Konrad-Zuse-Zentrum für Informationstechnik Berlin

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1 Konra-use-entrum für Informationstechnik Berlin Christof Schutte Smoothe Molecular Dynamics For Thermally Embee Systems SC 95-4 (May 995)

2 Smoothe Molecular Dynamics For Thermally Embee Systems Christof Schutte Konra{use{entrum Berlin Heilbronner Str., D-7 Berlin, Germany 29. Mai 995 Abstract This paper makes use of statistical mechanics in orer to construct eective potentials for Molecular Dynamics for systems with nonstationary thermal embeing. The usual approach requires the computation of a statistical ensemble of trajectories. In the context of the new moel the evaluation of only one single trajectory is sucient for the etermination of all interesting quantities, which leas to an enormous reuction of computational eort. This single trajectory is the solution to a correcte Hamiltonian system with a new potential ~V. It turns out that ~ V can be ene as spatial average of the original potential V. Therefore, the Hamiltonian ynamics ene by ~V is smoother than that eecte by V, i.e. a numerical integration of its evolution in time allows larger stepsizes. Thus, the presente approach introuces a Molecular Dynamics with smoothe trajectories originating from spatial averaging. This is eeply connecte to time{averaging in Molecular Dynamics. These two types of smoothe Molecular Dynamics share avantages (gain in eciency, reuction of error amplication, increase stability) an problems (necessity of closing relations an aaptive control schemes) which will be explaine in etail. Keywors: smoothe molecular ynamics, eective potentials, averaging, nonstationary heat bath embeing, expectation values, ensemble averages

3 Contents Introuction 2 Smoothe MD an Averaging in Time 3 2. Basic Ieas an Problems : : : : : : : : : : : : : : : : : : : : Reuction of Error Amplication : : : : : : : : : : : : : : : : Aaptation of Stepsizes : : : : : : : : : : : : : : : : : : : : : : 8 3 Statistical formulation of Molecular Dynamics 8 3. Probability Density an Expectation Values : : : : : : : : : : Temperature an Heat Bath Embeing : : : : : : : : : : : : 3.3 Initial Density : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 4 Smoothe MD an Spatial Averaging 3 4. Separable Densities : : : : : : : : : : : : : : : : : : : : : : : : Probability Density for Thermally Embee Systems : : : : : Average Potentials for MD Calculations : : : : : : : : : : : : 8 5 Conclusion 9 References 9

4 Introuction In Molecular Dynamics (MD) we are intereste in a escription of the ynamical behaviour of a (macro)molecular system in the scope of classical mechanics. Therefore we are concerne with Hamiltonian functions of the form H(q; p) = 2 pt M? p + V (q) (.) which lea to the following Hamiltonian equations of motion: t q = D ph(q; p) = M? p t p =?D qh(q; p) =?DV (q); (.2) where D p an D = D q are the ierential operators with respect to p 2 R 3N an q 2 R 3N, the momenta an space coorinates of the N atoms of the consiere molecular system. We assume that the potential V is given, an that it is a \goo classical moel" for the system. Then, if aequate initial conitions (q; p)() = (q ; p ) (.3) are given, a solution of (.2) escribes the motion of the molecular system without interaction with any other system. Thus, we only have to care for an ecient, accurate, an stable numerical solution of (.2). This is the iealize situation. There are several serious problems. Three of them will be explaine in the following. They make up the starting point for the consierations in this paper. Problem : Even if we accept that the exact solution of (.2) gave us the answers we are intereste in, we are often not able to compute its numerical solution on a suciently large time scale. In the typical situation the potential V contains parts which stan for the bon interaction between bone atoms in the molecule: V (q) = U(q) + 2 mx k= k g k (q) (.4)

5 with e.g. harmonic moels for g k for the k = : : : m ierent bon-types: g k (q) = X i;j2b (kq i? q j k? L k ) 2 ; where L k is the equilibrium length of bon type k, q i 2 R 3 the vector of the space coorinates of the ith atom, an the summation runs over all bone pairs. This g{part of the potential V causes highly oscillatory motions of the bone atoms. Because of the typical magnitue of the k, these bon vibrations appear on a timescale of about femtosecon an are the \fastest egrees of freeom" of the molecule. Careful investigations have shown that the bon vibrations are an essential part of the nonlinear ynamics of the molecule, i.e. they cannot simply be eliminate or moelle [3][8]. Thus, if we are intereste in the accuracy of the numerical solution of (.2), we have to resolve this timescale, i.e. we have to choose stepsizes fs in the time iscretization. An even if accuracy is less important we have to use fs in orer to ensure numerical stability for the iteration of the iscretization (at least for all conventional explicit iscretizations []; for most implicit methos similar stepsize bouns result from the requirement of unique convergence of the iterative solution of the nonlinear equations in each step). The typical time length of an MD simulation is t max ps. Therefore we have to make a large number t max = of time steps, i.e. only the large computational eort of a typical MD calculation strictly limits its time length. In Section 2 the iea of \smoothe ynamics" will be iscusse as a proposal for reucing the eort of those computations. Problem 2: Normally, we are not intereste in the motion of the molecular system without interaction with any other system. It is frequently esirable to simulate a system uner conitions of constant temperature T, since this is the conition uner which most experiments are performe. Hence, we have to moel the heat bath embeing of the consiere molecular system, \simply" solving (.2) is not enough. But \temperature" an \heat bath embeing" can only be ene in a statistical sense, i.e. for an ensemble of ientically prepare systems or in a stochastic theory for the single system. In aition, we are not intereste in the pure equilibrium theory but in the motions, reactions, an structural changes of the molecular system embe- 2

6 e in an environment with constant temperature. Hence Section 3 shortly presents the statistical formulation of (.2) an a enition of an ensemble of molecular systems with nonstationary embeing in heat baths of temperature T, which in particular is ierent from the \canonical" equilibrium theory. Problem 3: Another crucial point is the requirement for aequate initial conitions. The initial state of the molecular system is typically known from measurement. Therefore there is a funamental uncertainty about the initial conition of (.2) if they are aapte to experimental realizations. More precisely, the coorinates q can only be given with a (small) egree of uncertainty q k () 2 fx : jx? qkj k g 8k = : : : 3N; (.5) whereas the initial momenta p are typically not etermine experimentally. As a consequence aequate initial conition can only be given in the framework of, again, an ensemble formulation. The statistical version of (.2) (cf. Section 3) can be use for a concise enition of the \initial conitions", in particular for the initial momenta. Problems 2 an 3 eman for a statistical formulation of (.2). But the ynamical behaviour of the statistical ensemble is \rich", the computational eort for its full simulation is far too large. One main aspect of the following sections is the reuction of this eort. This is realize by reucing the rich ensemble{ynamics to the far simpler evolution of a single Hamiltonian system with a new Hamiltonian ~ H. The other main aspect (construction of a smoothe MD) is eeply connecte to this because the ~ H-trajectories are smoother than the corresponing H-trajectories. 2 Smoothe MD an Averaging in Time As state above bon vibrations are the fastest motions in typical MD situations an lea to har restrictions for the stepsize in (explicit) numerical integration methos. Mostly, we o not want to compute all these \unessential" oscillatory etails. But we want to get correct information about the physically relevant ynamical behaviour of the consiere system, i.e. we 3

7 cannot ignore the bon ynamics. 2. Basic Ieas an Problems The iea of smoothe ynamics is to compute only the \running average"!! (t) q(t) p(t) := A q p of the exact solution (q; p) T of (.2). The average operator A is given by (A x)(t) := t? s w x(s) s with an appropriate weight function w with lim t! w(t) =, e.g. R w(x) = [?=2;=2] (x) = ( :?=2 x =2 : otherwise : Another possibility may be to choose w in a way which makes A a low pass lter with cut{o frequency O(=). Now, the task is to euce a ierential equation for (q; p) from (.2). We can use the fact that to get using (.2) that t A x = A t q t p A M? p?dv (q) 8x A (2.) The trajectories of (2.) woul be smooth, if A was chosen in a way which lets the \fastest oscillations" of (q; p) occur on a time scale t fs, i.e. that the numerical integration of (2.) allows stepsizes t fs (cf. Figure ). Unfortunately, we have to know the solution q(t) of (.2) to compute the right han sie DV (q) of (2.), because the nonlinear function DV (q) an A o not commute: DV (q) = A DV (q) 6= DV (A q) = DV (q): (2.2) 4

8 t[fs] Fig.. Typical ynamics with bon vibrations (top) an its running average with averaging on ierent time scales. An worse, up to now there is no way to euce a function v with DV (q) = v (q) (2.3) using mathematical means only. Thus, we have to look for a physical alternative: we may construct v by using aitional physical insight in the ynamics, e.g. in form of aitional postulates. [9] an [] may be taken as examples for this approach. In [9], a result from statistical mechanics (equipartition of energy for ergoic systems embee in a heat bath of xe temperature) is use as such a postulate. Then, it is shown that v can be written as v = D ~ V with a correcte potential ~ V, if the parameters k in (.4) are large enough in comparison to all changes in the forces eecte by U (gap conition): k max t D 2 U(q(t)): In the new potential ~ V the bon interaction part V? U is cancelle an a \smoother" correction term occurs, which moels the inuence of the bon motions on the \rest" of the motions. Conclusively, a statistical postulate allows to construct a smoother potential which moels the inuence of bon ynamics instea of containing it explicitly. How can such a smoother potential be constructe irectly using a statistical formulation of (.2)? Section 4 gives an answer to this question. 5

9 2.2 Reuction of Error Amplification Potentials with steep graients can lea to a strong amplication of numerical errors along the trajectories of the corresponing Hamiltonian system. For highly oscillatory trajectories a \successful smoothing" can eect an essential reuction of this error amplication. This shoul become clear if one consiers the following 4-imensional test system: H(q ; q 2 ; p ; p 2 ) =! p p !2 (q 2? q ) 2 + V (q ) (2.4) with a strong harmonic part (! ) an the morse potential (cf. Figure 2) V (q) = 2 (? exp(?a q))l with a > an L 2 N: (2.5) It can be shown (using perturbation analysis or the results of [9]) that for! a (gap conition) an time intervals not too large the smoothe evolution of (2.4) is approximately given by the solution of the 2{Hamiltonian system with H(q; p) = 2 p2 + V (q): This allows us to compare the error propagation in the two systems: Assume that we have mae an (numerical) error for the state (q; p) of a system at t =. How strong will it be amplie by the evolution of the system? Let t be the phase ow of (2.4), i.e. t x is the solution of (2.4) with initial conitions x = (q; p)() = (q ; p ). Then we are intereste in the interval conition number [; t] := max s2[;t] sup k t (x + )? t x k ; (2.6) kk in an arbitrary norm k k (for the theoretical backgroun of this concept see [2]). Figure 3 shows the evolution of [; t] in time t for the original an for the smoothe system. We observe that, rstly, the amplication of errors can be strong for collision potentials like (2.5) an that, seconly, this amplication can be reuce by smoothing the ynamics. Thus, smoothing techniques will not only help to increase stepsize an eciency but also allow accurate MD integration on larger time intervals. 6

10 Potential V V x Original ynamics Fig. 2. Potential V with resolution of the fast oscillations. q t from (2.5) for a = 4 an L = 8 an corresponing initial ynamics q component of smoothe ynamics.5 3 Conition of smoothe ynamics t q component of original ynamics t 5 Conition of original ynamics t t Fig. 3. Interval conition numbers [; t] an q{components of the original (bottom) an the smoothe (top) ynamics of (2.4) with! =, a = 4, an L = 8. Fast oscillations of original ynamics graphically not resolve (cf. Fig.2). Note that [; t] is times larger for the original system. 7

11 2.3 Aaptation of Stepsizes Figure 3 shows another important aspect of the smoothe ynamics: For the original, highly oscillatory solution an for each iscretization scheme there is a xe stepsize which is overall optimal with respect to eciency ( = a xe fraction of the average perio of the oscillation), i.e. stepsize control cannot increase eciency. For the corresponing solution of the smoothe ynamics this is not the case, because the oscillations are \cancelle" (cf. Figure : you can make large timesteps except in the region of the two jumps). Thus, in orer to be ecient, smoothe MD requires stepsize control schemes. Up to now, it has not become clear how to solve this problem most eciently: in the scope of explicit, symmetric extrapolation schemes (cf. [4] or [2]) or by use of symplectic iscretizations [5][6]. 3 Statistical formulation of Molecular Dynamics This Section is concerne with the question of how to give an ensemble formulation of (.2) an of the aitional heat bath embeing of the molecular system. 3. Probability Density an Expectation Values We consier a statistical ensemble of ientically prepare molecular systems which are escribe by the Hamiltonian H from (.). The basic concept of the formulation is the introuction of a (phase space) probability ensity f : R 3N R 3N R! [; ]: for this ensemble. f(q; p; t) must be interprete as the relative frequency of systems in the ensemble which occupy state (q; p) at time t. The equation of motion for f is the well{known Liouville t f = [H ; f] = D q H D p f? D p H D q f (3.) = DV (q) D p f? D q f M? p 8

12 with the Poisson brackets [; ]. Let us assume that a normalize initial ensity f(; ; ) = f : R 3N R 3N! [; ] with R 6N f (q; p) q p = is given (see Section 3.3). Let t again be the phase ow of (.2), i.e. t (q ; p ) is the solution of (.2) with initial conitions (q; p)() = (q ; p ). Then, the formal solution of (3.) can be given: f t (q; p) ; t = f (q; p) : (3.2) From (3.2) we see that (3.) escribes the transport of the initial ensity along the integral curves of (.2). Moreover, it is obvious that a solution of (3.) is equivalent to the evaluation of the total ow t, i.e. equivalent to the solution of an innite number of initial value problems with (.2) as ierential equation. Fortunately, we are not intereste in f itself but in the expectation values of physical observables with respect to f, i.e. with respect to our ensemble. An observable is a suciently smooth an f{integrable function A : R 3N R 3N! R m ; m 2 N an its expectation value is ene as hai(t) = R 6N A(q; p) f(q; p; t) q p: So far, this can be foun in textbooks on Statistical Mechanics, e.g. [7]. Now, Liouville's equation (3.) gives us equations of motion for the expectation values (via partial integration): t hai = hd qa M? pi? hd p A DV i: (3.3) In particular, the equation of motion for the position an momenta observable A(q; p) = (q; p) T hqi t hpi A 9 M? hpi?h DV (q) i A (3.4)

13 an we observe the same funamental problem of noncommutativity as we ha in Section 2 (eq. (2.2)): hdv (q)i 6= DV (hqi); (3.5) i.e. equation (3.4) is not close, we nee knowlege about f for the evaluation of its right han sie. More precisely, we o not nee f but only the reuce ensity because of F (q; t) := h DV (q) i = R 3N f(q; p; t) p; (3.6) R 3N DV (q) F (q; t) q: (3.7) But again, we are not able to euce this knowlege mathematically without solving (3.) an we have to construct it using a physical moel, i.e. we have to construct a closing relation hdv (q)i) = D ~ V (hqi) for the statistical equation of motion (3.4). For the case of (nonequilibrium) thermal embeing ~ V can be constructe via a heuristical moel for f (cf. Section 4). But before going into etails we must give some comments on the enition of \temperature", \heat bath embeing", an the initial ensity f (\solving" Problem 2 an 3 from the introuction). 3.2 Temperature an Heat Bath Embeing All ensities f(q; p) = (H(q; p)) with a smooth an suciently ecreasing function : R +! [; ] are stationary solutions of Liouville's equation (3.). One of these, the well{known canonical ensemble f c (q; p) = Q exp(?h(q; p)) with Q = R 6N e?h p q; (3.8) is use to ene \temperature": f c is the probability ensity of our ensemble i the ensemble is in equilibrium with a heat bath of temperature T = k B ; k B : Boltzmann constant:

14 This statistical way of ening temperature has an interesting consequence for Hamiltonian of the form (.): If hi is the expectation value with respect to f c it is hpi = an with the yaic prouct (p p) kl = p k p l we n: hp pi = M = M k BT; if M = iag(m k ) is iagonal (what we assume in the following). Together we have hp pi? hpi hpi = M: (3.9) In particular, the eviation of the measurement of hp l i in the canonical ensemble is controlle by the temperature: (p l ) := hp 2 l i? hp l i 2 = m l = m lk B T: A concrete computation of an expectation value hai with respect to f c remains a very har problem, because a careful approximation of the corresponing high{imensional integrals (e.g. in the evaluation of Q) prouces ramatically large computational eort. Moreover, in the typical MD context, we often are not intereste in escribing the equilibrium state of the molecular system. Certainly, we want to simulate the system in interaction with a heat bath of xe temperature but not necessarily in equilibrium with it. Thus, our question is how to construct a ensity which escribes this situation? This is a crucial point. Let us be careful an therefore precise. The solution f of Liouville's equation with initial conition f(; ) = f escribes an ensemble of single system of type S, which all are totally characterize by the Hamiltonian H, i.e. the evolution of each single systems is totally etermine by H an the corresponing initial conition for this system. But H oesn't inclue the heat bath: systems of type S are free, i.e. they are not interacting with a heat bath. Then, what is the meaning of the statement \f c escribes equilibrium heat bath embeing of the S{ensemble"? It states, that there is a particular initial ensity f = f c which moels the situation of heat bath embeing of S in the sense that the expectation values with respect to the corresponing solution f(; t) = f c of Liouville's equation are correct escriptions for S in thermal equilibrium! This shows the importance of the initial ensity in this statistical approach, i.e. the importance of the

15 initial preparation of the ensemble. If f 6= f c we o not know how to moel \thermal embeing". We may go the way of changing the Hamiltonian H! ^H, e.g. by aing aitional stochastic forces. Or we may construct a moel for the ensity f which, then, must no more full Liouville's equation for H but a \correcte" one. This last approach is realize herein (cf. Section 4). As a rst step it shoul be note that (3.9) is not equivalent to the canonical ensemble, i.e. if (3.9) is fullle for the expectation values with respect to a ensity f this oes not imply f = f c. We may use this freeom an e- ne that a ensity f which fulls (3.9) escribes an ensemble in interaction with a heat bath with temperature T = =k B (in local equilibrium). 3.3 Initial Density What is the \right" initial ensity f = f(; ; ) for (3.) if we are in the situation explaine in Problem 3 in the introuction? Equation (.5) leas us to the following moel for the \spatial part" of f :! 3NY q k? qk f (q; p) = (p) w k k k= where must still be ene an the w k : R! [; ] are suitable weight functions with R w(x)x =. If we assume normal istribution for the error of the spatial measurements we will e.g. use w k (x) = p exp(?x 2 ): If f is the initial ensity of an ensemble in interaction with a heat bath of temperature T = =k B, the usual moel for is normal istribution with a variance controlle by temperature (p) = exp? 2 pt M? p! with so that f is normalize, i.e. f (q; p) = exp? 2 pt M? p 2! 3NY w k l=! q k? qk k (3.)

16 with = p! 3N 2 3NY p p k mk : k= This construction guarantees that f fulls (3.9) an that hpi = an hq k i = q k: (3.) In particular, the form of the spatial part shows that f 6= f c, i.e. the initial preparation of our ensemble given by the spatial measurement (.5) oes not t in the context of thermal equilibrium (see above). 4 Smoothe MD an Spatial Averaging In \stanar" MD{approaches the statistical nature of our problem may be taken into account by computing a representatively large number of trajectories with ierent, f {istribute initial values an (.2) as equation of motion. Then, interesting expectation values are compute as mean values over all these trajectories. If this is one carefully it prouces an enormous computational eort. Is it possible to evaluate these expectation values from one trajectory only, for instance the solution of our statistical equations of motion (3.4)? This woul only be possible if we foun a closing relation hdv (q)i = D ~ V (hqi) for (3.4) an aequate initial values. In the following such a closing relation an initial values are constructe from a moel for the nonstationary probability ensity for thermally embee systems which ts to the initial conitions (3.). The reformulate equation of motion will again be Hamiltonian with a smoothe eective potential ~ V leaing to smoother trajectories. But let us start proving some useful properties of \separable" ensities. 4. Separable Densities In this subsection we assume that the consiere ensity f is separable, i.e. for all t it hols f(q; p; t) = Q(q; t) P (p; t) (4.) 3

17 with both, Q an P, being normalize an with lim P (p; t) = lim jp k j! In this situation the following theorem hols: Q(q; t) = 8k = ; : : : ; 3N: (4.2) jq k j! theorem. Let hi be the expectation value with respect to a separable ensity. Then the equations of motion (3.4) for the position an momenta expectation can be written in close form an as a new Hamiltonian system t hqi = D ~ p H(hqi; hpi) = M? hpi t hpi =?D ~ q H(hqi; hpi) =?D V ~ (hqi); with a correcte Hamiltonian (4.3) ~H(q; p) = 2 pt M? p + ~ V (q): (4.4) Thus, the closing relation for system (3.4) is euce: hdv (q)i = ~ DV (hqi). The new potential ~ V only epens on the ol one V an on the initial ensity f(; ; ). Proof. (4.) implies for the reuce ensity (3.6) an, in particular: F (q; t) = Q(q; t) p P (p; t) p = hpi(t): (4.5) R 3N In this situation we can euce an equation for F = Q alone. Therefore, integrate (3.) over p an use (4.2) to t F =? D q F M? p P (p; t) p R 3N =? D q F M? hpi: (4.6) 4

18 If hpi(t) an the initial reuce ensity F = F (; ) are known, the solution of (4.6) can be written as F (q; t) = t q? M? an we can use our general formula (3.4) to get an from this t M? hpi(s) sa ; hpi(s) s = hqi(t)? hqi() F (q; t) = F (q? hqi(t) + hqi()) : If we switch to the centere initial ensity we nally have ~F (q) := F (q + hqi()) (4.7) F (q; t) = ~ F (q? hqi(t)); (4.8) i.e. the initial probabilities F ~ (q) are transporte along the curves hqi(t) of the spatial expectation value (cf.(3.2)). With (4.8) the esire consequences for the equation (3.4) follow irectly: From (3.7) we get hdv (q)i(t) = = R 3N R 3N DV (q) F (q; t) q DV (q) ~ F (q? hqi(t)) q = DV (q + hqi(t)) F ~ (q ) q R 3N = D q V ~ (hqi(t)) with a new potential ~V (q) := R 3N V (q + q) ~ F (q ) q : (4.9) 5

19 This implies the statement of the theorem. Our theorem has aitional nice consequences. If we consier an arbitrary spatial observable A = A(q) one can show by the same calculations starting with (3.3) that t hai = D A(hqi) ~ M? hpi (4.) with a new function ~A(q) := R 3N A(q + q) ~ F (q ) q ; i.e. solving (4.3) makes the computation of all spatial expectation values possible. 4.2 Probability Density for Thermally Embee Systems We want to construct a ensity f for an ensemble which escribes (nonstationary) thermal embeing. This ensity shall allow us to n a closing relation for (3.4). Theorem states that we can euce the esire closing relation if f is separable. Therefore, consier the following ensity: with f (q; p; t) = F (q; t) exp =? 2 (p? hpi(t))t M? (p? hpi(t)) 3NY l= s 2 ml :! (4.) It is separable in the sense of (4.), nonstationary, fulls the initial conition (3.) with an has the property F (q; ) = F (q) = an hpi() = ; 3NY l= k w k hp pi? hpi hpi = M;! q k? qk k (4.2) 6

20 which is (3.9), our ening equation for nonstationary embeing in a heat bath of temperature. Thus, f moels the situation which we wante to escribe with the \correcte spatial Liouville equation" t F =? D q F M? hpi (4.3) t hpi =? R 3N DV (q) F (q; t) p as equations of motion (cf. the proof of Theorem ) an (4.2) as initial conition. potential x q t q t Fig. 4. Original (soli line) an smoothe potential (otte line) an the corresponing Hamiltonian ynamics. Alreay in this simple example numerical integration of the smoothe ynamics nees 3 times less steps than in the original case. Finally, we know from Theorem, that the separability of f, the initial conitions (4.2), an the new potential ~ V from (4.9) give us new an close Hamiltonian equations of motion for the expectation values of ensembles 7

21 moelling nonstationary thermal embeing: t hqi = M? hpi hqi() = (q k ) k=;:::;3n t hpi =?D ~ V (hqi) hpi() = : (4.4) 4.3 Average Potentials for MD Calculations Accoring to (4.4) we have to use (4.9) to compute the new potential ~ V with a centere ensity ~ F given by (4.7) an (4.2): ~V (q) = R 3N V (q + q ) 3NY l= k w k! q k k q q 3N: (4.5) This means that ~ V is constructe from V by weighte (spatial) averaging on scales l, e.g. with Gaussians ~V (q) = R 3N V (q + q ) 3NY l= p k q k k! 2 A q q 3N : (4.6) Therefore, the new system (4.4) has smoothe trajectories in comparison to the original system (.2) an its integration prouces less computational eort (cf. Figure 4). Because of this, things explaine in Sections 2.2 an 2.3 are vali for (4.4), too. For typical MD potentials V the explicit evaluation of the integrals in (4.5) is possible, because they are sums of \simple" potentials V (k) : V (q) = mx k= X (j :::j Nk )2B k V (k) q j ; : : : ; q jnk ; wherein N k is small for all types k = ; : : : ; m. Realization of this evaluation an ecient numerical integration of (4.4) for realistic molecular systems will be subject of further work. 8

22 5 Conclusion Typical MD simulations for (macro)molecular systems with nonstationary thermal embeing necessarily require the computation of a representatively large (statistical) ensemble of trajectories an expectation values as corresponing ensemble averages. In aition, the evaluation of each of these trajectories prouces large computational eort because har stepsize limitations are emane in orer to ensure stability of the time integration. We have presente an approach which leas to a reuction of computational eort in both cases: The construction of a moel for the nonstationary probability ensity for the consiere situation allows us to euce a closing relation for the equation of motion (3.4) which is the equation for the (q; p){observable. Thus, we are able to evaluate the expectation values of all spatial observables via (4.) by computing only one trajectory (hqi; hpi) as the solution of a new Hamiltonian system (4.4) with uniquely etermine initial values. The potential ~V of this Hamiltonian system is etermine as a weighte spatial average of the original potential V. Thus, in comparison to the original trajectories, our single trajectory (hqi; hpi) is smoother, i.e. it allows larger stepsizes an has also the other avantages of a smoothe MD (gain in eciency, reuction of error amplication, increase stability). References [] M.M. Chawla. On the orer an attainable intervals of perioicity of explicit Nystroem methos for y = f(t; y). SIAM J. Numer. Anal., 22:27{3, 985. [2] P. Deuhar an F. Bornemann. Numerische Mathematik II. De Gruyter, Berlin, New York, 994. [3] M. Fixman. Classical statistical mechanics of constraints: a theorem an applications to polymers. Proc. Nat. Aca. Sci., 7:35{353, 974. [4] E. Hairer, S.P. Nrsett, an G. Wanner. Solving Orinary Dierential Equations I, Nonsti Problems. Springer Verlag, Berlin, Heielberg, New York, Tokyo, 2n eition, 993.

23 [5] E. Hairer an D. Stoer. Reversible long{term integration with variable step sizes. Report, 995. [6] W. Huang an B. Leimkuhler. The aaptive Verlet metho. Report KITCS95-3-, University of Kansas, 995. [7] R. Kurt. Axiomatics of Classical Statistical Mechanics. Pergamon Press, Oxfor, New York, 98. [8] M.R. Pear an J.H. Weiner. Brownian ynamics stuy of a polymer chain of linke rigi boies. J. Chem. Phys., 7:22{224, 979. [9] S. Reich. Smoothe ynamics of highly oscillatory Hamiltonian systems. Report SC 94-28, Konra use entrum Berlin, 994. [] Ch. Schutte. A quasiresonant smoothing algorithm for the fast analysis of selective vibrational excitation. Impact of Computing in Science an Engineering, 5:76{2,