Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Lecture 39: Review for FinalLecture 39: Review for FinalChapters 1-10, Monday April 21Chapters 1-10, Monday April 21stst•Announcements•Exam 3 statistics•Review third exam•Quiz (not necessarily in this order)•Review Chapters 3 & 4Reading: Reading: Chapters 1-10 (pages 1 - 207)Chapters 1-10 (pages 1 - 207)Final: Wed. 30th, Chs 5:30-7:30pm in Final: Wed. 30th, Chs 5:30-7:30pm in herehereExam will be cumulativeExam will be cumulativeExam 3 statisticsExam 3 statistics20 40 60 80 100024Mean = 64%Median = 60%1 perfect scoreQ1 - 5.1Q2 - 7.5Q3 - 6.5Number of studentsScore (%)Exam statistics with 1 dropExam statistics with 1 drop20 40 60 80 1000246Mean = 75%Median = 76%2 scores > 100Number of studentsScore (%)scale factors:Exam 1 1.18Exam 2 1.00Exam 3 1.08Review of Review of Chapters 3 & 4Chapters 3 & 4Classical and statistical probabilityClassical and statistical probabilityClassical probability:•Consider all possible outcomes (simple events) of a process (e.g. a game).•Assign an equal probability to each outcome.Let W = number of possible outcomes (ways)Assign probability pi to the ith outcome1 1& 1i iip p WW W= = � =�Classical and statistical probabilityClassical and statistical probabilityStatistical probability:•Probability determined by measurement (experiment).•Measure frequency of occurrence.•Not all outcomes necessarily have equal probability.•Make Make N N trialstrials•Suppose Suppose iithth outcome occurs outcome occurs nnii times timeslimiiNnpN��� �=� �� �0 1 2 3 4 5-3.0-2.5-2.0-1.5-1.0-0.50.0N1 0.510 0.15100 0.041000 0.013210000 0.00356100000 0.00145log( )log(N)( ) ( )log log0.516a N bas = +=-Statistical fluctuationsStatistical fluctuations1/ 2Ns-�The axioms of probability theoryThe axioms of probability theory1. pi ≥ 0, i.e. pi is positive or zero2. pi ≤ 1, i.e. pi is less than or equal to 13. For mutually exclusive events, probabilities add, i.e.•Compound events, (Compound events, (ii + + jj): this means either event ): this means either event ii occurs, or event occurs, or event jj occurs, or both. occurs, or both.•Mutually exclusive: events Mutually exclusive: events ii and and jj are said to be mutually exclusive are said to be mutually exclusive if it is impossible for both outcomes (events) to occur in a single if it is impossible for both outcomes (events) to occur in a single trial.trial.1 2........rp p p p= + + +•In general, for In general, for rr mutually exclusive events, the probability that one mutually exclusive events, the probability that one of the of the rr events occurs is given by: events occurs is given by:Independent eventsIndependent eventsExample:What is the probability of What is the probability of rolling two sixes?rolling two sixes?Classical probabilities:Classical probabilities:166p =Two sixes:Two sixes:1 1 16,66 6 36p = � =•Truly independent events always satisfy this property.•In general, probability of occurrence of r independent events is:1 2........rp p p p= � � �nixiStatistical distributionsStatistical distributions87 9 106, wherei iiiin xx N nN= =��Mean:Statistical distributionsStatistical distributionsnixi16, where limii i iiNnx p x pN��= =�Mean:N � �Statistical distributionsStatistical distributionsnixi16( ) ( )2 2i iix p x xs = D = -�Standard deviation( )221( ) exp22x xp xss p� �-� �= -� �� ��Statistical distributionsStatistical distributionsGaussian distribution(Bell curve)Statistical Mechanics – ideas and Statistical Mechanics – ideas and definitionsdefinitionsA quantum state, or microstateA quantum state, or microstate•A unique configuration.A unique configuration.•To know that it is unique, we must specify it To know that it is unique, we must specify it as completely as possible...as completely as possible...Classical probabilityClassical probability•Cannot use statistical probability.Cannot use statistical probability.•Thus, we are forced to use classical Thus, we are forced to use classical probability.probability.An ensembleAn ensemble•A collection of separate systems prepared in A collection of separate systems prepared in precisely the same way.precisely the same way.Statistical Mechanics – ideas and Statistical Mechanics – ideas and definitionsdefinitionsThe microcanonical ensemble:The microcanonical ensemble:Each system has same:Each system has same:# of particles# of particlesTotal energyTotal energyVolumeVolumeShapeShapeMagnetic fieldMagnetic fieldElectric fieldElectric fieldand so on....and so on................These variables (parameters) specify the These variables (parameters) specify the ‘macrostate’ of the ensemble. A macrostate is ‘macrostate’ of the ensemble. A macrostate is specified by ‘an equation of state’. Many, many specified by ‘an equation of state’. Many, many different microstates might correspond to the same different microstates might correspond to the same macrostate.macrostate.Ensembles and quantum states Ensembles and quantum states (microstates)(microstates)Cell volume, Cell volume, VVVolume Volume VV10 particles, 36 cells10 particles, 36 cells10161363 10ip-� �=� �� �� �Ensembles and quantum states Ensembles and quantum states (microstates)(microstates)Cell volume, Cell volume, VVVolume Volume VV10 particles, 36 cells10 particles, 36 cells10161363 10ip-� �=� �� �� �EntropyEntropyBoltzmann hypothesis: the entropy of a system is related to the probability of its being in a state.( ) ( )1np S f W WWf= � = =( )lnBS k W=Rubber band modelRubber band modeld( )( )! !,! ! ! !N NW N nn n n N n++ - + += =-( ) ( )ln ln ln lnW N N n n N n N n+ + + += - - - -Sterling’s approximation: ln(Sterling’s approximation: ln(NN!) = !) = NNlnlnNN  NN1 1 1 1ln ln2 2 2 2x x x xN+ + - -� �� � � � � � � �=- +� �� � � � � � � �� � � � � � �