Computing properties in simulations

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Jonah Booker
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1 Computing proprtis in simulations ChE210D Today's lctur: how to comput thrmodynamic proprtis lik th tmpratur and prssur, and kintic proprtis lik th diffusivity and viscosity, from molcular dynamics and othr simulations Equilibration and production priods Oftn w start our simulation with initial vlocitis and positions that ar not rprsntativ of th stat condition of intrst (.g., as spcifid by th tmpratur and dnsity). As such, w must quilibrat our systm by first running th simulation for an amount of tim that lts it volv to configurations rprsntativ of th targt stat conditions. Onc w ar sur w hav quilibratd, w thn prform a production priod of simulation tim that w usd to study th systm and/or comput proprtis at th targt stat conditions. How do w know if w hav wll-quilibratd our systm? On approach is to monitor th tim-dpndnc of simpl proprtis, lik th potntial nrgy or prssur. Th following is takn from a 864-particl molcular dynamics simulation of th Lnnard-Jons systm. Initially, th atoms ar placd on an fcc lattic and th vlocitis ar sampld from a = 2.0 (rducd units) distribution. Th crystal mlts to a liquid phas U / N instantanous prssur instantanous tmpratur molcular dynamics tim For th abov systm, an quilibration tim might b ~0.2 tim units. Aftr quilibration, many quantitis will still fluctuat and should fluctuat if w ar corrctly rproducing th proprtis of th statistical mchanical nsmbl of intrst (hr, th NVE nsmbl). M. S. Shll /18 last modifid 4/17/2013

2 At a basic lvl, w want th quilibration tim to b at last as long as th rlaxation tim of our systm, broadly dfind hr as th largst tim scal for molcular motion. On approach for stimating th rlaxation tim is to us diffusion cofficints or othr masurs of molcular motion. In a bulk liquid, for xampl, w might think of a rlaxation tim scal as that corrsponding for on molcul to mov a distanc qual to on molcular diamtr (). If w know th diffusion cofficint, w can comput a rlaxation tim rlax from rlax Thus our quilibration tim should at last xcd rlax svral tims ovr. Notic that can vary with stat conditions (.g., tmpratur), and this should b takn into account if w prform multipl simulations at diffrnt conditions. Simpl stimators What kinds of proprtis or obsrvabls can w comput from th production priod of our simulation? Th following discusss som variabls commonly of intrst. Each of ths involvs avrags ovr th simulation duration. Enrgis Th avrag kintic and potntial nrgis in our simulation ar givn by: = 1 = 1 whr w sum indpndnt sampls of th instantanous kintic and potntial nrgis at diffrnt tim points in th simulation. Rmmbr, th statistical bhavior of ths sums shows that th rror in our stimat gos as. Tmpratur Thr is no rigorous microscopic dfinition of th tmpratur in th microcanonical nsmbl. Instad, w must us macroscopic thrmodynamic rsults to mak a connction hr. Namly, 1 =!." = # $ lnω!." It can b shown (using th quipartition thorm) that th avrag kintic nrgy rlats to th tmpratur via: M. S. Shll /18 last modifid 4/17/2013

3 = DOF # $ 2 Hr, DOF is th numbr of dgrs of frdom in th systm. For a systm of * atoms that consrvs nt momntum, DOF = 3* 3 Howvr, for larg nough systms th subtraction of th 3 cntr of mass dgrs of frdom has littl ffct sinc it is small rlativ to 3*. If rigid bonds ar prsnt in our systm (tratd in a latr lctur), w also los on dgr of frdom pr ach. Thus w can mak a kintic stimat of th tmpratur: = 2 # $ DOF Not that w can dfin, oprationally, an instantanous kintic tmpratur: inst = 2 # $ DOF = inst Bcaus fluctuats during a simulation, inst also fluctuats. Not that this is an stimator of th tmpratur in that w must prform an avrag to comput it. Th sam idas about indpndnt sampls also apply hr. Although it is not as frquntly usd, w can also comput a configurational stimat of th tmpratur, drivd by [Butlr, Ayton, Jpps, and Evans, J. Chm. Phys. 109, 6519 (1998)]: # $ config = 4 5" 5 " 5 "8 This stimat dpnds on th forcs and thir drivativs (via th dnominator). Sinc th forcs dpnd only on th atomic positions, and not momnta, this is trmd a configurational stimat. W must also avrag ovr multipl configurations and corrlation tims in ordr to comput this tmpratur accuratly. Both th kintic and configurational tmpraturs ar qual in th limit of infinit simulation tim and quilibrium. Vlocity rscaling Typically w want to prform a simulation at a spcifid tmpratur. For an NVE simulation, this mans that w want to adjust th total nrgy such that th avrag tmpratur is qual to th on w spcify. W can adjust th total nrgy asily by changing th momnta. M. S. Shll /18 last modifid 4/17/2013

4 Th most common approach is to rscal th vlocitis at priodic tim intrvals basd on th dviation of th instantanous tmpratur from our st point tmpratur. This is a form of a thrmostat. If w rscal all of th vlocitis by: W want: 9 " nw = ;9 " = = ; 9 # $ DOF whr is our stpoint tmpratur. Solving for ;, ; =? # $ DOF = 9 Typically this rscaling is not don at vry tim stp but only priodically (.g., vry tim stps). Tchnically spaking, rscaling should b don with th vlocity autocorrlation tim, discussd blow. On problm with vlocity rscaling is that it affcts th dynamics of th simulation run and is an artificial intrruption to Nwton s quations of motion. In particular, vlocity rscaling mans that th total nrgy is no longr consrvd, and that transport proprtis cannot b accuratly computd. An altrnativ and prhaps bttr approach is: 1. First quilibrat th systm using priodic vlocity rscaling at th dsird tmpratur. 2. Run a short production phas with vlocity rscaling. Du to th rscaling, will fluctuat. Comput an avrag total nrgy. 3. Turn off vlocity rscaling. 4. Rscal th momnta such that th total nrgy quals. That is, givn th currnt configuration with potntial nrgy, rscal th momnta and kintic nrgy to satisfy =. 5. Th simulation can thn b volvd in tim normally (NVE dynamics) and should avrag to th dsird tmpratur, to within th rrors in dtrmining. Thr ar mor sophisticatd ways of prforming tmpratur rgulation but th abov approach is prhaps th simplst. Morovr, this approach prsrvs th tru NVE dynamics of th systm, th only truly corrct dynamics. M. S. Shll /18 last modifid 4/17/2013

5 Prssur To comput th prssur, w oftn us th virial = 1 3A B3*# $+5 D E = 1 3A B2 +5 D E This xprssion is drivd for th canonical nsmbl (constant NVT), but it is oftn applid to molcular dynamics simulations rgardlss (NVE). For larg nough systms, th diffrnc btwn th two is vry small. Th xprssion abov is not gnrally usd for systms of pairwis-intracting molculs subjct to priodic boundary conditions. Instad, w can rwrit th forc sum: 5 D = F5 G H D G = 5 G D,G = 5 G D +5 G D JG KG = 5 G D +5 G D G JG JG = 5 G LD D G M JG = NOLP GM NP JG P G Thus, Q NOLP GM NP = 1 2 +Q 3A Hr, Q is calld th virial. Notic that Q involvs a sum of pairwis intractions. W thrfor nd to comput it in our pairwis loop, alongsid th nrgis. Oftntims, th calculations w us for th pairwis nrgis can b r-usd in th loop. Tak th Lnnard-Jons systm for xampl, in dimnsionlss units: P G M. S. Shll /18 last modifid 4/17/2013

6 = 4LP G P G T M JG Q = 24L2P G P G T M JG For systms involving rigid bonds (discussd latr), th forcs acting to hold th bonds rigid must b computd and addd to th ovrall virial. Hat capacity On way w might masur th hat capacity is to prform multipl simulations at diffrnt tmpraturs and thn numrically stimat U! = N N W+ΔY WY Δ Altrnativly, w can stimat U! from a singl simulation using nrgy fluctuations. For th canonical nsmbl (and approximatly th microcanonical on), w can writ: U! = NW+Y N = DOF# $ 2 = DOF# $ 2 = DOF# $ 2 + N N + # $ + Z # $ That is, w can masur th hat capacity from th varianc in th potntial nrgy. Th last trm in this quation is oftn trmd th configurational hat capacity. Othr quantitis It is rlativly asy to masur th nthalpy, which stms from a mchanical avrag: [ = +@A On th othr hand, it is vry challnging to comput ntropic or fr-nrgtic quantitis, lik,\,],^. W will discuss advancd simulation approachs for dtrmining ths quantitis latr in th cours. Unlik th quantitis w hav studid so far, fr-nrgtic quantitis rquir computation of distributions of simulation obsrvabls, not just avrags of thm. M. S. Shll /18 last modifid 4/17/2013

7 Statistics of avrags Basic avrags Considr th computd avrag potntial nrgy of a simulation. For a production priod of MD tim stps, w could comput ` _ = 1 In this sction, w will us an ovrbar to indicat an stimat dducd from a singl, finitduration simulation. It will b mor informativ for now if w nglct th discrtizd natur of our solutions to th dynamic trajctoris and instad rprsnt this as an intgral: a _ = 1 d tot c WbYNb This xprssion isn t spcific to th potntial nrgy. For any obsrvabl \ for which w want to comput th avrag, \ = 1 d tot c \WbYNb Ths avrags of obsrvabls corrspond to finit-duration simulations. Thr ar two ways in which w might xpct to s rrors in our rsults: Th simulation tim is not long nough to prvnt statistical rror in \. Only in th limit will w rigorously masur th tru, statistical-mchanical avrag that w xpct from thrmodynamics. In practic, w rally only nd to tak this intgral to a modrat numbr of corrlation tims of th proprty \, which w discuss blow. Th simulation is not at quilibrium. In this cas, w nd to xtnd th quilibration priod bfor computing this intgral. In what follows, w will us th following notational dfinitions. Lt \ = 1 d tot c \WbYNb \ = lim d tot j 1 c \WbYNb d tot That is, \ dnots a simulation avrag, whil \ dnots th tru statistical-mchanical quilibrium avrag for \ that w would xpct in th limit of infinit simulation tim, in which our systm is at quilibrium. M. S. Shll /18 last modifid 4/17/2013

8 Corrlation tims Assum w can prform a simulation that initially is fully quilibratd at th dsird quilibrium conditions. If w wr to prform multipl trials or runs of our simulation, w would gt an stimat for \ that would b diffrnt ach tim bcaus of th finit lngth for which w prform thm. W could obtain a numbr of masurmnts from diffrnt runs: \,\,\ k, W want to know what th xpctd varianc of \ is, rlativ to th tru valu \. This is th squard rror in our masurmnt of th avrag using finit simulation tims: m = W\ \Y Hr, th brackts indicat an avrag ovr an infinit numbr of simulations w prform. W can simplify this xprssion: m = \ \ = 4 1 d tot nc \WbYNb d tot d tot d tot onc \WbYNb = 4 1 b c c \WbY\Wbp YNb p tot = 1 d tot d tot o8 \ Nb8 \ c c \WbY\Wbp YNb p Nb \ In th last lin, w movd th avrag into th intgrand. Notic that w ssntially hav a doubl summation of all \WbY\Wb p Y pairs at diffrnt tim points. For two spcific tim points, b = b and b p = b, th idntical products \Wb Y\Wb Y and \Wb Y\Wb Y both appar as th intgrand variabls pass ovr thm. This nabls us to considr only th uniqu tim point pairs of b,b p for which b p < b, multiplying by two: m = 2 c c \WbY\Wbp YNb p Nb \ d tot d W simplify things hr bcaus Nwton s quations ar symmtric in tim. First, th avrag \WbY\Wb p Y should not dpnd on th absolut valu of th tims, but only thir rlativ valu, bcaus at quilibrium w can look at our simulation at any two rlativ points in tim and w would xpct to gt th sam avrag. Thrfor, w can shift this avrag in tim by b p : M. S. Shll /18 last modifid 4/17/2013

9 m = 2 c c \Wb bp Y\W0YNb p Nb \ Sinc th simulations start at quilibrium, w hav d tot d \ = \W0Y \ = \W0Y m = \ \ = \W0Y \W0Y Notic that m (without ovrbar on th \) givs th quilibrium varianc of \, or that xpctd from a singl, long quilibrium simulation. It is diffrnt from m, which stimats th varianc in th avrag \, or th squard rror in th avrag w comput from run to run. W xpct m to b finit, constant, and charactristic of th quilibrium fluctuations, whil w xpct m to approach zro as w mak our simulations longr and longr. With ths idas, w can rwrit this xprssion as: m = 2 m d tot d nc c U mwb b p YNb p Nbo Hr, U m is th tim autocorrlation function for \. Its formal dfinition is U m WbY \WbY\W0Y \W0Y\W0Y \W0Y\W0Y \W0Y\W0Y = \WbY\W0Y \W0Y\W0Y m Physically, it masurs how corrlatd th variabl \ is at som tim b with its valu at initial tim 0. By th dfinition abov w s that U m Wb = 0Y = 1 U m Wb Y = 0 Schmatically, th corrlation function may look somthing lik this: M. S. Shll /18 last modifid 4/17/2013

10 U m WbY Autocorrlation functions dcay with tim, sinc at long tims, a masurmnt is uncorrlatd from its valu at arlir tims. W can dfin an autocorrlation tim as: j m c U m WbYNb If th total simulation lngth is longr than this tim, m, th xprssion for th varianc in \ can b rwrittn approximatly as: m 2 m d tot nc mnbo = 2 m m W can dfin an ffctiv numbr of indpndnt sampls m such that: m 2 m m = m m This rsult is an xtrmly important on. It says svral things: Th squard rror in any quantity for which w avrag in simulation dcrass as on ovr th ffctiv numbr of indpndnt sampls. Sampls that w us in our avrag to comput \ ar only indpndnt if w pick thm to b spacd at last 2 m units apart in tim. W will not gt bttr statistical accuracy by avraging th valu of \ for vry singl tim stp in our simulation. W gt just as good accuracy by avraging th valu of \ for tims spacd 2 m units of tim apart. b M. S. Shll /18 last modifid 4/17/2013

11 Block avraging W want to mak sur that w ar including nough indpndnt sampls in our stimats of diffrnt proprty avrags. A vry basic approach would b to stimat th largst tim scal in our systm, th rlaxation tim, and mak sur w prform th simulation for a larg numbr of ths tims. This is prhaps th most common approach. Altrnativly, w could comput m. Indd, thr ar procdurs for stimating corrlation functions from simulations. W could prform a vry long simulation, comput th corrlation function, and stimat m using th intgral dfinition of it. Howvr, it can b a significant programming ffort to intgrat corrlation functions into our simulations. Instad, w can us a simpl block avraging approach to dtrmin, approximatly, th corrlation tim for a givn variabl. Th ida of this analysis is to plot: m as a function of m for simulations of diffrnt lngths. Th slop of this lin givs twic th corrlation tim, pr th quation m m =2 m In practic, w tak a long simulation trajctory and first comput th following: m =varianc of \ ovr ntir simulation trajctory Thn, w subdivid th trajctory into diffrnt, nonovrlapping tim sgmnts or blocks. W can thn comput th othr quantitis abov: \ =avrag \ for ach block { m =varianc of th \ =lngth of ach block { By prforming th block avrags for diffrnt numbrs of blocks, and hnc diffrnt, w ar abl to find th slop corrsponding to m abov. Multipl trials Whil it is vry important to prform avrags for lngths that xcd corrlation tims in a singl simulation, it is common practic to also prform multipl trials of th sam run and avrag th rsults not only in tim but also across th diffrnt trials. Th us of multipl M. S. Shll /18 last modifid 4/17/2013

12 trials can hlp to produc rsults that ar mor statistically indpndnt. Each trial should b sdd with a diffrnt random initial vlocity st. Notation In th rmaindr of ths nots and in latr lcturs, w will drop th notation \ and us \ to dsignat both tru quilibrium, statistical-mchanical avrags and finit-duration simulation avrags. Kp in mind, though, that any avrag computd from simulation will b subjct to th statistical proprtis dscribd abov. Transport proprtis As NVE molcular dynamics simulations follow th Nwtonian volution of th atomic positions, thy giv ris to trajctoris that accuratly rprsnt th tru dynamics of th systm. Thus, ths simulations can b usd to comput kintic transport cofficints in addition to thrmodynamic proprtis. Slf-diffusivity: Einstin formulation Th slf-diffusion constant is dfind as th linar proportionality constant btwn th mass/molar flux of a spcis and th concntration gradint (Fick s law). For a uniform diffusion constant (with spac, as in a homognous bulk phas), th following quation dfins volution of th concntration (molculs pr volum) with tim: (D, b) b = (D,b) W can rwrit this quation in trms of th probability dnsity that w will find a molcul at som point in spac. Ltting (D;b) b this probability, w thn hav (D,b)= (D;b)* c (D;b)ND=1 Making this substitution, (D; b) b = (D;b) Imagin that a molcul is known to initially start at a givn point D=D in spac at b=0. Thn, th solution to (D;b) is givn by M. S. Shll /18 last modifid 4/17/2013

13 (D;b)=(b) k xp D D 4b W can comput from this th man-squard displacmnt with tim: D D = c WD;bY ND = 6b In othr words, th man-squard displacmnt grows linarly with tim with a cofficint of 6. This quation is an Einstin rlation, aftr Albrt Einstin s sminal work in diffusion. Importantly, it givs us a way to masur th diffusion constant in simulation: 1. At tim b = 0, rcord all particl positions D ". 2. At rgular intrvals b, comput th man squard displacmnt avragd ovr all atoms, D D. 3. Find th diffusion constant from th limit at larg tims: D D = lim d j 6b or, bttr, from th slop of th man squard displacmnt at long tims: = 1 6 lim N d jnb D D Som logistical aspcts must b kpt in mind: Th tim at which th diffusion cofficint is masurd should b a numbr of rlaxation tims of th systm. For bttr statistics in computing th man-squard displacmnt curv (vs. tim), it is oftn usful to hav multipl tim origins,.g., D ",D ",D ", rfrnc positions takn at statistically indpndnt tim intrvals (i.., a rlaxation tim). Thn, at ach tim b on can mak updats to th avrag man-squard displacmnt curv at tims b b,b b,b b, using th rspctiv rfrnc coordinats. If a systm consists of multipl atom typs, ach can hav its own slf diffusion cofficint and th quations will involv sparat man squard displacmnt calculations for th rspctiv atoms of ach typ. Th following shows th man-squard displacmnt curvs for oxygn atoms in liquid silica (SiO 2 ), takn from [Shll, Dbndtti, Panagiotopoulos, Phys. Rv. E 66, (2002)]: M. S. Shll /18 last modifid 4/17/2013

Notic th log-log plot. W s linar bhavior in th curvs (xpctd for random-walk diffusion according to a diffusion constant) aftr som initial tim priod has passd.

14 Notic th log-log plot. W s linar bhavior in th curvs (xpctd for random-walk diffusion according to a diffusion constant) aftr som initial tim priod has passd. Thr ar diffrnt rgims in particl diffusion: ballistic rgim At vry short tims, particls do not fl ach othr, DV9b, and th man squard displacmnt simply scals as ΔD ~ˆb. On th plot abov, w would xpct to s a slop of 2 at short tims, ln ΔD ~2lnb. diffusiv rgim At long tims, particls hav lost mmory of thir initial positions and ar prforming a random walk according to th diffusion constant, ΔD ~6b. W should only us data from this rgim whn computing th diffusion constant. Notably, th slop in this rgim on th abov plot should b 1, ln ΔD ~lnb. cagd rgim At intrmdiat tims, th man squard displacmnt may not follow ithr of ths scaling laws. Oftn, ΔD will appar to platau for som tim priod. This bhavior is typical of sluggish dynamics in viscous liquids and polymrs. Slf-diffusivity: Grn-Kubo formulation It is ntirly possibl to transform th Einstin xprssion for th slf-diffusivity, in trms of th man squard displacmnt, into a form that rlats to th atomic vlocitis instad, using d D,D Š c9wbynb M. S. Shll /18 last modifid 4/17/2013

15 W substitut this xprssion into th quations abov and simplify using idas similar to thos dvlopd in th tim-corrlation sction. This approach givs a Grn-Kubo rlation that conncts th diffusivity to th vlocity autocorrlation function: U 9 (b)= 9(b) 9(0) 9(0) 9(0) 9(0) 9(0) 9(0) 9(0) =9(b) 9(0) 9(0) 9(0) 9 Th avrags hr ar prformd for particls of th sam typ and ovr multipl tim origins for rcording th initial vlocity 9(0). Th diffusion constant rlats to th intgral of U 9 : = 9 j 3 c U 9(b)Nb = In othr words, th diffusion constant rlats to th corrlation tim of th vlocity. In practic, th autocorrlation function is approximatd by a discrtizd array (th indx corrsponding to a tim bin) and computd in a similar mannr as th man-squard displacmnt using multipl tim origins. This function typically dcays to nar zro in a finit lngth of tim and thus th intgral only nds to b computd up until this point. Somtims spcial tchniqus ar ndd to coars-grain tim in ordr to trat th statistical fluctuations around zro in th tails of th computd autocorrlation function. Othr transport cofficints A vry gnral thory shows that Grn-Kubo rlations can b formulatd for any transport cofficint that is a linar constant of proportionality btwn a flux and a gradint. Som xampls includ th bulk viscosity, shar viscosity, th thrmal conductivity, and th lctrical conductivity. Exprssions for ths can b found in standard txts. Th bulk viscosity, for xampl, is givn by: Œ! = j! A# $ c U!(b)Nb whr U! (b) is th corrlation function for fluctuations in th M. S. Shll /18 last modifid 4/17/2013

16 Structur-basd avrags Radial distribution functions (RDFs) Th radial distribution function (RDF) or pair corrlation function is a masur of th structur of a homognous phas, such as a liquid, gas, or crystal. Givn that a particl sits at th origin, it givs th dnsity of particls at a radial distanc P from it, rlativ to th bulk dnsity. Ž(P) 1 Formally, th pair corrlation function for a monoatomic systm in th canonical nsmbl is dfind by: P Ž(D,D )= A (* 1) * ZLD M ND k ND ND " (,A,*) whr (,A,*) is th canonical partition function. In an isotropic mdium, this function dpnds only on th rlativ distanc btwn two atoms, not thir absolut position: For an idal gas with (D " )=0, Not that, Ž(P ) Ž(P )= * 1 * V1 (larg *) c(4p NP) Ž(P)=* 1 On can also dfin a radial distribution function for atoms of diffrnt typs,.g., btwn hydrogn and oxygn atoms in liquid watr. In this cas, w can dfin M. S. Shll /18 last modifid 4/17/2013

17 for two atom typs \ and. Ž m$ (D,D )=A ZLD M ND k ND ND " (,A,* m,* $ ) RDFs can b computd using histograms of th pairwis distancs btwn particls. For a monatomic systm with just on kind of particl, th rcip is th following: 1. At priodic intrvals in th simulation, xamin all pairwis *(* 1)/2 distancs of th * particls. On dos not nd to xamin vry tim stp, but only thos approximatly spacd by th rlaxation tim in th systm, or a modrat fraction throf. Lt th numbr of ths intrvals b obs. 2. Lt dnot an array of histogram counts for th total numbr of tims a pairwis distanc P G is obsrvd, whr {δ P G < W{ +1Yδ and δ is th width of th histogram bins. 3. Aftr sufficint data collction, th RDF can b approximatd at discrt intrvals {š. For atoms of th sam typ: Ž mm W{šY = obs * m W* m 1Y/2 A 4š k 3 WW{+1Yk { k Y For atoms of diffrnt typs: Ž m$ W{šY = obs * m * $ A 4š k 3 WW{+1Yk { k Y Enrgy and prssur from RDFs For pair potntials, intgrals of an RDF can b usd to comput th potntial nrgy and prssur: = * j 2 c œ4p ŽWPY OWPYNP j = 2* A c P ŽWPYOWPYNP = *# $ A * 6A c œ4p ŽWPY P NOWPY NP NP = *# $ A 2* 3A c Pk ŽWPY NOWPY NP NP j M. S. Shll /18 last modifid 4/17/2013

18 Th lattr quation is mrly an xtnsion of th virial xprssion for th prssur. If thr ar multipl atom typs in th systm, thn w will hav multipl Ž(P) functions that nd to b intgratd. For xampl, for two typs \ and : = mm + $$ + m$ j = 2 A c P œ* m Ž mm (P)O mm (P)+* $ Ž $$ (P)O $$ (P)+2* m * $ Ž m$ (P)O m$ (P) NP Th cofficint of two in front of th AB trms coms from th fact that ths intractions ar not doubl countd whn prforming th usual intgral. A convnint way to xprss this is through a doubl sum ovr all atom typs (with ž total typs): j = 2 A c P Ÿ* * Ž (P)O (P) NP a a A similar xprssion can b drivd for th # $* tot A 2 j 3A c Pk Ÿ * * Ž WPY NO WPY NP NP a a M. S. Shll /18 last modifid 4/17/2013

Brief Introduction to Statistical Mechanics

Brief Introduction to Statistical Mechanics Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.