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# Thermal Physics PHYSICS 415

UW

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This 4 page Class Notes was uploaded by Nichole Keebler on Thursday September 17, 2015. The Class Notes belongs to PHYSICS 415 at University of Wisconsin - Madison taught by Ludwig Bruch in Fall. Since its upload, it has received 46 views. For similar materials see /class/205215/physics-415-university-of-wisconsin-madison in Physics 2 at University of Wisconsin - Madison.

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Date Created: 09/17/15

Lecture 2 Gauss and Poisson 23 Jan 09 Admin The library reserves course page for Physics 415 is located at this URL httpswwwlibrarywisceducourse pagesviewershow268 you will need to login to MyUW to view the contents A Review and extend Binomial distribution 1 Postulate the probability of a state occurring is proportional to the number of ways that it can be formed7 ie7 equal weight for each distinct con guration 3 Treat a 1D random walk of N steps of length L with 71R steps to the right 7 and 71L steps to the left 7 Or other random binary choices7 77 eg7 updown7 yesno Random the direction of successive steps is uncorrelatedindependent Allow for a bias in the walk each 7 has probability p 7 q 1 7 p A speci c sequence of steps has probability anan Even without large N7 a lot can be done explicitly a The total number is N 71R 71L b The net displacement is z 71R 7 nL and n3 lN965l271L N 7 Hal2 c The number of sequences many sequences of steps give the same x with 71R and 71L is NlannLl d The probability of the nR7nL grouping for displacement z is Pz ananNlnRXnLl indeed normalized 7 shown next 9 F N n N7n N Ewigpq m7pq The probability is PNn p 1 7pN NlnlN 7 1 Evaluate the generating function Fpq using the binomial theorem U rb H 3 9 Evaluate the factorial N by brute force 10 a Hence7 evaluate the normalization 2210 PNn 1 b The average mean and mean square are denotes A average ltngt ZnPNUquot paFaplq717p Np ltn2gt anPNn p55pp55PF N N 7 1p2 Np c The dispersion is W 7 7 12 Np1 7 p Npq The rms value is Then 13116 of text Several pieces of the mathematics that carry over to thermal physics 1 many microscopic ways to get the same net effect net z here 2 form average mean and dispersion variance 3 approximations to WNn that become sharply peaked and simpler at large N saddle point integration method of steepest descents next Stirling s approximation math aside 36 gtlt 106 50 7 30 gtlt 1064 65 82 gtlt 10907 7 exponentially large numbers Look for an accurate approximation to Nl Estimate the factorial by ln N NlnN 7 N Start from N ln Nl Z lnj j1 use a histogram like picture to get upper and lower bounds N N NlnN7N lnxdx 1an lnx1dz N11nN17N 1 0 For a more accurate estimate7 and a type of approximation used in thermal physics7 start from a generating function text AppA3 A6 Nl A00 zN exp7x dx E 000 expfx dx where f 7s Nlns f varies more slowly and is easier to approximate a Approximate the last integral by nding the maximum of f and making a second order Taylor series near that point dfdz 71 is zero at 0 N The second and third derivatives evaluated at x0 are dedxz 7N2 71N and dgfdxs 2Nx3 2N2 d The Taylor series then is AA 0 7 VV 95 3 950 WW 7 950 w 9mg A D V The integrand is then a small fraction of its peak value when x 7 x0 5xN 5 is a few and the third order term is still negligible Such statements depend on N being large A H Stirling7s approximation is obtained by plugging the second order Taylor series into the integral also Appendix A2 N 2 expf0 OOO exp7z 7 zo22N dx 2 expfz0 27TN or lanlenN7N ln27rN C Gaussian distribution H Use Stirling7s approximation in the form ln n nlnn 7 n to simplify the binomial probability distribution WNn pnqN NlnlN 7 anNn nlnp N771 lnqlan 7lnnl 7lnN7nl D Since dln zldz 2 ln x the maximum of ln WN is at the zero of dln WNndn 2 lnpq 7 lnn lnN 7 n at 0N n0PqP1 P HONP lenWNndn2 2 71n 7 1N 7 71Npq lt Oatn no The functional dependence near the peak maximum probability is is ln WNn 2 constant 7 n 7 n022Npq 3 A normalized approximation is WNW 27TNPQ 12 eXpHn NP22NPQl Write as Pz 1 exp7x22027 the bell curve77 or normal distribution 95 of the probability lies in the range 720 lt z lt 207 a piece of arithmetic which gets translated into 95 con dence limits77 in statistical applications D Poisson distribution EX19 1 Both the Poisson and Gaussian distributions are approximations to the binomial distribution for large N The Poisson case has the additional restriction p ltlt 1 The Poisson distribution arises in low counting rate77 experiments The algebra is with lnq ln1 7p 2 7p a anNn nlnp N 7 n lnq lan 7 lnnl 7 lnN 7 n b Stirling7s approximation for the factorials involving N is lan7lnN7n12 NlnN7NN7n7N7n lnN7n 2 7n 7 Nln1 7 nlnN 7 n 2 nlnN c Combine pieces anN 2 nlnp7 lnnl 7 NpnlnN nlan7 Np7 lnnl 2 This gives a normalized distribution n A WNW EGXPVAL A N10 Homework for January 26 1 P13 the probability of a speci c sequence of the digits is 5 gtlt 5 then count the number of sequences 2 P14 Can N odd give zero net displacement 3 P114 use the normalized Gaussian as in Eq1519 or write a short program to calculate ln W V 4 P118 form 27 and the average ofthe square7 using 714 07239 31 j

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