.03 Microscopic Theory of Trasport (Fall 003) Lecture 6 (9/9/03) Static ad Short Time Properties of Time Correlatio Fuctios Refereces -- Boo ad Yip, Chap There are a umber of properties of time correlatio fuctios which follow from their defiitios, sometimes i a straightforward maer ad other times ot so simply. It is importat to make ote of them because ot oly do they have useful physical meaigs, but also they are ofte used to formulate model descriptios of time correlatio fuctios. Actually we have already ecoutered several examples i our discussio of diffusio, for example, the relatio betwee the log-time behavior of the mea-squared displacemet fuctio ad the diffusio coefficiet. Static Correlatio Fuctios Sice a time correlatio fuctio is a fuctio of time, oe ca always ask about its value at time t = 0, also kow as the iitial value. This value is the correlatio of two variables (which may be differet or the same) at the same time but at differet spatial positios. Correlatio fuctios which have spatial variatios but do ot deped o time are called static. We have see that the iitial value of the Va Hove self correlatio fuctio is just the delta fuctio, (4.5). It is a iterestig exercise to work out the correspodig iitial value for the desity correlatio fuctio Grt (,). Settig t = 0 i (5.5) we fid Gr (,0) = δ () r + gr [ () ] (6.) where gr δ r Ri δ R j i, j () = < ( ) ( ) > = (6.) ( ) 3 3 U(0, r, R3,, R )/ kbt dr3 dre Q The fuctio g(r) is well-kow i liquid-state theory as the equilibrium pair distributio or the radial distributio fuctio (RDF). This is the quatity that oe obtais from a x- ray or eutro diffractio measuremet. Recall from the previous lecture that G is the sum of G s ad G d. Eq.(6.) shows that the first term is the iitial value of G s as we already kow from (4.5). The secod term is therefore the iitial value of G d.
Because g(r) is the fudametal quatity i the equilibrium theory of liquids [see McQuarrie], we digress somewhat from the preset discussio of time-depedet desity correlatio fuctios to examie its relatio to the caoical distributio fuctio. Suppose we defie βu ( ) 3 3 e 3 3 P ( r... r ) d r... d r = d r... d r Z (6.3) as the probability that atom is i d 3 r about r,, atom is i d 3 r about r. Itegratig this over the positios of atoms + through gives ( ) 3 3 βu P ( r... r)... d r... d r e + Z = (6.4) gives the probability that atom is i d 3 r about r,, atom is i d 3 r about r irrespective of the positios of atoms + through. ow the probability that ay atom is i d 3 r, ay other atom is i d 3 r, ay atom other tha the first two is i d 3 r 3,, ad ay atom other tha the previous - atoms is i d 3 r is ( )! ( ) ( r... r) = P ( r... r ) (6.5) ( )! where!/( )! is the umber of ways oe ca pick objects out of possible choices. For example, the umber of ways of pickig two atoms out of a -particle system is!/(-)! = (-). Give (6.5), we have V 3 () d r ( r) = ρ (6.6) V which is the umber desity of the system (usually deoted as ad take to be a ( ) costat). We ext defie the -particle correlatio fuctio g ( r... r ) by writig ( ) ( ) r r = g r r (... ) (... ) (6.7) ( ) otice that if the atoms are ot correlated, the ( ) =, ad g. This is what we mea operatioally whe we say there is o spatial correlatio amog the -particles i the system. Combiig (6.4), (6.5), ad (6.77) we obtai a defiitio of the -particle correlatio fuctio i terms of the caoical distributio, V! g r r d r d r e ( ) 3 3 βu (... ) =...... + ( )! Z (6.8) which is the desired relatio.
To reduce to the pair distributio fuctio g(r) we ote that i a liquid composed () spherical atoms, g ( r, r ) depeds oly o the separatio distace r = r r. Thus we ca write g () ( r, r) g( r), or ( r, r ) = g( r) (6.9) () which is a defiitio of g(r) i terms of the -particle correlatio fuctio (58) with =. It the follows from the above probability distributios that 3 g() r d r = probability (ot ormalized) of fidig a atom i 3 dr about r give a atom is at the origi To fid the ormalizatio we take 3 d r g() r = g()4 r π r dr = 0 (6.0) Thus, π g r dr = 4 r ( ) umber of atoms i a shell of radius r ad thickess dr cetered about a atom at the origi A sketch of g(r), which is also kow as the pair correlatio fuctio or the radial distributio fuctio, for a typical liquid shows its most sigificat features - a promiet peak at the distace of earest eighbor separatio, ad a weaker peak at the secodearest separatio, see Fig. 6.. Below a certai separatio distace g(r) vaishes, sigifyig a hard core i the iteratomic iteractio which gives rise to a excluded volume i the spatial distributio of particles. I other words, particles repeal each other at very short rage, a basic property associated with the existece of a certai desity of the system. At the opposite extreme of large separatios, g approaches uity, sigifyig the loss of spatial correlatio (particles are ucorrelated whe they are far apart). It is useful to compare the behavior of g(r) to that of a typical pair potetial like the Leard- ( ) Joes. Ideed at low desities a virial expasio of g(r) gives gr () e βv r, where V(r) is the pair potetial. This leads to the defiitio of a 'potetial of mea force' V ( r) g( r)/ β which oe ca apply eve to hard-sphere fluids. Sice the static eff structure factor S(k) ad g(r) are related by Fourier trasform, a osciallatory behavior i oe fuctio meas the other fuctio will also show oscillatios. The small wiggles i g(r) at small r are umerical artifact of the Fourier trasformatio. The value of S(k) i 3
the limit of zero k is the compressibility factor, which becomes large i the viciity of the critical poit. Image Removed Yarell et al., Physical Review A7, p.30 (973) top graph is fig 6 i paper; bottom graph is fig 4. Fig. 6.. Radial distribtutio fuctio g(r) of liquid argo as obtaied by Fourier trasform of the measured static structure S(Q). Data poits are eutro diffractio measuremets, the curve is the result of molecular dyamics simulatio usig the Leard-Joes potetial (Yarell et al., Phys. Rev. A7, 330 (973)). The oscillatory shape of g(r) has a simple physical meaig. Suppose we sit o ay oe of the particles ad look aroud to see how the eighborig particles are distributed. We ca do this by drawig a sphere of radius r aroud us, with some thickess dr, ad ask how may of the eighbors lie iside this spherical shell, doig this repeatedly with icreasig r. Sice for a give r ad dr, the umber of eighbors is just 4 π rg( rdr ), the fact that g(r) is a oscillatory suggests that the eighbors are ot uiformly distributed, rather they like to arrage themselves i a shell-like structure. Thus the first peak i g(r), which is by far the strogest, sigifies the positio of the earest-eighbors, the secod peak, the secod earest eighbors, ad so o. The peaks are see to damp out fairly quickly, this is a reflectio of the fact that there is oly local order i a liquid ad o log-rage order. Oe expects that for a crystal the correspodig g(r) would be a set of very sharp lies, each sittig precisely at the positio for the particular eighbors. 4
Returig to the desity correlatio fuctios, we ote that just as g(r) ca be measured by eutro ad x-ray diffractio, G ad G ca be directly measured by eutro ielastic s scatterig ad light scatterig. eutro ielastic scatterig experimets actually give the double Fourier trasforms of the two desity correlatio fuctios, (, ω) 3 i( k r ωt ) (, ) Sk dt dre Grt = (6.) ad the same relatio exists betwee S ( k, ω ) ad G (,) r t. Skω (, ) is geerally kow as s s the dyamic structure factor, which is reasoable sice the Fourier trasform of g(r), Sk ( ) = + dre [ gr ( ) ] (6.) is called the static structure factor. To see the coectio betwee (6.) ad (6.), we itroduce aother quatity which is related to Skω (, )(ad therefore Grt (,)) by writig (6.) as iωt Sk (, ω) = dte Fkt (, ) (6.3) so that the itermediate scatterig fuctio F is just the Fourier spatial trasform of G(r,t) (for isotropic systems such as a simple fluid, G depeds oly the magitude of r ), F kt = dre Grt (, ) (, ) with * 3 =< (0) ( t) > ( π) δ( k) k i () t k i= k (6.4) ik R ( t) = e (6.5) We ow ask what is the iitial value of F(k,t). At t = 0, (6.4) gives where F( k,0) = d re G( r,0) (6.6) δ δ, ' G(,0) r =< (' r R (0)) ( r R (0)) > (6.7) Settig r ' = 0 without ay loss of geerality, we ca write out the double sum explicitly, 5
G(,0) r = < δ( R ) δ( r R ) >+ ( ) < δ( R ) δ( r R ) > 3 3 βu ( ) 3 3 βu = dr... dre δ( R ) δ( r) dr... dre δ( R ) δ( r R) Q + Q () r g () r = V δ + (6.8) Isertig this result ito (67) we obtai F( k,0) = + d re [ g( r) ] S( k ) (6.9) Sice the iformatio eeded to calculate the desity correlatio fuctios are the timedepedet particle positios, the output of MD simulatios, it follows that MD provides a direct meas of determiig the correlatio fuctios ad their Fourier trasforms. I this way, MD is able to provide results that ca be used to iterpret experimets, as well as results that ca be used to test various dyamical models of atomic motios i gases ad liquids. Examples of both kids of applicatios will be discussed i class. Short-Time Behavior ad Sum Rules The iitial value of a time correlatio is the leadig term i a Taylor series expasio i time. Because the Fourier trasform of a time correlatio fuctio is ofte also of iterest, we ca make use of the coectio betwee the coefficiets of the Taylor expasio ad the frequecy momets of the correspodig Fourier trasform of the time correlatio fuctio. Take the example of the desity correlatio fuctio where we write F kt = dre Grt (, ) (, ) The Taylor series expasio is i t = dte ω S( k, ω) (6.0) Fkt Fk t t dfkt (, ) dfkt (, ) (, ) = (,0) + +... dt t= 0! dt t= 0 + (6.) while from (6.0) we have 6
( it) Fkt d Sk it d Sk d Sk! (, ) = ω (, ω) + ωω (, ω) + ωω (, ω)... + (6.) Matchig terms with the same power of t i (6.) ad (6.) gives relatio betwee the frequecy momets of S, kow as sum rules, to the time derivatives of F, the coefficiets i the Taylor series i (6.). The first few such relatios are: ω o ( k) = dωs( k, ω) π = S(k) (6.3) o ωs = dωss( k, ω) π = (6.4) ω ωω ω ( k) = d S( k, ) = ( kv o) π = ωs( k) = dωω Ss( k, ω) π (6.5) where v k T/ m, with v beig the thermal speed of the particle. o = B o The relatios betwee the coefficiets of the time expasio ad the frequecy momets of the Fourier trasform of the time correlatio fuctio is quite geeral; it is applicable to ay pair of fuctios which are Fourier trasforms of each other. Sice we are dealig with classical time correlatio fuctios, they are eve fuctios i time ad frequecy. So terms that are odd i t vaish i (6.) ad (6.). There are a umber of properties of classical correlatio fuctios which do ot carry over to quatum mechaical defiitios of correlatio fuctios. We do ot take the time to discuss them here except to refer the reader to Sec.,7 Liear Respose Theory for some of the details. To fiish up this lecture, we ote that there are properties such as log time behavior ad relatio to trasport coefficiets, the cocept of a memory fuctio, liear respose theory, ad kietic theory which are treated i Chap of Boo ad Yip. We will retur to them at various times i future lectures. 7