BoltzDis

# BoltzDis - Boltzman Distribution and the Most Probable...

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Boltzman Distribution and the Most Probable Distribution ε ε ε ε ε ε oo oooooo oo o 3 5 4 2 1 n =2 n =6 n =1 n =2 n =0 1 2 3 4 5 N = i n i ε = i n i E i sum over all states, i ε ε ε ε ε ε ooooo ooo oo o 3 5 4 2 1 n =5 n =3 n =2 n =1 n =0 1 2 3 4 5 U-U(0) = n i E i n i for most probable distribution W = N ! n 1 ! n 2 ! n 3 !... ln W = ln N ! – ln n i ! d(lnW) = lnW n i dn i Constraints: d N = dn 1 + dn 2 + dn 3 + . .. = dn i = 0 d ε = E 1 dn 1 + E 2 dn 2 + = 0 0 = lnW n i dn i + α dn i β E i dn i α and β undetermined multipliers 0 = lnW n i + α β E i dn i now n i 's are independent! lnW n i + α β E i = 0 Sterling's Formula: ln x! = x ln x - x ln W = N ln N – N (n j ln n j - n j ) n i = N so ln W = N ln N n j ln n j lnW n i = – n i ln n i n i + ln n i = –

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BoltzDis - Boltzman Distribution and the Most Probable...

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