Statistical mechanics canonical ensemble

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1 canoncal ensemble system n thermal equlbrum wth bath of system probablty of mcro state T = k/ bath system p = 1 Z e E Z = e E average energy of system he = p E = what about? P E E e P e E ln

2 replcate many tme ensemble of N replcas

3 replcate many tme ensemble of N replcas number of replcas n mcro-state n = Np p =1

4 replcate many tme ensemble of N replcas number of replcas n mcro-state n = Np total number of mcro-states N = N! n 1!n 2!...n!... p =1

5 replcate many tme ensemble of N replcas number of replcas n mcro-state n = Np total number of mcro-states N = N! n 1!n 2!...n!... of ensemble p =1 apple N! S N = k ln N = k ln n 1!n 2!...n!...

6 replcate many tme ensemble of N replcas number of replcas n mcro-state n = Np total number of mcro-states N = N! n 1!n 2!...n!... of ensemble apple N! S N = k ln N = k ln n 1!n 2!...n!... Strlng approxmaton: lm ln x! =x ln x x x!1 p =1

7 replcate many tme ensemble of N replcas number of replcas n mcro-state n = Np of ensemble p =1 apple N! S N = k ln N = k ln n 1!n 2!...n!...

8 replcate many tme ensemble of N replcas number of replcas n mcro-state n = Np of ensemble p =1 apple N! S N = k ln N = k ln n 1!n 2!...n!... S N = k " N ln N N (n ln n n ) #

9 replcate many tme ensemble of N replcas number of replcas n mcro-state n = Np of ensemble p =1 apple N! S N = k ln N = k ln n 1!n 2!...n!... S N = k " N ln N N (n ln n n ) # S N = k " N ln N N N (p ln[np ] p ) #

10 replcate many tme ensemble of N replcas number of replcas n mcro-state n = Np of ensemble S N = k " p =1 N ln N N N (p ln[np ] p ) #

11 replcate many tme ensemble of N replcas number of replcas n mcro-state n = Np of ensemble S N = k " p =1 N ln N N N (p ln[np ] p ) # " # S N = k N ln N N N ln N N p ln p N

12 replcate many tme ensemble of N replcas number of replcas n mcro-state n = Np of ensemble S N = k " p =1 N ln N N N (p ln[np ] p ) # " # S N = k N ln N N N ln N N p ln p N S N = Nk p ln p

13 replcate many tme ensemble of N replcas number of replcas n mcro-state n = Np of ensemble p =1 S N = Nk p ln p

14 replcate many tme ensemble of N replcas number of replcas n mcro-state n = Np of ensemble p =1 S N = Nk p ln p of replca S = k p ln p

15 canoncal ensemble system n thermal equlbrum wth bath of system T = k/ bath system S = k p ln p

16 canoncal ensemble system n thermal equlbrum wth bath of system T = k/ bath system S = k p ln p Boltzmann dstrbuton p = 1 Z e E Z = e E 1 kt

17 canoncal ensemble system n thermal equlbrum wth bath of system T = k/ bath system S = k p ln p Boltzmann dstrbuton p = 1 Z e E Z = e E 1 kt substtutng and rearrangng S = k Z e E E + k Z e E ln Z

18 canoncal ensemble system n thermal equlbrum wth bath of system T = k/ bath system S = k p ln p Boltzmann dstrbuton p = 1 Z e E Z = e E 1 kt substtutng and rearrangng S = k Z e E E + k Z e E ln Z an almost famlar expresson S = 1 T he + k ln Z

19 canoncal ensemble system n thermal equlbrum wth bath free energy of system mcroscopc T = k/ bath system S = 1 T he + k ln Z

20 canoncal ensemble system n thermal equlbrum wth bath free energy of system mcroscopc T = k/ bath system S = 1 T he + k ln Z mcroscopc free energy kt ln Z = he TS

21 canoncal ensemble system n thermal equlbrum wth bath free energy of system mcroscopc T = k/ bath system S = 1 T he + k ln Z mcroscopc free energy kt ln Z = he TS A = kt ln Z

22 canoncal ensemble system n thermal equlbrum wth bath free energy of system mcroscopc T = k/ bath system S = 1 T he + k ln Z mcroscopc free energy kt ln Z = he TS A = kt ln Z macroscopc free energy A = U TS

23 canoncal ensemble system n thermal equlbrum wth bath free energy of system mcroscopc T = k/ bath system S = 1 T he + k ln Z mcroscopc free energy kt ln Z = he TS A = kt ln Z macroscopc free energy A = U TS from mcro to macro: generate partton functon Monte Carlo molecular dynamcs smulatons

24 p z A =4 p 2 p x p y dp p x = hn x 2L p y = hn y 2L p z = hn z 2L n x,n y,n z =1, 2, 3,... V (p + p) =4 p 2 dp

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015 Lecture 2. 1/07/15-1/09/15 Unversty of Washngton Department of Chemstry Chemstry 453 Wnter Quarter 2015 We are not talkng about truth. We are talkng about somethng that seems lke truth. The truth we want