and reconizin that we obtain the followin equation for ~x (s): ~q(s) = ~x = dt e?st q(t) dt e?st x (t) dt e?st x (t) = s ~x (s)? sx ()? _x () or (s +!

Transcription

1 G5.65: Statistical Mechanics Notes for Lecture 4 I. THE HARMONIC BATH HAMILTONIAN In the theory of chemical reactions, it is often possible to isolate a small number or even a sinle deree of freedom in the system that can be used to characterize the reaction. This deree of freedom is coupled to other derees of freedom (for example, reactions often take place in solution). Isomerization or dissociation of a diatomic molecule in solution is an excellent example of this type of system. The deree of freedom of paramount interest is the distance between the two atoms of the molecule { this is the deree of freedom whose detailed dynamics we would like to elucidate. The dynamics of the \bath" or environment to which is couples is less interestin, but still must be accounted for in some manner. A model that has maintained a certain level of both popularity and success is the so called \harmonic bath" model, in which the environment to which the special deree(s) of freedom couple is replaced by an eective set of harmonic oscillators. We will examine this model for the case of a sinle deree of freedom of interest, which we will desinate q. For the case of the isomerizin or dissociatin diatomic, q could be the coordinate r? hri, where r is the distance between the atoms. The particular denition of q ensures that hqi =. The deree of freedom q is assumed to couple to the bath linearly, ivin a Hamiltonian of the form H = p m + (q) + X " p m + m! x + q where the index runs over all the bath derees of freedom,! are the harmonic bath frequencies, m are the harmonic bath masses, and are the couplin constants between the bath and the coordinate q. p is a momentum conjuate to q, and m is the mass associated with this deree of freedom (e.., the reduced mass in the case of a diatomic). The coordinate q is assumed to be subject to a potential (q) as well (e.., an internal bond potential). The form of the couplin between the system (q) and the bath (x ) is known as bilinear. Below, usin a completely classical treatment of this Hamiltonian, we will derive an equation for the detailed dynamics of q alone. This equation is known as the eneralized Lanevin equation (GLE). # II. DERIVATION OF THE GLE The GLE can be derived from the harmonic bath Hamiltonian by simply solvin Hamilton's equations of motion, which take the form _q = p m _p X x? m! q _x = p m _p =? x? q This set of equations can also be written as second order dierential equation: mq x? m x =? x? q X q In order to derive an equation for q, we solve explicitly for the dynamics of the bath variables and then substitute into the equation for q. The equation for x is a second order inhomoeneous dierential equation, which can be solved by Laplace transforms. We simply take the Laplace transform of both sides. Denote the Laplace transforms of q and x as

2 and reconizin that we obtain the followin equation for ~x (s): ~q(s) = ~x = dt e?st q(t) dt e?st x (t) dt e?st x (t) = s ~x (s)? sx ()? _x () or (s +! )~x (s) = sx () + _x ()? m ~q(s) s ~x (s) = s +! x () + s +! _x ()? m ~q(s) s +! x (t) can be obtained by inverse Laplace transformation, which is equivalent to a contour interal in the complex s-plane around a contour that encloses all the poles of the interand. This contour is known as the Bromwich contour. To see how this works, consider the rst term in the above expression. The inverse Laplace transform is i I se st ds s +! = i I se st ds (s + i! )(s? i! ) The interand has two poles on the imainary s-axis at i!. Interation over the contour that encloses these poles picks up both residues from these poles. Since the poles are simple poles, then, from the residue theorem: I se st ds i (s + i! )(s? i! ) = i! e i!t i +?i! e?i!t = cos! t i i!?i! By the same method, the second term will ive (sin! t)=!. The last term is the inverse Laplace transform of a product of ~q(s) and =(s +! ). From the convolution theorem of Laplace transforms, the Laplace transform of a convolution ives the product of Laplace transforms: dt e?st d f()(t? ) = ~ f(s)~(s) Thus, the last term will be the convolution of q(t) with (sin! t)=!. Puttin these results toether, ives, as the solution for x (t): Z x (t) = x () cos! t + _x () sin! t? t! dq() sin! (t? ) The convolution term can be expressed in terms of _q rather than q by interatin it by parts: Z t d q() sin! (t? ) = [q(t)? q() cos! t]? d _q() cos! (t? ) The reasons for preferrin this form will be made clear shortly. The bath variables can now be seen to evolve accordin to Z x (t) = x () cos! t + _x () sin! t + t! d _q() cos! (t? )? [q(t)? q() cos! t] Substitutin this into the equation of motion for q, we nd mq x () cos! t + p () sin! t + q() cos! t? X We now introduce the followin notation for the sums over bath modes appearin in this equation: d _q() cos! (t?)+ X q(t)?

3 . Dene a dynamic friction kernel (t) = X cos! t. Dene a random force R(t) =? X x () + q() cos! t + p () sin! t Usin these denitions, the equation of motion for q reads d _q()(t? ) + R(t) () Eq. () is known as the eneralized Lanevin equation. Note that it takes the form of a one-dimensional particle subject to a potential (q), driven by a forcin function R(t) and with a nonlocal (in time) dampin term?r t d _q()(t?), which depends, in eneral, on the entire history of the evolution of q. The GLE is often taken as a phenomenoloical equation of motion for a coordinate q coupled to a eneral bath. In this spirit, it is worth takin a moment to discuss the physical meanin of the terms appearin in the equation. III. PROPERTIES OF THE GLE Below we discuss the physical meanin of the terms appearin the GLE A. The random force term Within the context of a harmonic bath, the term \random force" is somethin of a misnomer, since R(t) is completely deterministic and not random at all!!! We will return to this point momentarily, however, let us examine particular features of R(t) from its explicit expression from the harmonic bath dynamics. Note, rst of all, that it does not depend on the dynamics of the system coordinate q (except for the appearance of q()). In this sense, it is independent or \orthoonal" to q within a phase space picture. From the explicit form of R(t), it is straihtforward to see that the correlation function h _q()r(t)i = i.e., the correlation function of the system velocity _q with the random force is. This can be seen by substitutin in the expression for R(t) and interatin over initial conditions with a canonical distribution weihtin. For certain potentials (q) that are even in q (such as a harmonic oscillator), one can also show that hq()r(t)i = Thus, R(t) is completely uncorrelated from both q and _q, which is a property we miht expect from a truly random process. In fact, R(t) is determined by the detailed dynamics of the bath. However, we are not particularly interested or able to follow these detailed dynamics for a lare number of bath derees of freedom. Thus, we could just as well model R(t) by a completely random process (satisfyin certain desirable features that are characteristic of a more eneral bath), and, in fact, this is often done. One could, for example, postulate that R(t) act over a maximum time t max at discrete points in time kt, ivin N = t max =t values of R k = R(kt), and assume that R k takes the form of a aussian random process: R k = NX j= ha j e ijk=n + b j e?ijk=ni where the coecients fa j and fb j are chosen at random from a aussian distribution function. This miht be expected to be suitable for a bath of hih density, where stron collisions between the system and a bath particle are essentially nonexistent, but where the system only sees feels the relatively \soft" uctuations of the less mobile bath. For a low density bath, one miht try modelin R(t) as a Poisson process of very stron collisions. Whatever model is chosen for R(t), if it is a truly random process that can only act at discrete points in time, then the GLE takes the form of a stochastic (based on random numbers) intero-dierential equation. There is a whole body of mathematics devoted to the properties of such equations, where heavy use of an It^o calculus is made. 3

4 B. The dynamic friction kernel The convolution interal term d _q()(t? ) is called the memory interal because it depends, in eneral, on the entire history of the evolution of q. Physically it expresses the fact that the bath requires a nite time to respond to any uctuation in the motion of the system (q). This, in turn, aects how the bath acts back on the system. Thus, the force that the bath exerts on the system presently depends on what the system coordinate q did in the past. However, we have seen previously the reression of uctuations (their decay to ) over time. Thus, we expect that what the system did very far in the past will no loner the force it feels presently, i.e., that the lower limit of the memory interal (which is riorously ) could be replaced by t? t mem, where t mem is the maximum time over which memory of what the system coordinate did in the past is important. This can be interpreted as a indicatin a certain decay time for the friction kernel (t). In fact, (t) often does decay to in a relatively short time. Often this decay takes the form of a rapid initial decay followed by a slow nal decay, as shown in the ure below: Consider the extreme case that the bath is capable of respondin innitely quickly to chanes in the system coordinate q. This would be the case, for example, if there were a lare mass disparity between the system and the bath (m >> m ). Then, the bath retains no memory of what the system did in the past, and we could take (t) to be a -function in time: Then (t) = (t) and the GLE becomes d _q()(t? ) = d _q(t? )() = _q + R(t) d () _q(t? ) = _q(t) This simpler equation of motion is known as the Lanevin equation and it is clearly a special case of the more eneralized equation of motion. It is often invoked to describe brownian motion where clearly such a mass disparity is present. The constant is known as the static friction and is iven by = dt (t) In fact, this is a eneral relation for determinin the static friction constant. The other extreme is a very sluish bath that responds slowly to chanes in the system coordinate. In this case, we may take (t) to be a constant (), at least, for times short compared to the response time of the bath. Then, the memory interal becomes and the GLE becomes d _q()(t? ) (q(t)? q()) (q) + (q? q ) + R(t) where the friction term now manifests itself as an extra harmonic term added to the potential. Such a term has the eect of trappin the system in certain reions of conuration space, an eect known as dynamic cain. An example of this is a dilute mixture of small, liht particles in a bath of heavy, lare particles. The liht particles can et trapped in reions of space where many bath particles are in a sort of spatial \cae." Only the rare uctuations in the bath that open up larer holes in conuration space allow the liht particles to escape the cae, occasionally, after which, they often et trapped aain in a new cae for a similar time interval. 4

5 C. Relation between the dynamic friction kernel and the random force From the denitions of R(t) and (t), it is straihtforward to show that there is a relation between them of the form hr()r(t)i = kt (t) This relation is known as the second uctuation dissipation theorem. The fact that it involves a simple autocorrelation function of the random force is particular to the harmonic bath model. We will see later that a more eneral form of this relation exists, valid for a eneral bath. This relation must be kept in mind when introducin models for R(t) and (t). In eect, it acts as a constraint on the possible ways in which one can model the random force and friction kernel. IV. MORI-ZWANZIG THEORY: A MORE GENERAL DERIVATION OF THE GLE A derivation of the GLE valid for a eneral bath can be worked out. The details of the derivation are iven in the book by Berne and Pecora called Dynamic Liht Scatterin. The system coordinate q and its conjuate momentum p are introduced as a column vector: A = and, in addition, one introduces statistical projection operators P and Q that project onto subspaces in phase space parallel and orthoonal to A. These operators take the form q p P = h:::a T ihaa T i? Q = I? P These operators are Hermitian and satisfy the property of idempotency: Also, note that P = P Q = Q P A = A QA = The time evolution of A is iven by application of the classical propaator: A(t) = e ilt A() Note that the evolution of A is unitary, i.e., it preserves the norm of A: ja(t)j = ja()j Dierentiatin both sides of the time evolution equation for A ives: da dt = eilt ila() Then, an identity operator is inserted in the above expression in the form I = P + Q: da dt = eilt (P + Q)iLA() = e ilt P ila() + e ilt QiLA() The rst term in this expression denes a frequency matrix actin on A: 5

6 e ilt P ila() = e ilt hilaa T ihaa T i? A = hilaa T ihaa T i? e ilt A = hilaa T ihaa T i? A(t) ia(t) where = hlaa T ihaa T i? In order to evaluate the second term, another identity operator is inserted directly into the propaator: Consider the dierence between the two propaators: If this dierence is Laplace transformed, it becomes which can be simplied via the eneral operator identity: Lettin we have or e ilt i(p +Q)Lt = e e ilt? e iqlt (s? il)?? (s? iql)? A?? B? = A? (B? A)B? A = (s? il) B = (s? iql) (s? il)?? (s? iql)? = (s? il)? (s? iql? s + il)(s? iql)? = (s? il)? ip L(s? iql)? (s? il)? = (s? iql)? + (s? il)? (s? iql? s + il)(s? iql)? Now, inverse Laplace transformin both sides ives e ilt = e iqlt + d e il(t? ) ip Le iql Thus, multiplyin fromthe riht by QiLA ives e ilt QiLA = e iqlt QiLA + d e il(t? ) ip Le iql QiLA Dene a vector F(t) = e iqlt QiLA() so that e ilt QiLA = F(t) + d hilf()a T ihaa T i? A(t? ) 6

7 Because F(t) is completely orthoonal to A(t), it is straihtforward to show that QF(t) = F(t) Then, hilf()a T ihaa T i? A = hilqf()a T ihaa T i? A =?hqf()(ila) T ihaa T i? A =?hq F()(iLA) T ihaa T i? A =?hqf()(qila) T ihaa T i? A Thus, e ilt QiLA = F(t)? Finally, we dene a memory kernel matrix: Then, combinin all results, we nd, for da=dt: =?hf()f T ()ihaa T i? A d hf()f T ()ihaa T i? A(t? ) K(t) = hf()f T ()ihaa T i? Z da t dt = i(t)a? d K()A(t? ) + F(t) which equivalent to a eneralized Lanevin equation for a particle subject to a harmonic potential, but coupled to a eneral bath. For most systems, the quantities appearin in this form of the eneralized Lanevin equation are =m i =?m! K(t) = (t)=m F(t) = R(t) It is easy to derive these expressions for the case of the harmonic bath Hamiltonian when (q) = m! q =. For the case of a harmonic bath Hamiltonian, we had shown that the friction kernel was related to the random force by the uctuation dissipation theorem: hr()r(t)i = hr()e ilt R()i = kt (t) For a eneral bath, the relation is not as simple, owin to the fact that F(t) is evolved usin a modied propaator exp(iqlt). Thus, the more eneral form of the uctuation dissipation theorem is hr()e iqlt R()i = kt (t) so that the dynamics of R(t) is prescribed by the propaator exp(iqlt). This more eneral relation illustrates the diculty of denin a friction kernel for a eneral bath. However, for the special case of a sti harmonic diatomic molecule interactin with a bath for which all the modes are soft compared to the frequency of the diatomic, a very useful approximation results. One can show that hr()e iqlt R()i hr()e ilconst R()i where il cons is the Liouville operator for a system in which the diatomic is held riidly xed at some particular bond lenth (i.e., a constrained dynamics). Since the friction kernel is not sensitive to the details of the internal potential of the diatomic, this approximation can also be used for diatomics with sti, anharmonic potentials. This approximation is referred to as the riid bond approximation (see Berne, et al, J. Chem. Phys. 93, 584 (99)). 7

8 V. EXAMPLE: VIBRATIONAL DEPHASING AND ENERGY RELAXATION Recall that the Fourier transform of a time correlation function can be related to some kind of frequency spectrum. For example, the Fourier transform of the velocity autocorrelation function of a particular deree of freedom q of interest C vv (t) = h _q() _q(t)i h _q i where v = _q, ives the relevant frequencies contributin to the dynamics of q, but does not ive amplitudes. This \frequency" spectrum I(!) is simply iven by I(!) = dt e i!t C vv (t) That is, we take the Laplace transform of C vv (t) usin s =?i!. Since C vv (t) carries information about the relevant frequencies of the system, the decay of C vv (t) in time is a measure of how stronly coupled the motion of q is to the rest of the bath, i.e., how much of an overlap there is between the relevant frequencies of the bath and those of q. The more of an overlap there is, the more mixin there will be between the system and the bath, and hence, the more rapidly the motion of the system will become vibrationally \out of phase" or decorrelated with itself. Thus, the decay time of C vv (t), which is denoted T is called the vibrational dephasin time. Another measure of the strenth of the couplin between the system and the bath is the time required for the system to dissipate enery into the bath when it is excited away from equilibrium. This time can be obtained by studyin the decay of the enery autocorrelation function: where "(t) is dened to be C "" = h"()"(t)i h" i "(t) = m _q + (q)? kt The decay time of this correlation function is denoted T. The question then becomes: what are these characteristic decay times and how are they related? To answer this, we will take a phenomenoloical approach. We will assume the validity of the GLE for q: and use it to calculate T and T. Suppose the potential (q) is harmonic and takes the form d _q()(t? ) + R(t) (q) = m! q Substitutin into the GLE and dividin throuh by m ives where q =?! q? (t) = (t) m d _q(t? )() + f(t) f(t) = R(t) m An equation of motion for C vv (t) can be obtained directly by multiplyin both sides of the GLE by _q() and averain over a canonical ensemble: h _q()q(t)i =?! h _q()q(t)i? d h _q() _q(t? )i() + h _q()f(t)i 8

9 Recall that h _q()f(t)i = m h _q()r(t)i = and note that also Thus, h _q()q(t)i = d dt h _q() _q(t)i = dc vv dt d h _q() _q()i = h _q()q(t)i? h _q()q()i = h _q()q(t)i h _q()q(t)i = Combinin these results ives an equation for C vv (t) d C vv () Z d t dt C vv(t) =? Z d t dt C vv(t) =? d K(t? )C vv () d?! + (t? ) C vv () which is known as the memory function equation and the kernel K(t) is known as the memory function or memory kernel. This type of intero-dierential equation is called a Volterra equation and it can be solved by Laplace transforms. Takin the Laplace transform of both sides ives However, it is clear that C vv () = and also s ~ Cvv (s)? C vv () =? ~ Cvv (s) ~ K(s) Thus, it follows that ~K(s) =! s + ~(s) s Cvv ~! (s)? = s + ~(s) ~C vv (s) s ~C vv (s) = s + s~(s) +! In order to perform the inverse Laplace transform, we need the poles of the interand, which will be determined by the solutions of s + s~(s) +! = which we could solve directly if we knew the explicit form of ~(s). However, if! is suciently larer than ~(), then it is possible to develop a perturbation solution to this equation. Let us assume the solutions for s can be written as Substitutin in this ansatz ives s = s + s + s + (s + s + s + ) + (s + s + s + )~(s + s + s + ) +! = Since we are assumin ~ is small, then to lowest order, we have 9

10 so that s =!. The rst order equation then becomes s +! = s s + s ~(s ) = or Note, however, that s =? ~(s ) =? ~(i!) ~(i!) = dt (t)e i!t = dt [(t) cos!t i(t) sin!t] (!) i (!) Thus, stoppin the rst order result, the poles of the interand occur at Dene Then s i (! + (!))? (!) s + = i? (!) s? =?i? (!) ~C vv (s) s (s? s + )(s? s? ) i? (!) and C vv (t) is then iven by the contour interal C vv (t) = i I se st ds (s? s + )(s? s? ) Takin the residue at each pole, we nd which can be simplied to ive C vv (t) = s +e s+t (s +? s? ) + s?e s?t (s?? s + ) C vv (t) = e? (!)t= cos t? (!) sin t Thus, we see that the GLE predicts C vv (t) oscillates with a frequency and decays exponentially. From the exponential decay, we can directly read o the time T : = (!) T = (!) m That is, the value of the real part of the Fourier (Laplace) transform of the friction kernel evaluated at the renormalized frequency divided by m ives the vibrational dephasin time! By a similar scheme, one can easily show that the position autocorrelation function C qq (t) = hq()q(t)i decays with the same dephasin time. It's explicit form is C qq (t) = e? (!)t= cos t + (!) sin t

11 The enery autocorrelation function C "" (t) can be expressed in terms of the more primitive correlation functions C qq (t) and C vv (t). It is a straihtforward, althouh extremely tedious, matter to show that the relation, valid for the harmonic potential of mean force, is C "" (t) = C vv(t) + C qq(t) +! _ C qq(t) Substitutin in the expressions for C qq (t) and C vv (t) ives so that the decay time T can be seen to be C "" (t) = e? (!)t (oscillatory functions of t) = (!) = (!) T m and therefore, the relation between T and T can be seen immediately to be T = T The incredible fact is that this result is also true quantum mechanically. That is, by doin a simple, purely classical treatment of the problem, we obtained a result that turns out to be the correct quantum mechanical result! Just how bi are these times? If! is very lare compared to any typical frequency relevant to the bath, then the friction kernel evaluated at this frequency will be extremely small, ivin rise to a lon decay time. This result is expect, since, if! is lare compared to the bath, there are very few ways in which the system can dissipate enery into the bath. The situation chanes dramatically, however, if a small amount of anharmonicity is added to the potential of mean force. The ure below illustrates the point for a harmonic diatomic molecule interactin with a Lennard-Jones bath. The top ure shows the velocity autocorrelation function for an oscillator whose frequency is approximately 3 times the characteristic frequency of the bath, while the bottom one shows the velocity autocorrelation function for the case that the frequency disparity is a factor of 6.

12 FIG..