Statistical mechanics handout 4
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1 Statstcal mechancs handout 4 Explan dfference between phase space and an. Ensembles As dscussed n handout three atoms n any physcal system can adopt any one of a large number of mcorstates. For a quantum system se mcrostates are varous quantum levels system can be n, whle for a classcal system se mcrostates are varous combnatons of postons and veloctes atoms can have. Regardlessly, we refer to set of all se mcrostates as phase space. As dscussed n prevous handout statstcal mechancs s concerned wth calculatng probablty of beng n any gven mcrostate. The mcrostates that any physcal system can adopt wll depend on walls that are placed around t. If system s surrounded by walls that are mpermeable to heat, work and matter n system wll be confned to a set of mcrostates that all have same energy, volume and temperature. If system has walls that can exchange heat but that are mpermeable to work and matter n system wll be confned to a set of mcrostates that all have same volume and temperature. However, as long as y satsfy constrants on volume and number of atoms mcrostates wth all possble energes are permssble. We call set of states that are accessble to a system surrounded by walls. Clearly, set of mcrostates n any gven s a subset of set of mcrostates n phase space. Whch extensve fxed n mcrocanoncal. for What s probablty of beng n a partcular mcrostate Explan connecton between mcrocanoncal and a rmodynamc potental The mcrocanoncal (N V E) In statstcal mechancs when we develop models for solated systems we use mcrocanoncal. Ner materal, work nor heat can be exchanged wth solated systems so such systems have constant volume, constant number of atoms and constant nternal energy. As such se system can only be n mcrostates that have energy, E, volume, V and number of atoms N. In or words none of extensve rmodynamc varables need to be calculated by averagng and only constrant when we perform our constraned optmsaton s requrement of normalsaton. Consequently, f re are Ω mcrostates that have energy E, volume V and number of atoms N mcrocanoncal s gven by: Z mc (N, V, E) = Ω Furrmore, probablty of beng mcrostate s smply: p = Z mc (N, V, E) as long as mcrostate n queston has energy E, volume V and number of atoms N. The rmodynamc potental for ths s entropy, whch can be calculated from usng: S(N, V, E) = k B ln Z mc (N, V, E) To make connecton between classcal rmodynamcs and statstcal behavour of atoms we have to ncorporate effect of walls surroundng system. Walls fx values of certan extensve varables and wll thus force system to be n one of mcrostates n a subset of phase space ( ). The mcrocanoncal has fxed number of atoms, fxed volume and fxed nternal energy.
2 fxed n canoncal for canoncal Gve probablty mcrostate n canoncal Whch s rmodynamc behavour of canoncal Descrbe two ways n whch average of energy Explan how heat capacty can be calculated The canoncal (NV T ) In statstcal mechancs when we develop models for closed systems that cannot do PV work we use canoncal. Ner materal nor work can be exchanged wth closed systems that cannot do PV work so such system have constant volume and constant number atoms. The energy, however, has to be calculated as an average. Insertng se requrements nto expresson for that we derved on worksheet 3 we arrve at: E = λ () k B N dn λ () E dv + λ () d E Rememberng that we can also calculate by combnng frst and second laws of rmodynamcs (see handout 2) we arrve at: Equatng coeffcents of de gves us: = P T dv µ T dn + T de λ () = k B T = β Ths quantty k B T appears n many places n statstcal mechancs so t s thus gven specal symbol β. The canoncal s equal to: Z c (N, V, T ) = e βe where sum here runs over all mcrostates that have volume V and number of atoms N. The probablty of beng n any mcrostate wth volume V and number of atoms N s: e βe p = Z c (N, V, T ) By nsertng expresson above nto p and dfferentatng t wth respect to β t s possble to show that: E ( ) ln Zc (N, V, T ) E p = β (E E) 2 2 ) ln Z c (N, V, T ) β 2 In addton, we can relate logarthm of canoncal to Helmholtz free energy usng: F (N, V, T ) = k B T ln Z c (N, V, T ) Last but not least we can show that constant volume heat capacty (a response ) s related to average of fluctuatons n nternal energy va: C v = k B T 2 (E E)2 Closed systems are examned usng canoncal n whch volume and number of atoms are assumed fxed. The canoncal s connected to Helmholtz free energy. The nternal energy s calculated as an average. 2
3 fxed n sormalsobarc for sormalsobarc Gve probablty mcrostate n sormalsobarc Whch s rmodynamc behavour of sormalsobarc Descrbe two ways n whch average of energy/volume Explan how sormal compressblty/heat capacty can be calculated The sormal-sobarc (N P T ) In statstcal mechancs when we develop models for closed systems that can do PV work we use sormal-sobarc. Materal cannot be exchanged wth closed systems so such systems have constant number atoms. The energy and volume, however, have to be calculated as averages. Insertng se requrements nto expresson for that we derved on worksheet 3 we arrve at: k B = [ ] E λ () + λ (2) dn + λ () d E + λ (2) d V N N Rememberng that we can also calculate by combnng frst and second laws of rmodynamcs (see handout 2) we arrve at: = µ T dn + T de + P T dv Equatng coeffcents of de and dv gves us: λ () = k B T = β λ(2) = P k B T = βp The sormal-sobarc s equal to: Z (N, P, T ) = e βe βp V e where sum here runs over all mcrostates that have number of atoms N. The probablty of beng n any mcrostate wth number of atoms N s: βp e p = e βe V Z (N, P, T ) By nsertng expresson above nto p and dfferentatng t wth respect to (βp ) t s possble to show that: V ( ) ln Z (N, P, T ) V p = (βv ) (V V ) 2 2 ) ln Z (N, P, T ) (βv ) 2 In addton, we can relate logarthm of sormal-sobarc to Gbbs free energy usng: G(N, P, T ) = k B T ln Z (N, P, T ) Last but not least we can show that sormal compressblty (a response ) s related to average of fluctuatons n volume va: κ T = k B T V (V V )2 Notce also that we can (stll) also relate average of energy to a dervatve of logarthm of Z (N, P, T ) wth respect to β and that as such constant pressure heat capacty s related to fluctuatons n total energy for ths. Closed systems that can do PV work are examned usng sormal-sobarc only number of atoms s assumed fxed. 3
4 fxed n for Gve probablty mcrostate n grand canoncal Whch s rmodynamc behavour of Descrbe two ways n whch average of energy/number of atoms The (µv T ) In statstcal mechancs when we develop models for open systems that cannot do PV work we use grand canoncal. P V work cannot be exchanged wth such systems so y have constant volume. The number of atoms and energy, however, have to be calculated as averages. Insertng se requrements nto expresson for that we derved on worksheet 3 we arrve at: k B = [ λ () E + λ (2) N ] dv + λ () d E + λ (2) d N Rememberng that we can also calculate by combnng frst and second laws of rmodynamcs (see handout 2) we arrve at: = P T dv + T de µ T dn Equatng coeffcents of de and dn gves us: λ () = k B T = β λ(2) = µ k B T = βµ The s equal to: Z gc (µ, V, T ) = e βe e βµn where sum here runs over all mcrostates that have volume V. probablty of beng n any mcrostate wth volume V s: p = e βe e βµn Z gc (µ, V, T ) The By nsertng expresson above nto p and dfferentatng t wth respect to (βµ) t s possble to show that: N ( ) ln Zgc (µ, V, T ) N p = (βµ) (N N) 2 2 ) ln Z gc (µ, V, T ) (βµ) 2 In addton, we can relate logarthm of to Grand potental usng: Ω = k B T ln Z gc (µ, V, T ) Notce also that we can (stll) also relate average of energy to a dervatve of logarthm of Z gc (µ, V, T ) wth respect to β and that as such we can calculate an unnamed heat capacty from fluctuatons n total energy for ths. Open systems can be examned usng grand-canoncal n whch only volume s assumed fxed. The number of atoms and energy are calculated by averagng. 4
5 How s canoncal connected to mcrocanoncal How s sormalsobarc connected to canoncal Hos s connected to canoncal Relatonshps between s We can derve formula that relate s n varous s. For example, mcrocanoncal and canoncal s are related by: Z c (N, V, T ) = Z mc (N, V, E)e βe de You can arrve at ths formula by rememberng that mcrocanoncal Z mc (N, V, E) s just equal to number of mcrostates wth number of atoms N, volume V and energy E. Each of se mcrostates wll make a contrbuton of e βe to sum n canoncal and hence total contrbuton of Z mc (N, V, E) mcrostates wll be Z mc (N, V, E)e βe. The ntegral can be thought of contnuous lmt of a summaton and hence formula above. A smlar logc allows one to relate canoncal and sormal sobarc s usng: Z (N, P, T ) = Z c (N, V, T )e βp V dv and and canoncal by: Z gc (µ, V, T ) = Z c (N, V, T )e βµn dn What do we mean when we talk about term rmodynamc lmt The rmodynamc lmt We call lmt when number of atoms goes to nfnty (N ) rmodynamc lmt. In ths lmt fluctuatons around averages are zero and consequently all s are dentcal. The absence of any fluctuatons means that classcal rmodynamcs (especally Gbbs phase rule) s recovered n ths lmt. Why s classcal rmodynamc recovered n ths lmt. Integrals allow us to connect varous s. The rmodynamc lmt s lmt as N. In ths lmt all s are dentcal and classcal rmodynamcs s recovered. 5
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