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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

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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.
Transcribed Image Text:(*) 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.
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