course their sum N r +N l =N is fixed). Now the number of sequences of steps containing total N r steps right and N
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This note was uploaded on 05/04/2010 for the course CHEM 161 taught by Professor Shaklovich during the Spring '10 term at Harvard.

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Lecture2Jan28 - Section Times Monday 5pm-6pm Cabot Division...

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