Targe system 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 , 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 (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

Targe system 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 , 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 (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