Introduction fle SI 3 61 Non equilibrium Sttisticl Mechnics 2520 1 Techer : Tens It Brdrsonbrdrson@kthse Office 174 : 1049 ( open door policy Lecture 1 & The second lw nd origin of irreversibility Wht is equilibrium? Isolted systems left for long ( but not lwys rech stedy generlly stte which is independent nd uniform This stedy stte is for most systems which in sttisticl mechnics is described opertor by density g ~ e P # ( since rml stte for exmple glt glo [ g kjo E mp : Dog model ( Ambegokr Am Pugs nd Clerk 1068 ( 1999 Fles µ An ;k ( see Python code ~ n ftpfe?:fmmmhm to N fles rndomly Every step select fle tht n hops beten dogs E± Get Python code : work nd ply with t it Quickly get ech stedy stte with men number Nk fles on dog corresponding to T A rml stte with round men tht s N fluctutions decy 42
Mcroscopic 6 fle If Description ot of rmodynmics rml nd ( equilibrium 2 stte is topic equilibrium sttisticl mechnics We re not interested in this here nd I ssume you lredy re interested in nd In dog know lot bout this In this course resulting dynmics Wy stte which ws model? describing its evolution under stochstic non equilibrium sttes nd trnsport prepred system in nd observed fr from equilibrium very dynmics Exmple ( see Ch / ory hydrodynmics in Le Brlc et l for detils some more hve Suppose mcroscopic vribles tht correspond conserved such s number of quntity prticles to energy or momentum Such vribles re somes clled hydrodynmic vribles or relevnt urrsles cn not fr from equilibrium imgine Cn seprte spce equilibrium number into cells tht re M locl nd define density n ( F t density for exrmpe re tht such tht number of in cell is prticles Nq d3r fncritl V out The fux of Prticles of volume comes from nd is surrounding cf jcrit d5 Since DNA t
D DD Stt Phys We obtin ( course grined continuity eqution 3 The current is nd nd Ft On + Big 0 only becuse of difference in density if this difference is smll cn expnd obtin n jcrit n ( Fick s lw [ minus such tht D > 0 D clled diffusion constnt dncrttl t D I Znlrit Diffusion eqution Solve with Founv trnsform once tl k2n( K t dt NCE # Sdedftnck to e D " to eie E if inlxito 86+0 n nl I to e ik E nd tn//2exp( nce+ EDIT @ yjf I ( see e g Dniel Arous lecture notes on Noneq Note : This mcroscopic description does not tell s nything bout D; for this need more microscopic description ( will come lter
complex It @ Note tht diffusion efvlion is not reversl symmetric describes n irreversible process consistent with second lw of rmodynmics which for isolted systems reds ds 20 The sme is true for or mcroscopic lws might write down such s het conduction friction ete Thts good right? Well yes it Mens hve n rrow of s re used to ( splttering eggs ete But microscopic equtions of motion re reversl symmetric : ME Dee ihofltc > EHK etc Newton > Schillings ( F conjugtion Therefore if increses forwrd in it should lso increse bckwrds in!
5 How does rrow of or irreversible dynmics emerge from reversl symmetric dynmics? ( Boltzmnn Mxll pge 2 + 30 in Jck s notes
f ftz Increse of 6 Lets try to clculte increse in using clssicl mechnics First question : which? Lets use sttisticl s { Pnln Pn which cn be interpreted tht is missing when s mount of informtion only know probbilities Pn We define probbility density in phse spce g ( I Fit where I ( Gis F 1 P / % Por ] I ( of F Then tibbs s KB Sdx f ( x bnfcx equls rmodynmic in equilibrium wht bout evolution? Lionville eqution : # { Hit } 8 o hf T Poisson brcket
Sdx J DICFJ Detniz i (ftpto#lifsx @ [ Hmilton s equtions of motion Def TE ( FJ + ftpj 0 since tkiu fgttgp of g continuity eqution i e How in phse spce is incompressible Now ( put kbl 25 ot fdxlsilnf + f fdx th ( f J ( lufty FJ tz #f (^ integrtion by B If nd prts since system ii ( fdx is isolted fdx drop surfce terms ft 01 The Gibbs is constnt of motion!
3 67 this Why does Gibbs not increse? 8 Being sttisticl it mesures our lck of informtion bout system remins constnt during Hmiltonin evolution since it is reversible nd in derivtion kept trck of ll degrees of freedom In mcroscopic description only keep trck of few relevnt vribles ( mount of gs in left continer necessity irrelevnt but disregrd ( out of vribles ( complicted velocity correltions ete During evolution informtion flows from retest vribles to irrelevnt reby incresing disorder in relevnt vribles One cn define relevnt or Boltzmnn which mximizes sttisticl with constrints tht relevnt vribles ( Aict > Eilts obtin density mtrix slt t e Erin Zlt e to rini see Blin Am Phys 107849991 for detils