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- (*) Consider a large system of volume V containing N non-interacting particles. Take some fixed subvolume V « V. Calculate the probability to find N particles in volume V. Now assume that both N and V tend to oo, but in such a way that the particle number density is fixed: N/V →n = const. a) Show that in this limit, the probability py to find N particles in volume V (both N and V are fixed, N «N) tends to the Poisson distribution whose average is (N) = nV. Hint. This involves proving Poisson's limit theorem. b) Prove that ((N – (N))²)/2 (N) 1 V(N) (so fluctuations around the average are very small as (N) > 1). c) Show that, if (N) > 1, pN has its maximum at N = (N) = nV; then show that in the vicinity of this maximum, 1 e-(N-nV)²/2nV /2TNV Hint. Use Stirling's formula for N! (look it up if you don't know what that is). Taylor- expand In pN around N = nV.In addition to the "distribution law" - which you have just plotted - Wien also determined a "displace- ment law" which describes the color of the peak in the blackbody spectrum according to b Amax = 7 (1) Here b is some constant which we will determine. (a) Set 4 = 0 to find the wavelength Amax that corresponds to the maximum value of u(A), and use your result to determine the value of the constant b defined in equation (1) above. You should find that b = 0.2014hc/kB, or about 2.9 × 10-3 m · K, a value known as Wien's constant.In addition to the cosmic background radiation of photons, the universe is thought to be permeated with a background radiation of neutrinos (v) and antineutrinos (v ), currently at an effective temperature of 1.95 K. There are three species of neutrinos, each of which has an antiparticle, with only one allowed polarization state for each particle or antiparticle. For parts (a) through (c) below, assume that all three species are exactly massless. Derive a formula for the number of neutrinos per unit volume in the neutrino background radiation. Evaluate your result numerically for the present neutrino temperature of 1.95 K.