Lecture 3 : Classical Mechanics and. Micro canonical Ensemble

Transcription

1 Lecture 3 : Classical Mechanics and Micro canonical Ensemble

2 Last time Hamilton s Gns of Motion if Fpi is off Phase space is the coordinates describing everything about the system SO X (f) { of CH fatty N ( x ) is one function of x of hamiltonian / new terrain dynamics How do other HB CH P CH Bn and we showed that DIII o if the system follows quantities change with time? By the chain rule formula from last Hine : 3 N 3 N dacxydttec#gieetfpeofiat tee?fpiofioftqi if we define E a b3??? g?ip ftp?obg then we see that A conked quantity daydt feah o eq (Poisson Bracket ) one that doesnt depend on time L we shamed f 0 So EH H 30 Another example di d + a txt p Epi Ei * Mfg ; 2 { pi 43 so if net force is O momentum is conserved "

3 A " microstates " of a system is one particular point in phase space *CH for our system For a conservative system Hcxltik E canst So the system must remain an some canst E hypersurface Eg ie Haim " htt " radius of this circle depends a Lk on initial total energy x Ah " ensemble " is a collection of microstates all with the Same macroscopic characteristics We can express MET thin the points in the fraction of phase space a smell volume d # of point the function flex Hdx fcx H2O and fdxflxtt L Is by We can assert that the total number of members of the ensemble stays fined So that f stays normal iced

4 o If we draw a volume in phase space No Sources or sinks IF points in uolme flux through surface gtt/etitfoi?g?susrfaafnctiminrfdxfcxt can see how f charges by equating flux out with drldt This turns out to reduce to ( see chapter?s) OfCf t + diff { g in pin psn3 6 I if :c anti [ :}!± OH Fpi 0 Tlftxcthtt Ogi { II ii!f tefh3 Liouville ifgw equator ftpif dat in analogy w/ qm Call { H } I ih such that tilt formally ACH e itfw

5 Equilibrium means f is canst in time Off + at every point o { f H 30 in space this means f is a function of the Hamiltonian ie one way of restating this is if f TCH ) Iggy C series power expansion ) n then Ef?{ H 7 x y o dfydt only determines F to a canst factor op so flats FC HUH ZfdEFu where it counts the total number of microstates accessible in the ensemble Ensemble averaging it we knew Z C at Eq) LA ) fdf ACHI Ctkxlyz The form of Ii will depend an He ensemble ci#inhsfzagnamgeearcise This is called the partition function and

6 Eqns of at SHHH Micro canonical Ensemble For an isolated macroscopic System with a given Set all microstates with the same macroscopic properties are equally likely ( " equal a priori probabilities postulate ) If we have N particles in a box of volume U with no exchange of Energy then its a conservative system & dynamics will follow Ham Motion Hence at equilibrium Filth E ) where SCH is the dirac delta function with the property f Scx fat flat I unit of The micro canonical partition function * " RENN E) ajdischch E ) counting mm points on hypersurface The proportionality const Carta set by comparison to expts but

7 What here we will just note it should have a YN! in it since classical particles are indistinguishable Correction to thermodynamics Suppose we have our system Diwthr " wite : is nu how System still in isolation States there are many Total RIU zu ze ) RUN El rcn " e NUN E) RCN yet Extensive functions double when the system size doubles etc function of D ke? would an extensive for ) a logs has this properly

8 Which Suppose we take our two systems not exactly identical and let only E flow whole system is isolated I µ! Ez Got E Slice energy can mare from are ride to the other E can be any ache from O to Etot value is most likely? Some as saying which value of E maximizes RC Efes A tnz 4th ) To Max / min A we do o DIE evidently O d %E b/c loser monotonically increasing odk#heiclogcehil(d9t9ii)nt " hair Cd " l ± d "" E tit " " u Nd Vz Thermodynamics says heat will flow from are part to the other until the temperatures are equal

9 So somehow we be related to the temperature Basic thermo will tell us ( next that hence if we associate TSCNigelkblcgrcnuFI we gut Hut 1 2 bodies in contact een lice HEB 2 Entropy is maximized far a closed system at egeilibnn This connects microscopic States to one macro observable L & others see next D