Stochastic Theory of the Classical Molecular Dynamics Method

Transcription

1 ISSN , Mathematical Moels an Computer Simulations, 0, Vol. 5, No. 4, pp. 05. Pleiaes Publishing, Lt., 0. Original Russian Text G.E. Norman, V.V. Stegailov, 0, publishe in Matematicheskoe Moelirovanie, 0, Vol. 4, No. 6, pp. 44. Stochastic Theory of the Classical Molecular Dynamics Metho G. E. Norman an V. V. Stegailov Joint Institute for High Temperatures, Russian Acaemy of Sciences, Moscow, Russia Moscow Institute of Physics an Technology (State University), Moscow, Russia Receive June 4, 0 Abstract The work is evote to funamental aspects of the classical molecular ynamics metho, which was evelope half a century ago as a means of solving computational problems in statistical physics an has now become one of the most important numerical methos in the theory of conense state. At the same time, the molecular ynamics metho base on solving the equations of motion for a multiparticle system prove to be irectly relate to the basic concepts of classical statistical physics, in particular, to the problem of the occurrence of irreversibility. This paper analyzes the ynamic an stochastic properties of molecular ynamics systems connecte with the local instability of trajectories an the errors of the numerical integration. The probabilistic nature of classical statistics is iscusse. We propose a concept explaining the finite ynamic memory time an the emergence of irreversibility in real systems. Keywors: molecular ynamics, instability of trajectories, numerical integration errors, statistical laws, irreversibility DOI: 0.4/S INTRODUCTION In 957, a paper was publishe by B.J. Aler an T.E. Wainwright [], who use newly emerging computers to integrate classical equations of motion for a system of har spheres. This material was presente in greater etail in []. The results were obtaine both for the equation of state an for the correlation functions. The consiere systems varie from several tens to several hunres of particles. The works were immeiately notice by I.Z. Fisher [, 4]. The approach propose in [, ], was name the molecular ynamics metho (MDM), as originally molecules rather than atoms were unerstoo by har spheres. As early as in [5, 6], particle systems with more realistic Born Mayer, Debye, an Lennar Jones potentials were consiere. Further, systems of interacting atoms were mostly consiere. The interatomic interaction potentials became increasingly sophisticate for escribing more an more iverse physical objects. The term atomistic simulation emerge an gaine increasing popularity but it also fails to embrace all the MDM applications because it has to cover, among other objects inclue in consieration, electrolytes an nonegenerate electronion plasma. The combination molecular ynamics metho has appeare very stable an remains the only term for this metho, although the wor molecules now refers not only an not so much to a molecule as such but as a rule to atoms, ions, an even electrons. MDM is applie to sufficiently ense an nonieal systems when it is no longer possible to introuce the concept of the mean free path an theories base on the expansion in the small parameter lose their meaning. At the same time, it is esirable to continue MD calculations until the parameters of the stuie system enable comparing with the results obtaine for rarefie systems. This makes it possible to cover the entire range of practical importance. MDM has now become one of the most important methos in the theory of conense matter. The interaction of particles with each other is escribe by ifferent pair an multiparticle potentials. Thermoynamic an transport properties were examine (equations of state, iffusion, viscosity, thermal conuctivity, etc.), structure, time correlation functions, an some relaxation processes for a wie variety of systems, incluing simple liqui metals, semiconuctors, ielectrics, covalent systems, collisional an usty plasma collois, glasses, polymers, biopolymers, an liqui crystals. Different aggregate states have been escribe. Along with homogeneous systems, phase equilibria an the ecay of metastable states were moele, as well as clusters, nanocrystals, processes on the surface, shock waves an solitons, the 05

2 06 NORMAN, STEGAILOV ynamics of raiation amage in crystals, fracture of materials, cracking, an various biomolecular systems. The progress in computer facilities enable researchers to consier an increasing number of particles. Calculations have alreay been performe for 00 billions [7] an a trillion [8] particles. The realism of the moels is ue to the choice of the multiparticle interaction potentials erive from quantum-mechanical calculations. A huge number of publications on this topic over the years has generate many surveys an monographs. We woul like to mention, for instance, [9 4]. Attempts have been mae to construct a quantum MDM [5] for the escription of chemical reactions, equilibrium an nonequilibrium systems with egenerate electrons, an some other problems. Despite the progress in this irection, it has not yet been possible to create quantum MDM as versatile as the classical MDM. In this review, we restrict ourselves to the classical MDM, focusing on the consistent exposition of the theoretical founations of the metho.the uneniable practical efficiency of the classical MDM an the simplicity of its unerlying assumptions seems to leave no room for oubt as to its theoretical valiity. In this connection, MDM is generally regare as a purely computational metho, an many researchers employing it usually o not give a secon thought to what is actually calculate by MDM, being satisfie with the agreement with the experimental ata provie by MDM an its preictive power. However, some theoretical physicists have long realize that MDM, espite its numerical character an utilitarian focus, concerns the founations of classical statistical physics an, in turn, requires a theory for its logical justification. MDM raises very serious questions. In the first place, the total energy in the MD calculation fluctuates. The law of energy conservation only hols on the average, in contrast to the equations of classical mechanics. This fact was note at the initial stage of the MDM evelopment but it ha not been properly recognize an was treate as a minor error. In the secon place, Henri Poincaré showe long ago that classical ynamical systems of many particles were systems with strong local instability [6]. These concepts were introuce in physics by N.S. Krylov [7] in book [8]. In connection with MDM, this circumstance was pointe out by E.E. Shnol [9]. These issues were consiere in [, 0 ]. The authors of [] referre to Ya. Sinai (the book [8] ha not yet been translate at the time), who analyze the results to clarify the exponential ivergence of the trajectories of particles in the Lennar Jones system. The local instability means that ifferent particle trajectories calculate for the same set of initial conitions, but, for example, with a ifferent step of numerical integration, by no means correspon to a single Newtonian ynamic trajectory an exponentially iverge from it, as well as from each other. An the metho evise for the solution of the Cauchy problem but leaing to a bunch of exponentially iverging trajectories is unacceptable for solving ifferential equations from the viewpoint of the theory of numerical methos. Despite the huge ifference in scale, MDM problems are similar to the restriction on the limit of the preictability of weather forecasting [4 6]. The limit of the preictability of particle trajectories was calculate in [0] for the Lennar Jones system. Starting from [], several authors pai attention to the increase number of numerical errors ue to the instability of the equations of motion [0, 9, 7 4]. A constraint was propose [8, 9] requiring MD trajectory to be close to the exact solution of Newton equations over the entire length use for statistical averaging. Since it was impossible to satisfy this requirement, E. Shnol [9] pointe out that MDM was not entirely vali, an R.F. Fox [8] even oubte MDM s reasonableness. Work [8] stimulate the precision calculations [9]. However, instea of one long MD trajectory, a set of short sections of the same total sum length was use for averaging. Each short section was calculate with sufficient accuracy, but the initial conitions for them were selecte from the canonical ensemble using the Monte Carlo metho. Therefore, work [9] was unrelate to the problem for which it was intene, which was note as early as in [4]. In the thir place, how can irreversibility arise in the metho base on the solution of the reversible Newton equations? It certainly emerges, as MDM perfectly escribes irreversible processes. As early as in 967, members of I. Prigogine s group calculate the evolution of the H-function for a system of 00 har iscs [4] an its recovery for time-reversible trajectories. It was foun that the recovery only took place for short trajectories in the range of relaxation times; the question of exponential instability was not iscusse. In the attempt to figure out the cause of the irreversibility, Prigogine returne to this work in 989 [44], rawing also to later numerical experiments [45, 46] on the emergence of correlations in a system of har isks as a result of collisions. Prigogine refutes persistent unsuccessful attempts to prove the

3 STOCHASTIC THEORY OF THE CLASSICAL MOLECULAR DYNAMICS METHOD 07 secon law of thermoynamics, the kinetic Boltzmann equation, an other irreversible laws relying solely on reversible Newton equations. This iscussion began over a hunre years ago starting from the arguments against L. Boltzmann voice by E. Zermelo an J. Loschmit (see, for instance, [6, 47, 48]. R. Feynman agree with Boltzmann s arguments in his iscussion with Zermelo an Loschmit an formulate them in the following way: Things are irreversible only in the sense that going one way is likely, but going the other way, although it is possible but accoring to the laws of physics, woul not happen in a million years [49, p.9]. Boltzmann s an Feynman s viewpoint is share by N. Bogolyubov, B.V. Chirikov, Sinai [48, 50, 5], J.L. Lebowitz [5], an W.G. Hoover [5]. Reasons are offere recalling the colossal number of atoms an molecules making up macroscopic boies, the practical impossibility of the complete mechanical escription of such boies, the nonlinearity of ynamical systems [5] an others [54, 55]. A.M. Yevseev [56] trie to work out a purely mechanical theory of thermoynamics by means of MDM. Krylov [8] an K.R. Popper [47] i not share the views of Boltzmann an his supporters. Boltzmann s concept, by virtue of Poincare s recurrence theorem, amits the existence of the worl with a ecreasing entropy, an, therefore, shoul be rejecte [47]. Objections to Boltzmann were also raise by J. von Neumann [57], L. D. Lanau [58], Prigogine [44, 59, 60], an B. B. Kaomtsev [6] assuming that only quantum mechanics coul be use to justify irreversibility. This viewpoint was most clearly articulate by Lanau [58] who pointe out that nonequivalence of the both time irections in quantum mechanics is manifeste in connection with the central process of interaction of the the quantum mechanical object with the system for quantum mechanics, obeying with a significant egree of accuracy the classical mechanics (see also [6] 47). Note that Lanau s view echoes a brief remark mae by Neumann as early as 99 [6, footnote on p. 7]. However, in the same work, Neumann remarke that it was still unclear whether the irreversibility of the measurement process ha anything to o with the irreversibility of what was actually taking place. It was note in [64 67] that in the equations of motion of any classical molecular system there shoul necessarily exist some aitional terms of the quantum nature. These terms are small but finite, have a stochastic nature, an trigger Lyapunov s instability. The roun-table iscussion Microscopic origins of macroscopic irreversibility was hel at the 0 th International Conference on Statistical Physics (Paris, 998). Two opposing points of view in front of two thousan scientists were efene by Lebowitz an Prigogine. A ry an brief escription of their emotional speeches is given in [5, 59]. In his verbal report, Lebowitz very extensively expoune MD s results [68] as an important argument in favor of his point of view. In [5] only a few lines were evote to this work, but Lebowitz still fails to see that the conclusions of [68] are erroneous, which will be pointe out in paragraph.5. Actually, MDM ran counter to some establishe ieas of classical statistical physics, which are set out, for example, in [49, 58]. The iscussion of these funamental issues was launche in [, 0, 4, 69]. They were evelope in [, ] an in this paper. We consier the place of MDM in the system of basic concepts of statistical physics an physical kinetics an where it complements these concepts in terms of the choice of number of particles, the probabilistic character of the results of classical statistics, etc. The original intent of the MDM is set out in Section, where the immeiately manifeste ifficulties are also inicate. In Section we analyze the set of particle trajectories that are actually calculate in MDM, i.e., the object whose properties are actually investigate by this metho. In paragraphs..4 the exponential ivergence of particle trajectories is treate, concepts of K-entropy an the ynamic memory time are introuce as applie to MDM problems, the connection of these notions with the total energy fluctuations is explaine, an the funamental restriction of the memory time is shown ue to the finiteness of the number length in a computer. Paragraphs.5 an.6 are evote to the conclusions from this analysis, i.e., the ynamic an stochastic properties of the set of particle trajectories calculate by MDM an the non-hamiltonian character of these trajectories. MDM problems are compare to the problem of ynamic chaos [48, 50, 5, 70, 7, 7]. In Section, we attempt to reformulate some of the postulates of classical statistical physics relying on the MDM results; we also briefly point out some physical analogs of numerical integration errors. In the conclusion, in particular, practical requirements for MD simulation are liste following from the performe analysis of the MDM funamentals.

4 08 NORMAN, STEGAILOV. ORIGINAL IDEA OF MDM.. Initial Equations The ynamics of N interacting atoms is escribe by the system of equations ri mi = F () i( r,, rn) t or vi ri mi = Fi( r,, rn), = vi, () t t where m i, r i, an v i are the mass, coorinate, an velocity of the ith particle ( i =,.., N) an Fi is the force acting on it, which is efine as U( r,, rn ) Fi( r,, rn) =. () ri The function U etermines the physical properties of the system uner stuy. The MD an Monte Carlo methos stimulate a sharp increase in interest in the evelopment of investigations in the U type. At the first stage, the simplest systems were examine such as, for instance, soli argon, using the existing pair potentials at that time, i.e., Lennar Jones an Buckingham. Their parameters were selecte so that they were consistent with the experimental ata about the properties of the consiere substances. The transition to the exact pair potentials erive from the beam experiments reveale the nee to take account of multiparticle interatomic interactions. In the general case N U( r,, r ) = u ( r, r ) + u ( r, r, r ) +. N i j i j k i< j i< j< k For moeling inert gases, it prove sufficient only to consier the contribution of triple interactions. The escription of the properties of semiconuctors an metals becomes much more accurate when multiparticle (up to several ozens) components are taken into account (see, for instance, [7 77]). Initially function U was foun empirically from the conition requiring that the set of the moel characteristics was consistent with the experimental ata [78 8]. At the same time, some properties of substances can, generally speaking, be escribe with the same accuracy not by one but by a whole family of potentials [8]. At present the stuy of interatomic interaction has passe to the area of ab initio approaches. The ynamics of a polyatomic system in the nonrelativistic approximation is escribe by the Schröinger equation with the Hamiltonian function of the form Pi pα ZZe i j Ze i H = e + + +, (5) el el el Mi m ri rj r i ij α rβ ri r α αβ iα α containing terms corresponing to the kinetic energy of the nuclei an electrons an the Coulomb energy of their interaction. The fact that electrons are much lighter than nuclei allows us to separate the movement of ions an electrons an consier the stationary problem for an electron subsystem in the aiabatic (Born Oppenheimer) approximation. In this case, the wave function can be represente as el el el ψ ({ ri},{ rα }) = Ψ({ ri}) Φ({ ri},{ r α }), where Ψ({ r i}) escribes the motion of the nuclei an Φ({ ri},{ rα }) epens parametrically on the coorinates of the nuclei { r i } an escribes the electron subsystem. Here the problem is split into two equations ˆ el el HelΦ ({ ri},{ rα })) = U({ ri}) Φ({ ri},{ rα }) an Hˆ iψ ({ ri}) = EΨ({ ri}), where pα ZZe i j Ze i Hel = e + +, (6) el el el m ri rj r ij α rβ ri r α αβ iα α i Hi = P + U({ ri}). (7) M i i The solution of the Schröinger equation for the electron subsystem at fixe positions of the nuclei etermines the eigenvalues U({ r i }), which are parametrically epenent on the nuclear coorinates. Thus, function U is the potential of the interatomic interaction in the groun state of the electron subsystem. However, the solutions of quantum mechanical equations for an electron subsystem are only possible for N (4)

5 STOCHASTIC THEORY OF THE CLASSICAL MOLECULAR DYNAMICS METHOD 09 systems with a small number of atoms (up to several hunre). In such a way, the form of U can be obtaine. By the use of multiparticle potentials of U of this type, investigations into the realistic moels of metals with the number of atoms of up to hunres of millions within the classical MD frame became possible (see, for instance, [84]). The transition to a fully classical escription of polyatomic system correspons to the assumption of the smallness of the e Broglie wavelength of atoms compare to the characteristic interparticle istance in the system that takes place, provie the temperatures are not too low for most of the chemical elements except the lightest ones... Numerical Schemes The most popular schemes are those in which the approximation error of the secon erivative ri t is subject to the first-orer ifference scheme ri( t +Δt) ri() t + ri( t Δ t) Fi() t (8) =, Δt mi where Δt is the numerical integration step. The simplest of such schemes is the implicit Euler scheme of the first orer of accuracy with respect to the coorinates an velocities Fi() t vi( t +Δ t) = vi() t + Δ t, ri( t +Δ t) = ri() t + vi( t +Δt) Δt. (9) mi The velocity Verlet scheme (of the secon orer of accuracy) is wiely use: i() t i( t Δt F v + ) = vi() t + Δt, ri( t +Δ t) = ri() t + vi( t + Δt) Δt, mi (0) i( t t) i( t t) i( t Δt F +Δ v +Δ = v + ) + Δt. mi The optimality of schemes like (8) (0) with respect to the accuracy/performance ratio was explaine in [85, 86], where the efficiency of schemes of ifferent orers of accuracy was investigate as applie to MD problems. The iea can be illustrate by the following arguments. We consier some segment of an MD trajectory. Let its calculation using the scheme (8) require M steps. Let the same accuracy (with respect to the fluctuation of the total energy, see Section.) be achievable by using a scheme of a higher orer over M steps, M < M. However, at each step the calculation of the forces acting on atoms (the most resource-consuming part of the program) is repeate m times. It turne out that for stanar, close to the maximum, values of numerical integration steps M < mm (for scheme (8) m = ) for all the schemes stuie in [85, 86]. That is why the schemes of type (8) prove to be the most cost-effective for the MD problems. The situation only changes when the numerical integration step ecreases several times or more, then mm < M. However, such egrees of accuracy are not use in the stanar MD calculation. When choosing the integration step Δt such physical characteristics of the system must be taken into consieration as the maximum frequency of oscillations arising in it. Another physical restraint is ue to the fact that the interaction energy must change little over the length, on which the particles are isplace uring time Δt. This is not an average characteristic as it concerns every particle. In this connection, a variable integration step can be use... MD Simulation... Molecular ynamics experiment. MDM can be classifie as a computer experiment; i.e., it can refer to numerical methos that, the same as real experiments, can be ivie into an experimental setup an iagnostic tools. The solution base on numerical schemes gives us comprehensive information on the trajectories of the particles, i.e., completely efines the system uner stuy; it is an analog of the experimental setup, which comprise a research object. In our case, such an object (a moel of a substance) is a set of particle trajectories that is calculate using MDM. Sometimes the moel of the substance is unerstoo as the mere set of interparticle interaction potentials. The calculate particle trajectories contain comprehensive information about the system an all the ata about its properties. It is only important to learn how to extract this information, i.e., to evelop iagnostic tools for every property. For this purpose, MDM uses the most general, strict relations of statistical physics an physical kinetics. Specific iagnostics epens on the experimental setup.

6 0 NORMAN, STEGAILOV The MD metho is employe to investigate a variety of systems an for their aequate escription, various interparticle potentials are use. Pure substances are simulate, as well as their mixtures, solutions, alloys, etc. All aggregate states are explore, both stable an metastable. Three main irections of moeling can be single out, i.e., equilibrium states, relaxation, an controlle MD.... Simulation of equilibrium states. In these calculations, the initial state is more or less arbitrary. Then, the system is brought into equilibrium. Since this transition segment of the MD trajectory is auxiliary an exclue from further consieration, the initial conitions are chosen to reuce this section as far as possible. With the same purpose in bringing the system into equilibrium, various workarouns are employe not relate to the real processes in the system uner stuy. The criteria for bringing simple systems into equilibrium can be quite trivial, i.e., achieving the constant value of the average particle temperature T an the stationarity of its fluctuations. The value T is relate to the average kinetic energy i= N in the absence of the motion of the mass center m However, we must ensure that the energy = i v i = 0. i (velocity) istribution of the particles shoul become Maxwellian, which woul enable us to erive its temperature. The agreement between the results of these two methos for obtaining the temperature inicates that the equilibrium has been establishe in the system with respect to this parameter. In the case of a crystal or liqui, we can aitionally calculate the ispersion of fluctuations over a certain section of the equilibrium trajectory. If we exten the MD calculation to the section of the same length, we can repeat the computation of the ispersion. The coincience of the two results will show that equilibrium has been also attaine with respect to fluctuations. We can ensure the establishment of equilibrium by checking the stationarity of all the calculate parameters. In the case of biomolecules, the criteria can be quite sophisticate [87]. After the system has been brought into equilibrium the generation of the MD trajectory is continue an this stage is consiere equilibrium. All information obtaine in the calculation is foun from this stage an only applies to the equilibrium state. The pressure is erive from the virial relation P = NT r U ij, () V r i i< j ij which follows from the general relations P = ( F V) T an F = T ln Z, where F is the free energy an Z is the statistical sum. There are, however, unresolve issues in the calculation of the pressure [88, 89]. The heat capacity is obtaine as the ifference between the mean square U an the square of the average of U, etc. The transport coefficients are calculate both by the Green Kubo formulas an from the Einstein Herzfel relations. For example, the iffusion coefficient D can be foun at the velocity autocorrelator D = (0) ( ), () v v t t 0 or irectly observing the particle isplacements r r = 6 Dt + C. (4) The agreement between the results obtaine by those two methos enables us to verify the operation of the program an the establishment of equilibrium in the system. Some subtleties arising in this case are illustrate by the example of the iffusion coefficient in [90 95]. More information about the formulas for other transport coefficients can be foun, for instance, in [96]. MDM allows us to stuy fluctuations. In the case of electron-ion nonegenerate plasma, fluctuations of ifferent nature are possible. Moeling within a single calculation, as in the real plasma, involves all kins of fluctuations. MDM provies a unifie approach to the fluctuations in the ensity of free charges, pair electron-ion fluctuations, electron states localize in the long-wavelength ensity fluctuations, an other possible fluctuations. However, investigation into the fluctuations of each type requires special iagnostics. A metho for the iagnostics of ensity fluctuations is presente in [97]. The pair electron-ion fluctuations (pair fluctuations or simply pairs) are consiere in [98 00]. It is iscusse there how these pairs pass into states that can be associate with highly excite atoms as they become less collisional an N i mi T = v ( N ) ()

7 STOCHASTIC THEORY OF THE CLASSICAL MOLECULAR DYNAMICS METHOD their bining energy an life time increase. The unerstaning of the spectrum part ajacent to the ionization limit is refine. Fluctuations of plasma pressure are stuie in [0]. The calculation of all quantities by MDM assumes that averaging is performe. Thus, the accuracy of averaging can simultaneously be etermine. The analysis of the iagnostic accuracy in MDM also points to the analogy with the real experiment. Stanar requirements to equilibrium MD calculations are briefly summarize in [0].... The proof of the ergoic hypothesis in MDM. In the strict expressions () (4) base on the first principles employe in MDM, averaging means averaging with respect to the Gibbs istribution. It ha been foun, however, rather long ago, that such averaging can be replace by averaging over time along the MD trajectory. In view of the funamental importance of such a replacement, its reliability was verifie in various ways. Thermoynamic quantities, such as pressure () an heat capacity can be calculate by the Monte Carlo metho, which only uses averaging with respect to the Gibbs istribution. The same values for the same multiparticle system an uner the same conitions can be foun by the MD metho using time averaging along the MD trajectory. The results compute using the two methos coincie with each other within the accuracy of the calculations, see, for instance, [0]. Averaging with respect to the Gibbs istribution can also be performe within MDM, see, for instance, [9, 04]. The results coincie with the time averaging. The space averaging (4) was compare with the time averaging () [05]. The choice of a particular averaging metho when MDM is use is etermine by the consieration of the calculations convenience in each specific case. Thus, while the ergoic hypothesis remains the subject of stuy in a rigorous mathematical theory [06, 07], the replacement of space averaging by time averaging along the MD trajectory has long become [05] a common iagnostic tool in MDM...4. Simulation of relaxation. In early works [ 6], MDM ha been esigne as an instrument for stuying equilibrium states. At the same time, by its very nature, MDM is a powerful tool for exploring the ynamics of relaxation an transition processes in ense meia, incluing highly nonequilibrium states of such meia that are inaccessible to theoretical approaches base on the smallness of the eviation from equilibrium. In the stuy of relaxation, all the information shoul be rawn from the initial, nonequilibrium part of the MD trajectory, i.e., just from the part that is truncate in the MD simulation of equilibrium states. In contrast, on approaching the MD equilibrium the calculation stops. Thus, the simulation of the real relaxation process starts with the physically aequate choice of the initial state, or, more precisely, of an ensemble of initial states, which iffer from each other microscopically but are equivalent macroscopically. The iagnostics tracks the time variation in all the system parameters, incluing those that can only be foun with the help of time averaging along the stationary MD trajectory at constant parameters. Stanar requirements for the MD simulation of relaxation are presente in [0]...5. Controlle molecular ynamic experiment. In this case, the initial state is brought into equilibrium, just as in the equilibrium MD. Further, this state is expose to a controlle physical action an the ynamics of the system s response to the applie action is stuie comprising variable action an incluing feeback [08 ]. By way of example, we recall the unfoling an foling of protein molecules, the ynamics of stretching an compression of other systems, an shock waves...6. Nonconservation of energy. The consistency of MDM s results with the experimental ata ha been maintaine for years, although MDM s capabilities were expaning with the evelopment of computer facilities an the progress in experimental techniques le to the enhancement of the atabase use for comparison. All this allowe the professionals working in the fiel of MDM applications not to pay attention to the funamental issues. At the same time, concerns were raise at the start of MDM s evelopment. It was foun that the total N energy of the system E = E kin + U fluctuate uring MD s calculation; Ekin = m is the kinetic = i v i i energy of the particles system. In some numerical schemes, the researchers even observe a rift of the total energy E. The conclusion was mae empirically without analyzing the causes; i.e., schemes were selecte base on (8), which i not involve any rift in E an in which E fluctuations with respect to the mean value are of a stationary character. Only in [] a requirement was foun for the ifference schemes for MDM, i.e., approximation of the secon erivative was to be even with respect to time.

8 NORMAN, STEGAILOV Thus, the massive popularity gaine in MDM by the schemes (8) (0) an their variations can be attribute not only to their best performance in solving MDM problems but also to their stationarity with respect to E. The nonconservation of E, however, points to the fact that the particle trajectories calculate by MDM o not satisfy Newton s equations, because, for these equations, the total energy shoul ientically be conserve along the trajectories obtaine from the solution of the Cauchy problem. Thus, it appeare that the real MDM was not consistent with its original intent. However, this fact was largely overlooke. The fluctuations of E were consiere a minor error of the metho an their value was often not mentione in publications. Fig.. Schematic illustration of the initial portion of the trajectory calculate in MDM from the initial state of system Γ 0. The soli curve shows the exact solution of the corresponing Cauchy problem. The otte curves show the Newtonian trajectories of the MD system, on which there falls the solution of the finite ifference approximation. Г 0. CONCEPTS OF SUBSTANCE DEVELOPED IN MDM.. Exponential Instability of Particle Trajectories MDM consists in solving a system of orinary ifferential equaltions () or () uner the given 0 0 initial conitions ( r, v ) (point Γ0 in 6N-imensional phase space), where r =( r,, r N ) an v =( v,, v N ). The forces acting on the particles are sufficiently smooth functions of the coorinates, while the theorem of the existence an uniqueness of the solutions of the Cauchy problem hol true. This implies that there exists an exact solution (trajectory) ( r (), t v () t ( esignates ynamical) shown in Fig. by curve 0 starting from point Γ 0. The solution of system () or () can only be foun numerically (starting from N ). The numerical scheme an the integration step uniquely etermine the function {( r kδt), v ( kδ t), k = 0,, }. The finiteifference solution is shown in Fig. by the polygonal line. Since the solution is approximate, point following point Γ 0 oes not lie on the original trajectory 0. Another trajectory can be rawn across point, i.e., the exact solution of the system, enote by the igit. Point falls on the new Newtonian trajectory, etc. The lines 0,, etc., cannot cross because of the uniqueness an unambiguity of the Cauchy problem solution. It is obvious from Fig. that the total energy must fluctuate along the MD trajectory because every subsequent point on this trajectory 0,,, etc. belongs to a new Newtonian trajectory, which correspons to a ifferent value of the total energy. In Section.5., we will stuy why an for which schemes the mean value of the total energy is still preserve. It becomes clear from Fig. how the coarse-graining proceure of the Newtonian trajectory takes place in MDM. An it is coarse-graining that is require for the emergence of irreversibility, which is one of the central questions; we will repeately return to ifferent aspects of this. All points on the MD trajectory are escribe by numbers, whose number of igital places is finite an etermine by the computer use. All the remaining points on trajectories 0 4 are irrational because they belong to the exact solutions of the system of equations (). Therefore, the points 0 4 an all subsequent points on the MD trajectory belong to ifferent solutions of the system of equations (). By itself, this fact woul be commonplace for the numerical metho, if the solution remaine in a small ε-neighborhoo of trajectory 0. However, ue to the Lyapunov instability, the Newtonian trajectories calculate from the initial conitions close to point Γ 0, but not coinciing with it exponentially, iverge with time. The same character of ivergence with the same value of the parameter can be expecte on average for the MD trajectories along the Newtonian trajectories.

9 STOCHASTIC THEORY OF THE CLASSICAL MOLECULAR DYNAMICS METHOD Δv 5 4 t' m Δr t' m 0 t Δt'/Δt Fig.. Normalize average ivergences of coorinates Δr (t) an velocities Δv (t) on two trajectories calculate from ientical initial conitions with the steps Δt = 0.00 an Δt' = (N = 64, ρ = 0.5, T = 0.44, Lennar Jones potential). Fig.. Depenence t m on Δt' Δt for ifferent values of Δt: () Δt = 0.0, () 0.005, () 0.00, (the leftmost point on the line correspons to Δt' = ); (N =64, ρ = 0.5, T = 0.44, Lennar Jones potential). While there is only a general statement about the ivergence of the Newtonian trajectories, the ivergence of the MD trajectories can be verifie irectly. We consier two trajectories calculate from the same initial conitions but with ifferent numerical integration steps Δ t an Δ t'. Let ( ri(), t vi()) t esignate the st an ( r' ' i( t), vi( t)) esignate the n trajectory. The trajectory-average ifferences of the coorinates (velocities) of the first an secon trajectories at coinciing instants of time t = ( kδ t) = ( k' Δt' ) N ' i i N r r i= N Δ r () t = ( () t ()), t Δ v () t = ( ' i() t i()) t (6) N v v i= are shown in Fig.. They exponentially increase with time Δ r ( t) = Aexp( Kt), Δ v ( t) = Bexp( Kt), tl < t < tm, (7) where t l is some transition time (of the orer of the inverse frequency of interparticle collisions), A, B, an tl epen on the numerical scheme, Δ t an Δt'. K are the values of K-entropy (Krylov Kolmogorov entropy, maximum Lyapunov inex average over the phase space). In this an the subsequent figures, the time is expresse in terms of ( mσ ε), where σ an ε are the parameters of the Lennar Jones potential NS m is the mass of the particles. ( mσ ε ) =.6 ps for argon. After the perio of time 6kT B t m ln( ) (8) K m B Δv () t becomes constant an Δr () t passes to the iffusion moe Δ v () t = v = vth, m m Δr () t Δr ( t ) ~ D( t t ), where v th = kb T m is the thermal velocity an D is the iffusion coefficient. Large-scale ivergence is no longer possible for Δv () t. (5) (9) (0)

10 4 NORMAN, STEGAILOV K The time t m (8) is the memory time, i.e., the 6 time, uring which the MD system remembers the fact that the initial configurations on the both trajectories coincie at the time instant t = 0. We 5 introuce the ynamic memory time t m that characterizes the perio of time, over which the correlation is lost between the solution of the finite-ifference approximation { r ( kδt), v ( kδt)} an the 4 n n exact solution of the system of ifferential equations () or () { r ( t), v ( t)} for the same initial N configuration. In orer to fin the value t m, calculations of t m are performe for one an the same Fig. 4. Depenence of the K-entropy on the number of value of Δ t an ifferent values of Δt': Δ t, Δ t 5, particles N (ρ = 0.5, T = 0.44 Lennar Jones potential). Δ t 0, etc. (Fig. ). The limiting value of tm at ( Δt Δt' ) 0 is the time t for this scheme an the K m selecte numerical integration step Δt. In the course of the numerical integration after time t m the system of particles completely forgets its 0 initial conitions an the calculate MD trajectory loses any correlation with the original Newtonian trajectory. Similar calculations were car rie out in [] for the embee atom metho 5 potential. 7 6 Calculations of the type shown in Figs. an 6 allow us to etermine the values of ˆK an t m for 5 various states of the system. The values of K weakly epen on N [4] starting from N ~ 0 4 (Fig. 4), because the etermining factor for quantity K is the strong close collisions. The same hap- pens also for t. As an example of the numerical m values, Fig. 5 shows the epenences of K on the ensity for a liqui an crystal calculate for the Lennar Jones system at N = 4000 in [5] for the region of equilibrium an metastable parameters for the stability bounaries. In the case of a ρ, m/σ crystal, the K-entropy changes monotonically an increases as it approaches the spinoal. For a Fig. 5. Depenence of the K-entropy on the ensity ρ at ifferent temperatures: crystal () an liqui () at T = 0.5; liqui, isotherm K( ρ) passes through the maximum at rather high temperatures. crystal () an liqui (4) at T =.; crystal (5) an liqui (6) at T =.0. The quantity Kt m increases logarithmically with the ecrease in the numerical integration n step. Such a result can be obtaine from (7) (9), assuming that A ( Δt), where n is the orer of the scheme s accuracy. Inee, at the time instant t = t m, 6kBT m = v = Δ v ( tm) = Aexp( Ktm), () an by taking the logarithm of () we obtain Ktm = n ln( Δ t) + const. () From the fact that the numerical integration is approximate, it also follows that energy E is only constant on average. The value of E from step to step fluctuates aroun the mean value. Therefore, the trajectory calculate in the MD metho is not on the surface E = const, as it shoul be for the solution of Newton s equation, but is locate in some layer of thickness ΔE >0near the surface E = const [, 4]. The

11 STOCHASTIC THEORY OF THE CLASSICAL MOLECULAR DYNAMICS METHOD 5 value ΔE is etermine by the accuracy an scheme of the numerical integration [, 4, 69, 6, 85, n 86], while ΔE Δt. It follows from () that Ktm ( E ) = ln Δ + const. () The calculation results for the Lennar Jones system [, ] an for the collisional plasma [7] illustrating the fulfillment of relations () an () are presente in Figs. 6 an 7. The goo agreement is obvious. Expressions () an () can be rewritten as Kt ( m tm) = nln( Δt Δ t) = ln ΔE ΔE (4), where tm an t m are the memory times for two values Δt an Δt. Expression (4) connects the K- entropy an ynamic memory time with the noise level in the ynamic system. This expression is consistent with the concepts of [7 0]. So far, we have consiere a single-component system. In [] plasma is stuie consisting of electrons an singly charge ions. For time less than t m for electrons, trajectories of both electrons an ions ha the same K-entropy. The value of K roppe sharply for ions at large times. Preliminary calculations for macromolecules an their systems show an exponential ivergence also for angular variables. In this case, there arise other constraints on the limits of such ivergence... Decay of Correlations In [48] the value K for monatomic systems is connecte with the rate of change in the entropy S as a result of the ynamic mixing of trajectories, an to the quantity K the meaning of the correlation ecay time τ is assigne (see also [, 5]). Let us compare these statements with MDM. Let the system preliminarily be brought into equilibrium. Further, we consier some small singly connecte element of the phase volume ΔΓ 0. We compare its evolution ΔΓ0( t) uring the motion of its points accoring to Newton s equations an along the MD trajectories. In the first case, by virtue of the Liouville theorem ΔΓ 0() t = ΔΓ0. (5) Trajectories, for which the initial conitions are locate close points insie ΔΓ 0, exponentially iverge with time. Since the value of the volume is preserve, its structure becomes increasingly rugge an stretche. The envelope of this structure encloses the increasing volume ΔΓ(). t A goo schematic rawing illustrating this process is presente in.5 [48]. The character of expansion of the phase volume envelope is also preserve in the case of motion Kt m Kt m ΔE/E 0 0 Fig. 6. Depenence of the quantity Kt m on the relative fluctuation of the total energy of the system ΔE / /E an integration step Δt (implicit Euler scheme (9)): crosses Lennar Jones system N = 64, ρ = 0.5, T = 0.44; crosses collisional plasma [] (plasma only correspons to the energy axis) Δt 0 ΔE/E Fig. 7. Depenence of the imensionless prouct Kt m on the relative fluctuation of the total energy of the system: circles an crosses Lennar Jones system an collisional plasma, respectively (implicit Euler scheme (9)); iamons Lennar Jones system (Runge Kutta scheme of the fourth orer of accuracy). The ashe line correspons to ().

12 6 NORMAN, STEGAILOV along the MD trajectories. However, the phase trajectories that are obtaine in the MD calculations are coarse-graine because of the numerical errors. Each point of such a trajectory passes through the new Newtonian trajectory. Therefore, ientity (5) is violate, the phase volume swept by MD trajectories grows, an the entropy increases. The process takes place even in the case where only one MD trajectory is consiere, i.e., when ΔΓ 0 = 0. The time t m restricts the perio of exponential ivergence of the trajectories. The quantity ΔΓ() t reaches the maximum value of ΔΓ max an the entropy S = kb ln ΔΓ( t) rises to its maximum. The reference point for this process can be chosen arbitrarily on the MD trajectory. For any segment of the trajectory with length tm the proceure of filling the volume ΔΓmax is repeate again an again. In other wors, the time t m is the time of filling the phase volume, in which a phase point is wanering, representing an MD cell at the given temperature T. An ΔΓ max is the size of the phase space region where the system remains in equilibrium most of the time. We can say that this region is an attractor for all MD trajectories. This region makes up a very small fraction of the phase volume. The MDM carries out significant sampling of that part of the phase space which etermines the properties of the system, just as the Monte Carlo metho performs the essential sampling in the coorinate space. By implication of the calculation, tm is also the time of the correlations ecay τ (in terms of [48]), i.e., τ=t m. Contrary to the assumption mae in [48] that K τ=, the values K an t m can iffer rastically. In aition to quantitative there is also some qualitative ifference between K an t m. The time K is a physical characteristic (such as pressure, iffusion coefficient, etc.) of the stuie multi-particle system an oes not epen on the accuracy or the scheme of the numerical integration. The values of t m are etermine by both the state an properties of the system an also by the numerical integration s accuracy, which in this case serves as the coarse-graining proceure. However, the latter epenence is very weak, logarithmic, as is evient from Figs. 6 an 7. Since the noise of the numerical integration can be compare with the real noise in physical systems (see below), the time also appears to be a physical characteristic although less is so far known about its value than for K. In [48], the roughening parameter ε was introuce an it was propose at the en of the erivation to let ε ten to zero. Moreover, it was assume that the result of the passage to the limit for the correlation ecay time i not epen on ε an remaine finite for ε 0. Let us recall that, accoring to [48], Kt ( m tm) = 0 for the correlation ecay time at small ε an is inepenent of the noise level. In MDM the role of ε is performe by the accuracy of the numerical integration. It is clear from relations () (4) an Figs. 6 an 7 that, unlike in [48], tm when ε 0. As in [48], the K-entropy is a metric invariant an oes not epen on the roughening metho uner ε 0. Along with the equilibrium states there can exist metastable ones. In this case the volume ΔΓ max practically ceases to be simply-connecte. Such states arise both for simple systems an for macromolecules. The latter ones require special consieration. t m.. Time of Computational Memory The results of Section. still hol out hope that by improving the accuracy of the numerical integration, it woul still be possible to exten, even if logarithmically slowly, the equilibrium trajectory satisfying Newton s equations. There is, however, another factor that limits this hope. An aitional error arises ue to the finite accuracy of the computer representation of real numbers [, 4]. For example, in the case of pair potentials, the force acting on the ith particle is the sum of the contributions mae by all its neighbors n i F = F, j G, i ij k k i k= where Gi is the orere sequence of inices. The structure Gi epens on the algorithm for particle sorting in calculating the forces. In computer summation with a finite accuracy of number representation (the fixe number of ecimal igits), ifferent orers of summation in (6) yiel ifferent result for F i. (6)

13 STOCHASTIC THEORY OF THE CLASSICAL MOLECULAR DYNAMICS METHOD Δr 0 40 Δv t t Fig. 8. Depenences on time t of the ivergences of the coorinates Δ r () t an the velocities Δv () t for two trajectories calculate from the same initial conitions uner ifferent summation orers of sums such as (6). The ata are presente for ifferent computer representations of real numbers: () float PC (4 bytes), () ouble Cray SV (8 bytes), () long ouble PC (8 bytes), (4) long ouble Cray SV (6 bytes). For the case of the most accurate representation of numbers (case 4), the ashe line shows the maximum an minimum ivergence of iniviual particles at a given time instant t: maxiδri () t an maxiδv i () t ( i =,..., N). Numerical rouning errors are a weak uncontrollable source of isturbance in the calculation of MD trajectories. In orer to stuy their effect on the calculation result, we use the proceure of artificial permutation of inices. By the ranom permutations of the orer of summation in (6), ifferent versions of rouning errors an, therefore, ifferent values of F i are implemente. Let two trajectories { r ( kδt), v ( kδ t), k = 0,, } an { r ( kδt), v ( kδ t), k = 0,, } be calculate from one an the same initial configuration, with one an the same numerical scheme, at one an the same step Δt, using one an the same computer, but let the orer of summation in the calculation of the sums (6) be ifferent. The trajectories iverge exponentially (Fig. 8). In this case we will refer to the memory time c as the time of computational memory t m. This time epens on the accuracy of the machine representation c of real numbers. From the conition tm = tm, we can fin the value of the numerical integration step which can provie the longest time of preserving the correlation of the MD trajectory with the true Newtonian trajectory for a given machine accuracy an finite-ifference scheme. Figure 9 is an example of the expecte limitation of the time t m by the finite accuracy of the machine representation of real numbers. For the values of numerical integration steps employe in MD calculations (of the orer of the inverse frequency of particle oscillations), the time t m is only several c factors less than t m. Thus, the finite accuracy of the computer representation of real numbers makes it hopeless in the foreseeable future to integrate Newton s equations exactly over times characteristic of MDM..4. Solution of Cauchy Problem in MDM. Irreversibility Divergence is shown in Fig. on a logarithmic scale. For most of the time less than t m, the absolute value of the ivergence is small an, on a linear scale, the ivergence woul be noticeable only in the vicinity of t m (Fig. 0). In contrast to Fig., where only one MD trajectory is shown, Fig. 0 presents four MD trajectories uner the same initial conitions Γ 0 ; their polygonal character is smoothe ue to the great number of the assume integration steps. In Fig. 0a the trajectories 4 are calculate with increasing

14 8 NORMAN, STEGAILOV Kt m ΔE/E Fig. 9. The expecte restriction of the limiting values Kt m by the number length in a computer:,, an correspon to 7, 6, an ecimal places in the machine representation of real numbers. 4 is taken from Fig. 7. accuracy at the expense, for instance, of a reuction in Δt or raising the orer of accuracy of the scheme. The logarithmically slow (Section.) increase in t m is schematically represente in Fig. 0a. In Fig. 0b the trajectories 4 calculate with the same accuracy iverge, for instance, ue to the ifferent orer of the forces summe in (6). In this case, the values t m prove the same for all trajectories. Figure 0 illustrates an important feature of the MDM, which was esigne as a solution to the Cauchy problem, i.e., instea of a single solution corresponing to the initial conitions Γ 0, MDM yiels a bunle of trajectories emanating from Γ 0. These trajectories exponentially iverge from each other over time t m. However, for times shorter than t m, the solution of the Cauchy problem is, nevertheless, practically obtaine. The fact that MDM only enables fining the solution of equations of motion for short perios of time τ was mentione in [9, 0, 5]. It ha still been unclear why MDM yiele such successful results by averaging over trajectories, the lengths of which were by several orers of magnitue greater than τ, although in [0] the way to the answer ha alreay been shown. Statements were repeately mae (see, for instance, [0, 40, 4, 5 7]) proclaiming that MDM simulate a microcanonical ensemble; i.e., the MD trajectories lay on a hypersurface of constant energy in the phase space. It was repeately state that if the equations of motion were correctly solve, the MD trajectories ha to be reversible in time [0]. However, from Fig. 0 it follows that trajectories calculate in MDM are irreversible if the length of the trajectories excees time t m. To prove this, it is sufficient to calculate the forwar trajectory with one accuracy an the backwar trajectory with a ifferent one. The research [68] was commissione by Lebowitz. Its authors, unfortunately, replace the concept of the reversibility of a trajectory by the reversibility of the MD algorithm. Earlier, a similar mistake ha been mae in [8]. In this connection, as early as in [], it ha been pointe out that accoring to some opinions, the trajectories obtaine by symmetrical ifference schemes were reversible; it is true if for reversing a trajectory not one finite point of the forwar trajectory is use but two, which was propose in [8, 68]. It shoul be remembere that in MDM, it is impossible to fin two neighboring points belonging to one an the same Newtonian trajectory. At the same time, if only one en point is use, as require by the Cauchy problem, for any ifference scheme trajectories will not be reverse ue to the instability of the

15 STOCHASTIC THEORY OF THE CLASSICAL MOLECULAR DYNAMICS METHOD 9 (a) 0 (b) Г 0 Г 0 Fig. 0. Schematic illustration of the linear scale of ivergence of MD trajectories (shown by fine lines) against the hypothetical exact solution of the Cauchy problem (continuous curve 0). MD trajectories are calculate with monotonically increasing accuracy when passing from trajectory to trajectory 4 (a) an with one an the same accuracy for all trajectories (b). Both the trajectory 0, an the trajectories 4 have the same initial conitions Γ 0. The moments of MD trajectories eviation from the exact solution correspon to ifferent ynamic memory times t m in the case (a) an to one an the same time t m in the case (b). The linear time scale was selecte; therefore, the initial coincience of MD trajectories an the exact solution illustrates the fact that the relative error of the coorinates an velocities is still very small, although it increases exponentially (see Fig. ). equations of motion. The authors of [68] coul verify the irreversibility of the MD trajectories by changing the step size for the backwar trajectory. The time t m is sometimes calle the horizon of preictability. In a ynamic problem of forecasting the weather, the preictability horizon limits the time of the reliable forecast [4 6]. The preiction of atomic trajectories is possible in the picosecon range, a reliable weather forecast is possible for a few ays, but the mathematical nature of limiting preictability is one an the same. At the same time, there is also a significant ifference. Of practical interest in the weather forecast is a single trajectory an the eviation of the preicte one ue to an unexpecte small isturbance is perceive by the general public as the preiction error. In MDM, on the contrary, the iniviual trajectory is not of practical interest but rather the result of averaging over the total istribution of the trajectories. An this result proves to be stable an oes not epen on the scheme, the step of numerical integration, or on the computer [0]. We also note the ifference between the problems of MDM an ynamic chaos [50, 48, 5, 70 7], which is efine as an irregular, aperioic state change (movement) of a ynamical system possessing the basic properties of a ranom process [9]. This result arises in the absence of ranom factors. In the problem of ynamic chaos, a unique trajectory is consiere to be etermine by the initial conitions. While in the problems of ynamic chaos, we can expect to obtain the characteristic of ranom processes after averaging istributions rigorously conserving energy over a single ynamic trajectory; all the more so, such istributions are attaine in the MDM when averaging is performe over an ensemble of tragectories only conserving energy on average an locate in the hyperlayer E ± ΔE. In star clusters an other systems of gravitating boies, the istances between the boies are huge compare to the size of the boies themselves. Therefore, the motion of these boies is etermine by the gravitational potential r. In molecular systems, there is also a long-range attraction potential, even if much

16 0 NORMAN, STEGAILOV weaker, i.e., r 6. The istances between particles in molecular systems are such that there necessarily occur close collisions resulting from short-range repulsion. It is these collisions that etermine the instability of the motion of particles an trigger the exponential ivergence of the trajectories. Krylov illustrate this by the simplest example of the collisions of balls [8]. In the theory of systems of gravitating boies the situation is the reverse. The motion of boies is usually escribe by stanar cycles an the search for exceptional conitions is conucte when, espite the absence of close collisions, the instability of trajectories can still arise in the system [6]. The irreversibility in the numerical solution of equations of motion of exponentially unstable multiparticle systems is use for the construction of encryption algorithms [0]..5. Non-Hamiltonian Nature an Statistical Meaning of the MDM We procee to the conclusions that can be rawn from the basic features of the numerical integration in MDM analyze in this work..5.. The accuracy of equilibrium MD calculations. The first conclusion [] is that MDM for equilibrium systems is a metho that (i) retains Newtonian ynamics only at times less than t m an (ii) conucts statistical averaging over the initial conitions along the MD trajectory. Hence, in particular, the ual meaning of the accuracy of MD calculations follows. On the one han, ynamically inepenent points on the MD trajectory are separate by time intervals close to t m. Thus, we estimate that the accuracy of averaging in the MD to be not worse than ( tm t0), where t0 is the length of the MD trajectory. On the other han, the quantity t m etermine the time interval over which the solution of the system of Newton s equations is close to the exact one. Thus, by improving the accuracy of the numerical integration, we increase t m but reuce the accuracy of averaging if we retain the same value of t 0. The value t m is uniquely connecte with the fluctuation of the total energy ΔE. This implies that the latter is also a measure of the integration accuracy. The comparison of the values of the ynamic memory time t m with the characteristic times of the velocity autocorrelation function (VACF) shows that the interval when the VACF values normalize to at excee 0 t =0 correspons to the times less than the memory time. Thus, the correlations in this area are ynamic correlations that follow from Newton s equations. Correlations for time greater than are alreay of a stochastic nature rather than a ynamic one. It woul be of interest to fin out whether the increase in the accuracy of the numerical integration affects the character of the correlations uring the transition from ynamic to stochastic correlations. The available computing capability only enable the reuction in ΔE by several orers of magnitue even when refine numerical schemes are use [6, 85, 86]. This woul only ouble t m. Thus, the area of stochastic correlations woul still be preserve in the time interval where VACF ecreases exponentially []..5.. Nonconservative equations of motion. Since Newtonian ynamics are only approximately retaine in the MDM ue to the impact of numerical errors, an inverse problem can be formulate consisting in fining equations satisfie by the trajectories of particles calculate by MDM [, 4, 69]. Thus, the secon conclusion is that these trajectories obey the equations vi ri mi = Fi( r,, rn) + ηi(), t = vi + ξi() t (7) t t for schemes with the explicit computation of velocity at each step or ri mi = F (8) i( r,, rn) + zi( t), i =,.., N, t for schemes with the implicit calculation of the velocity. The quantities ξ i, η i (or z i ) are aitional terms arising from the numerical integration of equations of motion () or (), m iξi + ηi = zi. The errors of the numerical integration are mae up of the scheme an rouning errors. The scheme errors are eterministic. Rouning errors in the computer are also formally eterministic. However, it is rouning that lays the founation of the algorithms for obtaining pseuoranom number sequences in ranom-number generators. The repetition perio lengths in these sequences are so large that ranom numbers actually turn out to be ranom, see, for instance, []. Therefore, for ξ i, η i an z i, a probabilistic escription can be introuce; i.e., we can use the concepts of a istribution function an correlation function. t m t m

17 STOCHASTIC THEORY OF THE CLASSICAL MOLECULAR DYNAMICS METHOD The presence of aitional terms having pseuoranom character makes the system of equations of motion (7) or (8) non-conservative. For the rift of the total energy to be absent an its fluctuations to be of a stationary character, the work of the force z i must average zero zṙ i i = 0. (9) From (9) there follows the evenness requirement for the approximation of the secon erivative [, 4, 69]; i.e., only schemes of an even orer are suitable for MDM. Those inclue Verlet schemes an leapfrog schemes. A stronger requirement for numerical schemes for MDM is their simplecticity, i.e., preserving the phase volume of the ynamical system (see, for instance, [ 6]). As applie to MD problems, the methos of geometrical numerical integration an backwar error analysis are evelope (see, for instance, [7]), enabling the construction of symplectic numerical schemes with stable preservation of the total energy of the system on average at large times. It shoul be note that the use of symplectic schemes cannot affect the presence of the exponential ivergence an stochastic entanglement of the MD trajectories. There is an opinion that the use of symplectic schemes is not always necessary in MD calculations [8]. Thus, the thir conclusion is that MD systems are non-hamiltonian. Attention is rawn in [9] to the fact that the system of ust particles in plasma is non-hamiltonian cause by the stochastic backgroun affecting the ynamics of ust particles an results from their collisions with electrons, ions, an atoms of the usty plasma. Dusty plasma is nothing exceptional in this regar. The stochastic backgroun is create alreay at the level of numerical integration errors (their physical counterparts are iscusse below in Section.4) an can only be enhance by other factors. Thus, any classical system of molecules, atoms, ions, these systems can be approximately Hamil- an electrons is non-hamiltonian. Only for time less than tonian..5.. Ensembles in MDM. The result of the solution of the system () or () woul be a phase trajectory Γ=Γ(; t Γ0), where Γ= ( r,.., rn, v,.., vn) is a point in the phase space an Γ0 = Γ(0) is the initial configuration. In this case, the total energy of the system woul be ientically conserve along the trajectory E = Ekin + U = const. The set of phase states of the system woul correspon to the microcanonical NVE ensemble. However, making use of the finite-ifference approximation for the numerical solution of equations of motion of a multi-particle system resulte in the fact that the value of the total energy E along the MD trajectory is not constant. The magnitue of the fluctuations ΔE epens on the selecte numerical integration step an usually is relatively small, ΔE E 0. %. However, the violation of the conition E = const significantly affects the ynamic properties of the MD system. Therefore, it makes sense to speak about a statistical ensemble NV( E ±ΔE). MDM generate this ensemble an operates within its frame. This is the fourth conclusion from the performe analysis of the numerical integration in MDM. The statistical ensemble NV( E ±ΔE) originate in the theory of MDM from the empirical analysis of particle trajectories calculate by this metho. However, it also has a complete analog in ergoic theory [07], in which we consier an ensemble of points in a phase space uniformly istribute in a thin layer near the surface of constant energy, i.e., the same ensemble NV( E ±ΔE). It is such an ensemble, instea of a microcanonical one, that is use in this theory to prove the equality of two averages of the functions epening on the coorinates an momenta of all the particles of an isolate system. The first average is taken over time along the trajectories of the system in the phase space. The secon average is the statistical mean over the ensemble of phase points in a thin layer ( E ±ΔE). The same metho is use in [58] Kt m t m 0 0 ΔE/E Fig.. Depenence (calculate by A.Yu. Kuksin) of the imensionless prouct Kt m on the relative fluctuation of the total energy of the system (Lennar Jones liqui, ρ =.0, T = 0.7, Verlet scheme in the velocity form). Circles esignate the Langevin thermostat, triangles enote the Berensen thermostat.

18 NORMAN, STEGAILOV for the proof of the Gibbs istribution in an isolate boy. Thus, the ergoic theory provies an aitional justification of the correctness of the MDM. Distributions in small subsystems of a MD cell containing a large number of particles naturally comply with the stanar statistical ensembles, i.e., canonical an gran canonical. In the stuy of both macromolecular structures an single-atom systems, broa use is mae of moifie MDM versions such as Langevin an stochastic MD proceures for maintaining a constant temperature, the NPT-ensemble, the Nose Hoover ensemble, an the Berensen thermostat. In all these cases, Newton s equations are supplemente with aitional terms maintaining the constancy of, say, pressure an/or temperature or some flow. Any aition further istorts the solution of Newton s equations in comparison with the stanar numerical schemes escribe above. However, the uration of the ynamic memory time t m can change insignificantly ue to the semilogarithmic epenence of the value of t m on the errors introuce in solving Newton s equations. In Fig. universal coorinates are selecte analogous to Fig. 7, for which the intensity of isturbing sources is expresse through the value of the relative fluctuations of the total energy generate by them. The semilogarithmic epenence prove similar to Fig. 7. With excessively large Δ E E, the values of t m are reuce so that the metho loses its ynamic character an its capabilities are restricte to the level of the Monte Carlo metho. The propose analysis makes it possible to estimate the reliability of the approximate MDM variants an limits of their applicability in the stuy of ynamic properties.. SOME CONCLUSIONS FROM THE MDM RESULTS MDM contains all the basic concepts of statistical physics an physical kinetics, i.e., irreversibility, the transition to the statistical escription, an for equilibrium systems it also features Gibbs istribution an equality between time an space averages. MDM results are consistent with a variety of experimental ata, incluing the macroscopic properties such as the equation of state, transport coefficients, uctility, an fracture ynamics, as well as microscopic properties, for example, the istribution function. MDM also emonstrate its preictive power. Thus, MDM provies an aequate an internally self-consistent escription of real substances. By escription we unerstan the set of particle trajectories calculate in MDM. This set can be calle a moel of a substance or an object that is create by this metho. Thereafter, within the MDM frame, various iagnostic tools are foun that are use to stuy the properties of the create object. The agreement between the representations of the substance that are actually evelope in the MDM an real substances suggests that these representations are a goo approximation to escribe real-worl properties, processes an phenomena of classical physics. Let us turn to the conclusions that follow from the analysis of representations of a substance which is evelope in the MDM. We will try to list therewith all items both consistent with the usual ieas of classical statistical physics [49, 58] an those that are at some variance with them... Impossibility of Solving System () for Times Exceeing It is generally assume that by treating a multi-particle system as a mechanical multi-particle system, setting up classical equations of motion for it, the number of which equals the number of egrees of freeom, an integrating these equations, you can get comprehensive information about the movement an properties of the system; however, this problem is practically unsolvable. MDM has shown that this problem is funamentally unsolvable, because even for three particles it is impossible to solve the system of Newton s equations uring the time require for averaging, as this time significantly excees t m. The trajectories of the particles obtaine in the MDM are close to the trajectories following from Newton s equations only over the time less than t m. Therefore, the time interval of possible ynamic correlations on a microscopic level is limite by the value of t m. This alreay suggests the inication of irreversibility of the solution for the MDM system of equations of motion. It is not surprising, therefore, that the ieas evelope in the MDM fit into the principles of statistical physics an its results both qualitatively an quantitatively agree with the experimental ata. That is why the above-mentione features of the trajectories calculate in MDM must be consistent with the specifics of particle trajectories in the real systems. t m

19 STOCHASTIC THEORY OF THE CLASSICAL MOLECULAR DYNAMICS METHOD To avoi misunerstaning, we note that at the macroscopic level, the result of averaging over the istribution of trajectories is of interest rather than the trajectory of a single particle. Since this result is stable, the time interval of possible ynamic correlations at the level of a continuous meium by many orers of magnitue excees t m for the trajectory of a single particle... Choice of Particle Number It is usually state that the original so-calle statistical laws, ue to the presence of the large number of particles making up the boy, cannot be reuce to purely mechanical laws. MDM has shown that this is true. However, the point here is not the large number of particles but the exponential Lyapunov instability an low noise. The bounary between the systems, for which a purely mechanical escription is possible, an systems, for which statistical laws shoul be use, is a system of three particles. This is because systems of three an more particles are characterize by an exponential instability, which leas to qualitatively new laws an patterns. We note that we are talking about statistical laws, which are manifeste when the system is monitore over times significantly exceeing t m or in the cases when averaging is performe over an ensemble of statistically inepenent initial conitions. The specificity of these patterns loses all content not only after transition to mechanical systems of two particles but also for systems with many egrees of freeom, when the ynamics of these systems is stuie over times much smaller than t m, an no averaging is conucte over inepenent initial conitions. Such problems, however, are beyon the scope of this survey. Since the Lyapunov instability occurs in systems of three an more particles, the stuy of specific statistical laws can be starte from three particles. Thus, for example, the equilibrium istribution can be obtaine for three (even two) particles in a limite volume when average over time along the MD trajectory, the length of which is much greater than t m. These istributions are getting increasingly closer to the Maxwellian istribution in a growing energy range with a rise in N [40, 4]. When similarly long trajectories are use, the properties of clusters an their epenence on N can be investigate, starting from three-particle ones [4]. Recently, the problem of clusters has significantly increase in importance in connection with the theoretical analysis an MD simulations of small clusters in ust plasma. The proximity to the Maxwellian istribution of particle velocities in such clusters has also been establishe experimentally (see [4] an references therein). In the stuy of homogeneous systems in perioic bounary conitions (PBC), the error of this approach can be etermine in relation to the infinite homogeneous systems an its epenence on the number of particles N (or N ) starting from N =. Certainly, for stuying a variety of properties, unlike obtaining Maxwellian istribution, the choice of a small N can prove utterly meaningless. Thus, the choice of N is base not on the funamental requirement of aherence to statistical laws, but is ictate by the size of the system characteristic of the problem to be explore; i.e., it is guie by specific physical consierations. At the same time, the increase in N enhances the wealth of properties an processes arising in the system. Accoringly, their investigation calls for an increasing variety of instruments. The selection of the value of N (or MD size of cell L) which woul be the minimum allowable for the pose problem is etermine by the scale of the space an time correlations an inhomogeneities characteristic of this problem. Let us consier these aspects in greater etail. Space constraints. If we calculate a pair correlation function gr (), then N nr c, where n is the concentration of particles an rc is the stuie range of istances. The quantity rc may correspon both to the istance to the first or some subsequent maximum or minimum, an to the region where gr () asymptotically tens to unity. The correlation raii r ci are etermine not only by g(r) but also by other space characteristics, as there is a hierarchy of correlations rc < rc < rc < in the system, which correspons to the hierarchy N < N < N < where N i = nrci. When choosing the value of N, we thereby cut the series of correlations to be stuie in this MD calculation. The choice N = nl limits the wavelengths λ < L of the equilibrium fluctuations; i.e., it fixes the range of wave vectors for which we can calculate the variance in ensity fluctuations, such as phonons in conense meia an plasma waves in collisional plasma. For long-range potentials, in particular the Coulomb potential, the contribution of particle interactions ΔU( r > L) at istances greater than L must be small compare to the interaction energy U. Other-

20 4 NORMAN, STEGAILOV T M T τ T t 4 τ T t 4 0 τ M t t Fig.. On the left epenence of the temperature on the time uner heating to the require temperature with impose restrictions on the movement of atoms. On the right three examples of MD trajectories with remove restrictions on the motion of atoms each yieling its lifetime τ j, j =,,..., M. The results are obtaine for the Lennar Jones system of N = 500 particles; the ensity is. σ. wise, nonphysical correlations can emerge in the system inuce by PBC. N must be increase until its value reaches ΔU( r > L) U. Time constraints. Alongsie the hierarchy rci, there is a hierarchy of correlation times τc < τc < τ c <. The choice of L cuts this series by two inequalities, i.e., 6 Dτci < L where D is the iffusion coefficient an asτci < L where as is the soun velocity. The limitations impose by the secon inequality were ientifie in [] for the velocity autocorrelation function calculate at ifferent N an one an the same ensity for the system of har spheres. In the transition to relaxation processes, there are aitional spatial an temporal characteristic scales an the corresponing requirements for N. The choice of N limits the range of the investigate characteristics for such cooperative phenomena as islocations, an cracking. The general conclusion is that the choice of the system size (number of particles) restricts the limiting values r c, τ c, λ, etc., an, consequently, the range of phenomena an processes that can be stuie. When MDM is applie using parallel computations on moern supercomputers, the time spent on the ata exchange between computing noes (cores an processors) is much smaller than the computation time within a single noe. Initially, therefore, the performance increases almost proportionally to the number of noes use, then the performance reaches its maximum an begins to ecrease with the further increase in the number of noes. M noes is the optimal number of noes, at which the performance is close to the maximum. The value of M epens on the architecture of the compute cluster or supercomputer an N, i.e., the stuie property or process in the investigate system [4]... Probabilistic Nature of Classical Statistics In the literature, it is still a matter of ebate whether the finings an preictions about the behavior of macroscopic boies base on statistics are of a probabilistic nature. The statistics might be ifferent from classical mechanics, the conclusions of which are quite unique. MDM results mae it possible to show