Then, to determine the “multiplicity” of the 500-500 “macrostate,” use Stirling’s approximation. Now, if the coin is fair, each microstate is equally probably, so the odds of getting n heads in Ntosses are (nH;[N n]T) all (2.1) The multiplicity multiplicity in this case) in the center surrounded by the other possible multiplicities. The multiplicity of a system of N particles is then : W N, D = N! Question: For A Two State System, The Multiplicity Of A Macrostate That Has N_1 Particles First State And N_2 Particles In The Second State Is Given By For This System, Using Stirling's Approximation, Show That The Maximum Multiplicity Results When N_2=N_1. 1.1.2 What is the Stirling approximation of the factorial terms in the multiplicity, N! We will look more closely at what is known as Stirling's Approximation . Large numbers { using Stirling’s approximation to compute multiplicities and probabilities Thermodynamic behavior is a consequence of the fact that the number of constituents which make up a macroscopic system is very large. with the entropy then given by the Sackur-Tetrode equation, V / 47mU3/2 S = Nk in + N 3Nh2 LG )) 1.1.1 How many nitrogen molecules are in the balloon? 500! For a single large two-state paramagnet, the multiplicity function is very sharply peaked about N ↑ =N / 2. a. The multiplicity function for a simple harmonic oscil-lator with three degrees of freedom with energy E n is given by g(n) = 1 2 (n+1)(n+2) where n= n x +n y +n z. = Z ¥ 0 xne xdx (8) This integral is the starting point for Stirling’s approximation. Taking n= 10, log(10!) ∼ 2 π n n e n. An improved inequality version of Stirling’s Formula is . Let ↑ N and ↓ N denote the number of magnet-up and magnet-down particles. but the last term may usually be neglected so that a working approximation is. It’s also useful to call the total number of microstates (which is the sum of the multiplic-ities of all the macrostates) (all). See Glazer and Wark (2001) for more details. Estimate the height of the peak in the multiplicity function using Stirling’s approximation. 1.1 Entropy We have worked out that the multiplicity of an ideal gas can be written as 1 VN (2mmU)3N/2 ΩΝ & N! (2) can be trivially rewritten for large N, Mbin(k) = N k 1! The entropy is the natural logarithm of the multiplicity ˙= lng(N;s) = ln N! (b) What is the probability of getting exactly 600 heads and 400 tails? shroeder gives a numerical evaluation of the accuracy of the. n! 2.6 (multiplicity of a two-state system) 2.9 (multiplicity of an Einstein solid) 2.14 (Stirling's approximation) 2.16 (Stirling's less accurate approximation for ln N!) ∼ eN[−p1log(p 2)−p log(p )] = eNS[p], [3] where an entropy functional of Shannon type [2] appears, S[p] = − WX=2 i=1 pi logpi. lnN "! to determine the "multiplicity" of the $500-500$ "macrostate," use Stirling's approximation. That is, Stirling’s approximation for 10! therefore has a multiplicity of 2. amongst a system of N harmonic oscillators is (equation 1.55): g(N;n) = (N+ n 1)! Problem 20190 The multiplicity of a two-state paramagnet is Applying Stirling's approximation to each of the factorials gives (N/e)N (N - - (N - up to factors that are merely large, Taking the logarithm of both sides gives N In N In NJ - (N - NJ) In(N - ND. ’NNe N p 2ˇN) we write 1000! EINSTEIN SOLIDS: MULTIPLICITY OF LARGE SYSTEMS 3 n! We can follow the treatment of the text on p. 63 to take the ln of this expression and apply Stirling' s approximation : lnW= ln N!-lnD!-ln N-D !ºNlnN-N - DlnD-D - N-D ln N-D - N-D 2 phys328-2013hw5s.nb for the multiplicity of this gas, analogous to the 3D expression. ∼ 2 π n n + 1 ∕ 2 e − n. The formula is useful in estimating large factorial values, but its main mathematical value is in limits involving factorials. Suppose you have 2 coins and you ip them. The final logarithm can be written ln[N(1 — NJ/ N)] In N + In(l — N I/N). Pages 3; Ratings 100% (1) 1 out of 1 people found this document helpful. = lnN! 2h2N. Question: For A Two State System, The Multiplicity Of A Macrostate That Has N_1 Particles First State And N_2 Particles In The Second State Is Given By For This System, Using Stirling's Approximation, Show That The Maximum Multiplicity Results When N_2=N_1. Let n be the macrostate. is within 99% of the correct value. Use Stirling's approximation to estimate… 3. is. Solution for For a single large two-state paramagnet, the multiplicity function is very sharply peaked about NT = N /2. ˇ 1 2 ln2ˇ+ N+ 1 2 lnN N: (3) This can also be written as N! Using Stirling approximation (N! So the peak in the multiplicity … The most likely macrostate for the system is N ↑ =N ↓ =N/2. Take the entropy as the logarthithm of the multiplicity g(N,s) as given in (1.35): N s s g N 2 2 σ( ) ≈log ( ,0) − for s <

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