10. LINEAR RESPONSE THEORYEquation Chapter 10 Section 1

Transcription

1 . LINEAR RESPONSE THEORYEquaion Chaper Secion.. Classical Linear Response Theory Correlaion funcions provide a saisical descripion of he dynamics of molecular variables; however, i remains unclear how hey are relaed o experimenal observables. You have probably sensed his from he perspecive ha correlaion funcions are complex, and how can observables be complex? Also, correlaion funcions describe equilibrium dynamics, bu from a realisic poin of view, exering exernal forces should move he sysem away from equilibrium. Wha happens as a resul? These quesions fall ino he realm of nonequilibrium saisical mechanics, an area of acive research for which formal heories are limied and approximaion mehods are he primary ool. Linear response heory is he primary approximaion mehod, which describes he evoluion away or oward equilibrium under perurbaive condiions. We will use linear response heory as a way of describing a real experimenal observable. Specifically his will ell us how an equilibrium sysem changes in response o an applied poenial. The quaniy ha will describe his is a response funcion, a real observable quaniy. We will go on o show how i is relaed o correlaion funcions. Embedded in his discussion is a paricularly imporan observaion. We will now deal wih a nonequilibrium sysem, bu we will show ha when he changes are small away from equilibrium, he equilibrium flucuaions dicae he nonequilibrium response! Thus knowledge of equilibrium dynamics is useful in predicing he oucome of nonequilibrium processes. So, he quesion is How does he sysem respond if you drive i away from equilibrium? We will examine he case where an equilibrium sysem, described by a Hamilonian H ineracs weakly wih an exernal agen, V(). The sysem is moved away from equilibrium by he exernal agen, and he sysem absorbs energy from he exernal agen. How do we describe he ime-dependen properies of he sysem? We firs ake he exernal agen o inerac wih he sysem hrough an inernal variable A. So he Hamilonian for his problem is given by H H f A (.) Here f() is he ime-dependen acion of he exernal agen, and he deviaion from equilibrium is linear in he inernal variable. We describe he behavior of an ensemble iniially a hermal equilibrium by assuming ha each member of he ensemble is subjec o he same ineracion wih he exernal agen, and hen ensemble averaging. Iniially, he sysem is Andrei Tokmakoff, /8/4

2 - described by H. I is a equilibrium and he inernal variable is characerized by an equilibrium ensemble average A. The exernal agen is hen applied a ime, and he sysem is moved away from equilibrium, and is characerized hrough a nonequilibrium ensemble average, A. A A as a resul of he ineracion. For a weak ineracion wih he exernal agen, we can describe expansion in powers of f. A by performing an A erms f erms f (.), (.3) A A d R f In his expression he agen is applied a, and we observe he sysem a. The leading erm in his expansion is independen of f, and is herefore equal o A. The nex erm in (.3) describes he deviaion from he equilibrium behavior in erms of a linear dependence on he exernal agen., R is he linear response funcion, he quaniy ha conains he microscopic informaion on he sysem and how i responds o he applied agen. The inegraion in he las erm of eq. (.3) indicaes ha he nonequilibrium behavior depends on he full hisory of he applicaion of he agen f and he response of he sysem o i. We are seeking a quanum mechanical descripion of R. Properies of he response funcion. Causal: Causaliy refers o he common sense observaion ha he sysem canno respond before he force has been applied. Therefore R, for, and he ime-dependen change in A is, A A A d R f (.4) The lower inegraion limi is se o o reflec ha he sysem is iniially a equilibrium, and he upper limi is he ime of observaion. We can also make he saemen of causaliy explici R,, where by wriing he linear response funcion wih a sep response: (.5)

3 -3. Saionary: Similar o our discussion of correlaion funcions, he ime-dependence of he sysem only depends on he ime inerval beween applicaion of he poenial and observaion. Therefore we wrie R, R and (.6) A dr f This expression says ha he observed response of he sysem o he agen is a convoluion of he maerial response wih he ime-developmen of he applied force. Raher han he absolue ime poins, we can define a ime-inerval, so ha we can wrie 3. Impulse response: Noe ha for a dela funcion perurbaion: A d R f (.7) f (.8) We obain A R (.9) Thus, R describes how he sysem behaves when an abrup perurbaion is applied and is ofen referred o as he impulse response funcion. An impulse response kicks he sysem away from he equilibrium esablished under H, and herefore he shape of a response funcion will always rise from zero and ulimaely reurn o zero. In oher words, i will be a funcion ha can be expanded in sines. Thus he response o an arbirary f() can be described hrough a Fourier analysis, suggesing ha a specral represenaion of he response funcion would be useful. The suscepibiliy The observed emporal behavior of he nonequilibrium sysem can also be cas in he frequency domain as a specral response funcion, or suscepibiliy. We sar wih eq. (.7) and Fourier ransform boh sides:

4 -4 A d A e i d d R f e i (.) i i Now we inser e e and collec erms o give or A f i i (.) A d d R f e e i i de f dr e (.) (.3) In eq. (.) we swiched variables, seing. The firs erm f ( ) is a complex frequency domain represenaion of he driving force, obained from he Fourier ransform of f( ). The second erm χ(ω) is he suscepibiliy which is defined as he Fourier Laplace ransform (single-sided Fourier ransform) of he impulse response funcion. I is a frequencydomain represenaion of he linear response funcion. Swiching beween ime and frequency domains shows ha a convoluion of he force and response in ime leads o he produc of he force and response in frequency. This is a manifesaion of he convoluion heorem: A B d A B d A B F A B (.4) Here refers o convoluion, an inverse Fourier ransform. Noe ha is complex: suscepibiliy A F A, F is a Fourier ransform, and F is R is a real funcion, since he response of a sysem is an observable. The i Since d R (.5) i e (.6) However, he real and imaginary conribuions are no independen. We have d R cos (.7) and d R sin (.8) and are even and odd funcions of frequency: (.9) (.)

5 -5 * so ha (.) Noice also ha eq. (.) allows us o wrie (.) i (.3) Example of a high frequency underdamped response funcion oscillaing as sin( ) and corresponding suscepibiliy Example of a low frequency overdamped response funcion and corresponding suscepibiliy Kramers Krönig relaions Since hey are cosine and sine ransforms of he same funcion,. The wo are relaed by he Kramers Krönig relaionships: d is no independen of P + (.4) P d (.5) These are obained by subsiuing he inverse sine ransform of eq. (.8) ino eq. (.7):

6 -6 Using cos sin sin sin d cos sin d L lim d cos sin d L ax bx a b x b a x, his can be wrien as (.6) cos L cos L lim P d L (.7) If we choose o evaluae he limi L, he cosine erms are hard o deal wih, bu we expec hey will vanish since hey oscillae rapidly. This is equivalen o averaging over a L / o monochromaic field. Alernaively, we can average over a single cycle: obain eq. (.4). The oher relaion can be derived in a similar way. Noe ha he Kramers Krönig relaionships are a consequence of causaliy, which dicae he lower limi of iniial = on he firs inegral evaluaed above. Example: Driven harmonic oscillaor One can classically model he absorpion of ligh hrough a resonan ineracion of he elecromagneic field wih an oscillaing dipole, using Newon s equaions for a forced damped harmonic oscillaor: x x xf m (.8) / Here he x is he coordinae being driven, is he damping consan, and k / m is he naural frequency of he oscillaor. We originally solved his problem is o ake he driving force o have he form of a monochromaic oscillaing source F F cos (.9) Then, equaion (.8) has he soluion F x ( ) sin m (.3) wih an (.3) This shows ha he driven oscillaor has an oscillaion period ha is dicaed by he driving frequency, and whose ampliude and phase shif relaive o he driving field is dicaed by is deuning from resonance. If we cycle-average o obain he average absorbed power from he field, he absorpion specrum is

7 -7 P F() x () avg F m (.3) To deermine he response funcion for he damped harmonic oscillaor, we seek a soluion o eq. (.8) using an impulsive driving force F F of his oscillaor o an arbirary force is. The linear response x d R F (.33) so ha ime-dependence wih an impulsive driving force is direcly proporional o he response funcion, x F R. For his case, we obain R exp sin (.34) m The reduced frequency is defined as (.35) 4 From his, we evaluae eq. (.6) and obain he suscepibiliy m i (.36) As we will see shorly, he absorpion of ligh by he oscillaor is proporional o he imaginary par of he suscepibiliy (.37) m The real par is (.38) m For he case of weak damping commonly encounered in molecular specroscopy, eq. (.36) is wrien as a Lorenzian lineshape by using he near-resonance approximaion (.39). (.4) m i /

8 -8 Then he imaginary par of he suscepibiliy shows asymmeric lineshape wih a line widh of full widh a half maximum. m /4 m /4 (.4) (.4) Nonlinear response funcions If he sysem does no respond in a manner linearly proporional o he applied poenial bu sill perurbaive, we can include nonlinear erms, i.e. higher expansion orders of A in eq. (.3). Le s look a second order: ;, A d d R f f (.43) Again we are inegraing over he enire hisory of he applicaion of wo forces f and f, including any quadraic dependence on f. In his case, we will enforce causaliy hrough a ime ordering ha requires () ha all forces mus be applied before a response is observed and () ha he applicaion of f mus follow f. Tha is or which leads o R ;, R (.44) ;, A d d R f f (.45) Now we will call he sysem saionary so ha we are only concerned wih he ime inervals beween consecuive ineracion imes. If we define he inervals beween adjacen ineracions (.46)

9 -9 Then we have, (.47) A d d R f f Readings. Berne, B. J., Time-Dependen Propeies of Condensed Media. In Physical Chemisry: An Advanced Treaise, Vol. VIIIB, Henderson, D., Ed. Academic Press: New York, 97.. Berne, B. J.; Pecora, R., Dynamic Ligh Scaering. R. E. Krieger Publishing Co.: Malabar, FL, Chandler, D., Inroducion o Modern Saisical Mechanics. Oxford Universiy Press: New York, Mazenko, G., Nonequilibrium Saisical Mechanics. Wiley-VCH: Weinheim, Slicher, C. P., Principles of Magneic Resonance, wih Examples from Solid Sae Physics. Harper & Row: New York, Wang, C. H., Specroscopy of Condensed Media: Dynamics of Molecular Ineracions. Academic Press: Orlando, Zwanzig, R., Nonequilibrium Saisical Mechanics. Oxford Universiy Press: New York,.

10 -.. Quanum Linear Response Funcions To develop a quanum descripion of he linear response funcion, we sar by recognizing ha he response of a sysem o an applied exernal agen is a problem we can solve in he ineracion picure. Our ime-dependen Hamilonian is H ˆ H H f A H V (.48) is he maerial Hamilonian for he equilibrium sysem. The exernal agen acs on he equilibrium sysem hrough Â, an operaor in he sysem saes, wih a ime-dependence f(). We ake V() o be a small change, and rea his problem wih perurbaion heory in he ineracion picure. We wan o describe he nonequilibrium response A, which we will ge by ensemble averaging he expecaion value of Â, i.e. A. Remember he expecaion value for a pure sae in he ineracion picure is A A I I I UI AIUI (.49) The ineracion picure Hamilonian for eq. (.48) is I f A V U V U I (.5) To calculae an ensemble average of he sae of he sysem afer applying he exernal poenial, we recognize ha he nonequilibrium sae of he sysem characerized by described by I is in fac relaed o he iniial equilibrium sae of he sysem hrough a ime-propagaor, as seen in eq. (.49). So he nonequilibrium expecaion value A is in fac obained by an equilibrium average over he expecaion value ofu A U : n I I I A p nu AU n (.5) n I I I Again n are eigensaes of H. Working wih he firs order soluion o i UI d f AI UI (.5) we can now calculae he value of he operaor A a ime, inegraing over he hisory of he applied ineracion f :

11 - A U AU I I I i i d f AI AI d f AI (.53) Here noe ha f is he ime-dependence of he exernal agen. I does no involve operaors in H and commues wih A. Working oward he linear response funcion, we jus reain he erms linear in f A i AI d f AI AI AI AI i AI d f AI, AI Since our sysem is iniially a equilibrium, we se and swich variables o he ime inerval A U AU obain and using I i A AI d fai, AI (.54) (.55) We can now calculae he expecaion value of A by performing he ensemble-average described in eq. (.5). Noing ha he force is applied equally o each member of ensemble, we have i A A d f AI, AI (.56) The firs erm is independen of f, and so i comes from an equilibrium ensemble average for he value of A. A p na n A (.57) n I n The second erm is jus an equilibrium ensemble average over he commuaor in AI(): A, A p n A, A n (.58) I I n I I n Comparing eq. (.56) wih he expression for he linear response funcion, we find ha he quanum linear response funcion is i R AI, AI (.59) < or as i is someimes wrien wih he uni sep funcion in order o enforce causaliy: i R AI, AI (.6)

12 - The imporan hing o noe is ha he ime-developmen of he sysem wih he applied exernal poenial is governed by he dynamics of he equilibrium sysem. All of he ime-dependence in he response funcion is under H. The linear response funcion is herefore he sum of wo correlaion funcions wih he order of he operaors inerchanged, which is he imaginary par of he correlaion funcion C i R AI AI AI AI i * CAA CAA C (.6) As we expec for an observable, he response funcion is real. If we express he correlaion funcion in he eigensae descripion: hen n mn i mn (.6) nm, C p A e R pn Amn sinmn nm, (.63) R can always be expanded in sines an odd funcion of ime. This reflecs ha fac ha he impulse response mus have a value of (he deviaion from equilibrium) a =, and move away from a he poin where he exernal poenial is applied. Readings. Mukamel, S., Principles of Nonlinear Opical Specroscopy. Oxford Universiy Press: New York, 995; Ch. 5.

13 -3.3. The Response Funcion and Energy Absorpion Le s invesigae he relaionship beween he linear response funcion and he absorpion of energy from he exernal agen in his case an elecromagneic field. We will relae his o he absorpion coefficien E / I which we have described previously. For his case, H H f A H E (.64) This expression gives he energy of he sysem, so he rae of energy absorpion averaged over he nonequilibrium ensemble is described by: H f E A (.65) We will wan o cycle-average his over he oscillaing field, so he ime-averaged rae of energy absorpion is T f E d A T T f d A d R f T Here he response funcion is R i elecromagneic field, we can wrie (.66), /. For a monochromaic i * cos i f E Ee Ee (.67) which leads o he following for he second erm in (.66): i * i i * i d R Ee Ee Ee Ee (.68) By differeniaing (.67), and using i wih (.68) in eq. (.66), we have T i * i i * i E A ft f d iee iee Ee Ee T 4T (.69) We will now cycle-average his expression, seing T. The firs erm vanishes and he cross erms in second inegral vanish, because T i i T i i T d e e and d e e. The rae of energy absorpion from he field is i E = E 4 E (.7)

14 -4 So, he absorpion of energy by he sysem is relaed o he imaginary par of he suscepibiliy. Now, from he inensiy of he inciden field, Readings /8 I c E, he absorpion coefficien is E 4 I c. McQuarrie, D. A., Saisical Mechanics. Harper & Row: New York, 976. (.7)

15 -5.4. Relaxaion of a Prepared Sae The impulse response funcion R describes he behavior of a sysem iniially a equilibrium ha is driven by an exernal field. Alernaively, we may need o describe he relaxaion of a prepared sae, in which we follow he reurn o equilibrium of a sysem iniially held in a nonequilibrium sae. This behavior is described by sep response funcion S. The sep response comes from holding he sysem wih a consan field H H fa unil a ime when he sysem is released, and i relaxes o he equilibrium sae governed by H H. We can anicipae ha he forms of hese wo funcions are relaed. Jus as we expec ha he impulse response o rise from zero and be expressed as an odd funcion in ime, he sep response should decay from a fixed value and look even in ime. In fac, we migh expec o describe he impulse response by differeniaing he sep response, as seen in he classical case. d R S (.7) kt d An empirical derivaion of he sep response begins wih a few observaions. Firs, response funcions mus be real since hey are proporional o observables, however quanum correlaion funcions are complex and * follow C C. Classical correlaion funcions are real and even, C C, and have he properies of a sep response. To obain he relaxaion of a real observable ha is even in ime, we can consruc a symmerized funcion, which is jus he real par of he correlaion funcion: S A A A A AA I I I I AA AA AA C C C (.73) The sep response funcion S defined as follows for. S AI, AI From he eigensae represenaion of he correlaion funcion, (.74)

16 -6 nm, n mn i mn C p A e (.75) we see ha he sep response funcion can be expressed as an expansion in cosines S pn Amn cosmn Furher, one can readily show ha he real and imaginary pars are relaed by dc C d dc C d nm, (.76) (.77) Which shows how he impulse response is relaed o he ime-derivaive of he sep response. In he frequency domain, he specral represenaion of he sep response is obained from he Fourier Laplace ransform AA i S ds e (.78) AA S C C AA AA AA e CAA Now, wih he expression for he imaginary par of he suscepibiliy, (.79) e C AA (.8) we obain he relaionship anh S AA (.8) This is he formal expression for he flucuaion-dissipaion heorem, proven in 95 by Callen and Welon. I followed an observaion made many years earlier (93) by Lars Onsager for which he was awarded he 968 Nobel Prize in Chemisry: The relaxaion of macroscopic nonequilibrium disurbance is governed by he same laws as he regression of sponaneous microscopic flucuaions in an equilibrium sae. x x x x Noing ha anh x e e e e high emperaure (classical) limi and anh x x for x>>, we see ha in he S AA (.8) kt

17 -7 Appendix: Derivaion of sep response funcion We can show more direcly how he impulse and sep response are relaed. To begin, le s consider he sep response experimen, H fa H (.83) H and wrie he expecaion values of he inernal variable A for he sysem equilibraed under H a ime = and =. H fa e H fa A A Z e (.84) Z e A A Z e H H (.85) Z If we make he classical linear response approximaion, which saes ha when he applied poenial fa is very small relaive o H, hen and Z Z, ha H fa H e e fa (.86) A A A f A (.87) and he ime dependen relaxaion is given by he classical correlaion funcion A f A A (.88) For a descripion ha works for he quanum case, le s sar wih he sysem under H a =, ramp up he exernal poenial a a slow rae unil =, and hen abruply shu off he exernal poenial and wach he sysem. We will describe he behavior in he limi.

18 -8 H H fae H (.89) Wriing he ime-dependence in erms of a convoluion over he impulse response funcion R, we have A lim dr e f (.9) Alhough he inegral over he applied force ( ) is over imes <, he sep response facor ensures ha. Now, expressing R as a Fourier ransform over he imaginary par of he suscepibiliy, we obain f i i A lim d d e e f dpp e i f i fc d e i i (.9) A more careful derivaion of his resul ha reas he quanum mechanical operaors properly is found in he references. Readings. Mazenko, G., Nonequilibrium Saisical Mechanics. Wiley-VCH: Weinheim, 6.. Zwanzig, R., Nonequilibrium Saisical Mechanics. Oxford Universiy Press: New York,.