Lecture 10 Ideas of Statistical Mechanics Chapter 4 Wednesday January 30th Finish Ch 3 Statistical distributions Statistical mechanics ideas and definitions Quantum states classical probability ensembles macrostates Entropy Definition of a quantum state Reading All of chapter 4 pages 67 88 Homework 3 due Fri Feb 1st Assigned problems Ch 3 8 10 16 18 20 Homework 4 due next Thu Feb 7th Assigned problems Ch 4 2 8 10 12 Statistical distributions 16 ni xi Mean nx x i N i i where N i ni Statistical distributions 16 ni xi Mean nx x i N i i where N i ni Statistical distributions ni N 16 xi Mean x i pi xi where ni pi lim N N Statistical distributions 16 ni xi Standard deviation s Dx 2 p x x i i i 2 Statistical distributions 64 Gaussian distribution Bell curve 2 1 x x p x exp 2 s 2p 2s Statistical Mechanics Chapter 4 What is the physical basis for the 2nd law What is the microscopic basis for entropy Boltzmann hypothesis the entropy of a system is related to the probability of its state the basis of entropy is statistical Statistics Mechanics Statistical Mechanics Thermal Properties Statistical Mechanics Use classical probability to make predictions Use statistical probability to test predictions Note statistical probability has no basis if a system is out of equilibrium repeat tests get different results How on earth is this possible How do we define simple events How do we count them How can we be sure they have equal probabilities probabilities REQUIRES AN IMMENSE LEAP OF FAITH Statistical Mechanics ideas and definitions A quantum state or microstate A unique configuration To know that it is unique we must specify it as completely as possible e g Determine Position Momentum Energy Spin of every particle all at once THIS IS ACTUALLY IMPOSSIBLE FOR ANY REAL SYSTEM Statistical Mechanics ideas and definitions A quantum state or microstate A unique configuration To know that it is unique we must specify it as completely as possible Classical probability Cannot use statistical probability Thus we are forced to use classical probability An ensemble A collection of separate systems prepared in precisely the same way Statistical Mechanics ideas and The microcanonicaldefinitions ensemble Each system has same of particles Total energy Volume Shape Magnetic field Electric field and so on These variables parameters specify the macrostate of the ensemble A macrostate is specified by an equation of state Many many different microstates might correspond to the same Statistical Mechanics ideas and 64 definitions An example Coin toss again width Ensembles and quantum states microstates Volume V 10 particles 36 cells Volume V 10 particles 36 cells 10 1 pi 36 16 3 10 Cell volume V Ensembles and quantum states microstates Volume V 10 particles 36 cells Volume V 10 particles 36 cells 10 1 pi 36 16 3 10 Cell volume V Ensembles and quantum states microstates Volume V 10 particles 36 cells Volume V 10 particles 36 cells 10 1 pi 36 16 3 10 Cell volume V Ensembles and quantum states microstates Volume V 10 particles 36 cells Volume V 10 particles 36 cells 10 1 pi 36 16 3 10 Cell volume V Ensembles and quantum states microstates Volume V 10 particles 36 cells Volume V 10 particles 36 cells 10 1 pi 36 16 3 10 Cell volume V Ensembles and quantum states microstates Volume V 10 particles 36 cells Volume V 10 particles 36 cells 10 1 pi 36 16 3 10 Cell volume V Ensembles and quantum states microstates Volume V 10 particles 36 cells Volume V 10 particles 36 cells 10 1 pi 36 16 3 10 Cell volume V Ensembles and quantum states microstates Volume V 10 particles 36 cells Volume V 10 particles 36 cells 10 1 pi 36 16 3 10 Cell volume V Ensembles and quantum states microstates any more states look like this but no more probable than the last on Volume V 10 1 pi 36 16 3 10 Cell volume V There s a major flaw in this calculation Can anyone see it It turns out that we get away with it Entropy Boltzmann hypothesis the entropy of a system is related to the probability of its being in a state 1 p W S f n W f W S k B ln W
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