p. 8 TIME-CORRELATION FUNCTIONS Time-correlatio fuctios are a effective way of represetig the dyamics of a system. They provide a statistical descriptio of the time-evolutio of a variable for a esemble at thermal equilibrium. They are geerally applicable to ay time-depedet process for a esemble, but are commoly used to describe radom or stochastic processes i codesed phases. We will use them i a descriptio of spectroscopy ad relaxatio pheomea. This work is motivated by fidig a geeral tool that will help us deal with the iheret radomess of molecular systems at thermal equilibrium. The quatum equatios of motio are determiistic, but this oly applies whe we ca specify the positios ad mometa of all the particles i our system. More geerally, we observe a small subset of all degrees of freedom, ad the time-depedet properties show radom fluctuatios ad irreversible relaxatio as a result of couplig to the surroudigs. It is useful to treat the eviromet with the miimum umber of variables ad iterpret i a statistical sese for istace i terms of temperature. Statistics Commoly you would describe the statistics of a measuremet i terms of the momets of the distributio. I fact, this was a postulate of quatum mechaics that the expectatio value is the mea value of the observable over may observatios. If A is a microscopic variable: N Average: A = 1 A i (9.1) N i=1 1 Mea Square Value: A = N AA i i N i=1 (9.) The ability to specify a value for A is captured i the variace of the distributio σ = A A (9.3) To characterize the statistical relatioship betwee two variables, we ca defie a correlatio fuctio C AB = AB A B (9.4)
p. 83 You ca see that this is the covariace the variace for a bivariate distributio. To iterpret this it helps to defie a correlatio coefficiet ρ = C AB. (9.5) σ A σ B ρ ca take o values from +1 to 1. If ρ = 1 the there is perfect correlatio betwee the two distributios. If the variables A ad B deped the same way o a commo iteral variable, the they are correlated. If o statistical relatioship exists betwee the two distributios, the they are ucorrelated, ρ = 0, ad AB = A B. It is also possible that the distributios deped i a equal ad opposite maer o a iteral variable, i which case we call them ati-correlated with ρ = 1. For time-correlatio fuctios we will be ivestigatig correlatio fuctios of the form (9.4), except rather tha two differet variables, we will be iterested i the value of the same iteral variable, although at differet poits i time. Equilibrium systems For the case of a system at thermal equilibrium, we describe the probability of observig a variable A through a equilibrium esemble average A. Classically this is A = dp dq A( pq, ;t) f (p, q ) (9.6) where f is the caoical probability distributio for a equilibrium system at temperature T β H f = e (9.7) Z Z is the partitio fuctio ad β=k B T. I the quatum mechaical case, we ca write A = p A (9.8) where p = e β E / Z (9.9) Equatio (9.8) may ot seem obvious, sice it is differet tha our earlier expressio A = m, a a A = Tr m m (ρ A). The differece is that i the preset case, we are dealig with a statistical mixture or mixed state, i which o cohereces (phase relatioships) are preset i the sample. To look at it a bit more closely, the expectatio value for a mixture
p. 84 A = p k ψ k A ψ k (9.10) k ca be writte somewhat differetly as a explicit sum over N statistically idepedet molecules 1 N () i () i A = ( a ) a m A m (9.11) N i=1, m Sice the molecules are statistically idepedet, this sum over molecules is just the esemble averaged value of the expasio coefficiets A = a a m, m A m (9.1) We ca evaluate this average recogizig that these are complex umbers, ad that the equilibrium esemble average of the expasio coefficiets is equivalet to phase averagig over the expasio coefficiets. Sice at equilibrium all phases are equally probable 1 π 1 m aa m d m 0 0 a a = π φ = a a π e iφ m dφ m (9.13) π where I have used a = a e iφ ad φ m = φ φ m. The itegral i (9.13) is quite clearly zero uless φ = φ m, givig β E a a = p = e m (9.14) Z Of course, eve at equilibrium the expectatio value of A for a member of esemble as a fuctio of time A t. Although the behavior of A i i () ( ) it geerally is observed to fluctuate radomly: t may be well-defied ad periodic, for mixed states Ai ( t) A t
p. 85 If we look at this behavior there seems to be little iformatio i the radom fluctuatios of A, but there are characteristic time scales ad amplitudes to these chages. We ca characterize these by comparig the value of A at time t with the value of A at time t later. With that i mid we defie a time-correlatio fuctio (TCF) as a time-depedet quatity, At ( ), multiplied by that quatity at some later time, A( t ), ad averaged over a equilibrium esemble: C AA (t, t ) A( t ) A( t ) (9.15) Techically this is a auto-correlatio fuctio, which correlates the same variable at two poits i time, whereas the correlatio of two differet variables i time is described through a crosscorrelatio fuctio Followig (9.6), the classical correlatio fuctio is C AB (t, t ) A( t ) B( t ) (9.16) C AA ( t t ) p pq ) (p q, ), = d dq A(, ;t A, ;t ') f (p q (9.17) while from (9.8) we ca see that the quatum correlatio fuctio ca be evaluated as C AA (t t, ) = p A( t ) A( t ). (9.18) So, what does a time-correlatio fuctio tell us? Qualitatively, a TCF describes how log a give property of a system persists util it is averaged out by microscopic motios of system. It describes how ad whe a statistical relatioship has vaished. We ca use correlatio fuctios to describe various time-depedet chemical processes. We will use μ (t) μ ( 0 ) -the dyamics of the molecular dipole momet- to describe spectroscopy. We will also use is for relaxatio processes iduced by the iteractio of a system ad bath: H SB (t ) H SB (0). Classically, you ca use if to characterize trasport processes. For istace a diffusio coefficiet is related to the 1 velocity correlatio fuctio: D = 3 dt v t v 0 0 () ( )
p. 86 Properties of Correlatio Fuctios A typical correlatio fuctio for radom fluctuatios i the variable A might look like: A C t AA (, t ') A ad is described by a umber of properties: t 1. Whe evaluated at t = t, we obtai the maximum amplitude, the mea square value of A, which is positive for a autocorrelatio fuctio ad idepedet of time. ( ) A( t ) C A AA (t, t ) = A t = 0 (9.19). For log time separatios, the values of A become ucorrelated lim C (t, t ') = A( t ) A( t ') = A (9.0) t AA 3. Sice it s a equilibrium quatity, correlatio fuctios are statioary. That meas they do ot deped o the absolute poit of observatio (t ad t ), but rather the time-iterval betwee observatios. A statioary radom process meas that the referece poit ca be shifted by a value T C AA (t, t ) = C AA ( t + T, t + T ). (9.1) So, choosig T = t, we see that oly the time iterval t t τ matters C AA (t, t ) = C AA (t t,0 ) = C AA (τ ) (9.) Implicit i this statemet is a uderstadig that we take the time-average value of A to be equal to the equilibrium esemble average value of A. This is the property of ergodic systems. More o Statioary Processes 1 The esemble average value of A ca be expressed as a time-average or a esemble average. For a equilibrium system, the time average is
p. 87 lim 1 T A = dt A i () t (9.3) T T T ad the esemble average is E A = e β A. (9.4) Z These quatities are equal for a ergodic system A = A. We assume this property for our correlatio fuctios. So, the correlatio of fluctuatios ca be writte lim 1 At () A ( 0 ) = T T T 0 ( ) ( ) d τ A t i +τ A i τ (9.5) β E () ( ) () ( ) = e or At A 0 At A 0 (9.6) Z 4. Classical correlatio fuctios are real ad eve i time: At ( ) At ( ) = At ( ) At ( ) C AA ( τ ) = C AA ( τ ) 5. Whe we observe fluctuatios about a average, we ofte redefie the correlatio fuctio i terms of the deviatio from average C A A () δ δ δ A A A () δ ( ) C () (9.7) (9.8) t = δ A t A 0 = AA t A (9.9) Now we see that the log time limit whe correlatio is lost lim C A A () time value is just the variace t δ δ t = 0, ad the zero C A δ δ ( 0 )= δ A A A = A (9.30) 6. The characteristic time-scale of a radom process is the correlatio time, τ c. This characterizes the time scale for TCF to decay to zero. We ca obtai τ c from τ = c 1 δ A dt δ A t 0 A 0 ()δ ( ) (9.31) which should be apparet if you have a expoetial form C( t ) = C ( 0exp ) ( t /τ c ).
p. 88 Examples of Time-Correlatio Fuctios EXAMPLE 1: Velocity autocorrelatio fuctio for gas. V x : xˆ Compoet of molecular velocity V x = 0 C t = V t V kt 0 C 0 = V 0 = VV () () ( ) VV ( ) ( ) x x x x x x x m Ideal gas: No collisios. Velocities are uchaged over t. C VV () t x x kt / m t Dilute gas: Ifrequet collisios V t V 0 for t < x ()= x ( ) x ()= x ( ) τ c V t V 0 ± δ for t >τ c τ c is related to mea time betwee collisios. After collisios, correlatio is lost. C VV () t x x kt / m τ c t
p. 89 EXAMPLE : Dipole momet for diatomic molecule i dilute gas: μ i. μ i = 0 (all agles are equally likely i a isotropic system) μ i = μ 0 û (the dipole has a magitude ad directio) C μμ () t = μ () t μ ( 0 ) = μ 0 uˆ () t uˆ ( 0) μ 0 collisioal dampig The correlatio fuctio projects the time-depedet orietatio oto the iitial orietatio oscillatio frequecy gives momet of iertia EXAMPLE 3: Displacemet of harmoic oscillator. mq = κq q = ω q q( t) = q ( 0cosωt ) Sice q ( 0 ) = kt mω C t = q t q 0 = q 0 cosωt qq () () ( ) kt = cosωt mω ( ) kt / mω dampig will cause Cqq to decay
p. 90 QUANTUM CORRELATION FUNCTIONS Quatum correlatio fuctios ivolve the equilibrium (thermal) average over a product of Hermetia operators evaluated two times. The thermal average is implicit i writig C AA ( t, t ) = A( t ) ( ) A t. Naturally, this also ivokes a Heiseberg represetatio of the operators, although i almost all cases, we will be writig correlatio fuctios as iteractio picture operators A t = e +ih t Ae I () 0 ih 0t. To emphasize the thermal average, the quatum correlatio fuctio ca also be writte as β H C t, t = A() t A( t ) AA ( ) e Z (9.3) If we evaluate this i a basis set, by isertig a projectio operator, this leads to our previous expressio C AA (t, t ) = p A( t ) A( t ) (9.33) with p = e β E Z. Give the case of time-idepedet Hamiltoia we ca also express this i the Schrödiger picture C t, t AA ( ) = = = m, = m, p p U HJJJJJJ J m ( t ) AU ( t ) U ( t ) AU ( t ) AU ( ) t t A p A m m A p A e ( t t ) i ω m e i ω ( t t ) e ( t t ) i ω m JJJJJJJJJJG A (9.34) Properties of Quatum Correlatio Fuctios There are a few properties of quatum correlatio fuctios that ca be obtaied usig the properties of the time-evolutio operator. First, we ca show the property of statioarity, which we have come to expect:
p. 91 At () At ( ) = U (t) A(0)U (t) U (t ) A(0)U (t ) ( ) () () ( ) = U t U t AU t U t A = U t AU t (t ) (t ) = At ( t ) A ( 0 ) A (9.35) Also, we ca show that ( ) ( 0) = A() t ( ) A( ) A( t ) A t A A 0 = 0 (9.36) or C AA (t) = C AA ( t ) (9.37) A( 0 ) A( t) = A(0)U AU = U AU A ( ) ( ) = A t A 0 (9.38) () ( ) At A 0 = U AU A = U AU A = A( 0) A( t ) (9.39) Note that the quatum C AA (t) is complex. You caot directly measure a quatum correlatio fuctio, but observables are ofte related to the real or imagiary part of correlatio fuctios, or other combiatios of correlatio fuctios. CAA ( t ) = C AA ( t ) + i C AA ( t ) (9.40) C () t = 1 C () t + C 1 t 1 = At (), A( 0 ) + A() t A ( 0 ) + A( 0) A( t) AA ( ) AA AA = (9.41) 1 1 C t AA ()= AA t t A t i C () C AA ( ) = i 1 = A () t, A ( 0 ) i () ( ) A 0 + 0 A( ) A( t ) We ca also defie a spectral or frequecy-domai correlatio fuctio by Fourier trasformig the TCF. (9.4) C ω t + AA ( ) = F C AA () = dt e i t C AA ω () t (9.43)
p. 9 For a time-idepedet Hamiltoia, as we might have i a iteractio picture problem, the TCF i eq. (9.34) gives C AA ( ω )= p A m δ ( ω ω m ). (9.44) m, This expressio looks very similar to the Golde rule trasitio rate from first order perturbatio theory. I fact, the Fourier trasform of time correlatio fuctios evaluated at the eergy gap gives the trasitio rate betwee states. Note that this expressio is valid whether the iitial states are higher or lower i eergy tha fial states m, ad accouts for upward ad dowward trasitios. If we compare the ratio of upward ad dowward trasitio rates betwee two states i ad j, we have C AA (ω ij ) p j β E ij C ( ω ) = p = e. (9.45) AA ji i This is oe way of showig the priciple of detailed balace, which relates upward ad dowward trasitio rates at equilibrium to the differece i thermal occupatio betwee states: C AA (ω) = e β=ω C AA ( ω). (9.46)