Many physical systems have the property that they can carry out oscillations at certain specific frequencies only. As a child (and hopefully also as an adult) you will have discovered that a swing in a playground will move only with a specific natural period, and that the force that pushes the swing is only effective when the period of the force matches the period of the swing. The patterns of motion at which a system oscillates are called the normal modes of the system. A swing has one normal mode, but you have seen in Section 13.6 that a simple model of a tri-atomic molecule has three normal modes. An example of a normal mode of a system is given in Figure 20.1 which shows the pattern of oscillation of a metal plate which is driven by an oscillator at a fixed frequency. The screw in the middle of the plate shows the point at which the force on the plate is applied. Sand is sprinkled on the plate. When the frequency of the external force is equal to the frequency of a normal mode of the plate, the motion of the plate is given by the motion that corresponds to that specific normal mode. Such a pattern of oscillation has nodal lines where the motion vanishes. These nodal lines are visible because the sand on the plate collects at these lines.

The last chapter has already familiarized us with the use of group theoretical methods for the construction and analysis of wavelets and gaborettes. We aim in this chapter to first indicate the general applicability of these techniques and then to look at the case of the two-dimensional continuous transform, using the SIM(2) group. Later, we look at general matrix groups of the type that can be used for constructing other types of wavelet transforms in two dimensions. We shall be led, in this manner, to studying a class of semidirect product type groups, certain coadjoint orbits of which are isomorphic to the group itself. In all these cases, the common features of such a matrix-group analysis will be: (a) the group will refer to a set of possible symmetry transformations which the signal may undergo; (b) the space over which the signals are defined (as L2-functions) is intrinsic to the group; (c) the parameters in terms of which the wavelet transform is expressed are the parameters of the group itself, i.e., symmetry parameters of the signal, and (d) these parameter spaces, which arise as coadjoint orbits of the group, are also identifiable with phase spaces of signals.

Referring back to the 2-D wavelet transform introduced in Chapter 2, we shall see that this transform is again related to a square integrable representation of a matrix group.

In physics and mathematics, coordinate transformations play an important role because many problems are much simpler when a suitable coordinate system is used. Furthermore, the requirement that physical laws do not change under certain transformations imposes constraints on the physical laws. An example of this is presented in Section 22.11 where it is shown that the fact that the pressure in a fluid is isotropic follows from the requirement that some physical laws may not change under a rotation of the coordinate system. In this chapter it is shown how the change of vectors and matrices under coordinate transformations is derived. The derived transformation properties can be generalized to other mathematical objects which are called tensors. In this chapter, only transformations of rectangular coordinate systems are considered. Since these coordinate systems are called Cartesian coordinate systems, the associated tensors are called Cartesian tensors. The transformation properties of tensors in Cartesian and curvilinear coordinate systems are described in detail by Butkov [24] and Riley et al. [87].

Coordinate transforms

In this section we consider the transformation of a coordinate system in two dimensions. In Figure 22.1 an old coordinate system with coordinates xold and yold is shown. The unit vectors along the old coordinate axis are denoted by êx,old and êy,old. In a coordinate transformation, these old unit vectors are transformed to new unit vectors êx,new and êy,new, respectively.

In physics one frequently handles the change of a property with time by considering properties that do not change with time. For example, when two particles collide elastically, the momentum and the energy of each particle may change. However, this change can be found from the consideration that the total momentum and energy of the system are conserved. Often in physics, such conservation laws are the main ingredients for describing a system. In this chapter we deal with conservation laws for continuous systems. These are systems in which the physical properties are a continuous function of the space coordinates. Examples are the motion in a fluid or solid, and the temperature distribution in a body. The introduced conservation laws are not only of great importance in physics, they also provide worthwhile exercises in the use of vector calculus introduced in the previous chapters.

General form of conservation laws

In this section a general derivation of conservation laws is given. Suppose we consider a physical quantity Q. This quantity could denote the mass density of a fluid, the heat content within a solid or any other type of physical variable. In fact, there is no reason why Q should be a scalar, it could also be a vector (such as the momentum density) or a higher order tensor. Let us consider a volume V in space that does not change with time. This volume is bounded by a surface ∂V. The total amount of Q within this volume is given by the integral ∫VQdV.

From this book and most other books on mathematical physics you may have obtained the impression that most equations in the physical sciences can be solved. This is actually not true; most textbooks (including this book) give an unrepresentative state of affairs by only showing the problems that can be solved in closed form. It is an interesting paradox that as our theories of the physical world become more accurate, the resulting equations become more difficult to solve. In classical mechanics the problem of two particles that interact with a central force can be solved in closed form, but the three-body problem in which three particles interact has no analytical solution. In quantum mechanics, the one-body problem of a particle that moves in a potential can be solved for a limited number of situations only: for the free particle, the particle in a box, the harmonic oscillator, and the hydrogen atom. In this sense the one-body problem in quantum mechanics has no general solution. This shows that as a theory becomes more accurate, the resulting complexity of the equations makes it often more difficult to actually find solutions.

One way to proceed is to compute numerical solutions of the equations. Computers are a powerful tool and can be extremely useful in solving physical problems. Another approach is to find approximate solutions to the equations. In Chapter 12, scale analysis was used to drop from the equations terms that appear to be irrelevant.

Introduction

In this chapter we examine the behavior of systems in equilibrium; in particular, we focus on measurements of thermodynamic properties and studies of spatial structure and organization. The treatment of properties associated with the motion of atoms – the dynamical behavior – forms the subject of Chapter 5.

While basic MD simulation methods – formulating and solving the equations of motion – fall into a comparatively limited number of categories, a wide range of techniques is used to analyze the results. Rarely is the wealth of detail embodied in the atomic or molecular trajectories of particular interest in itself, and the issue is how to extract meaningful information from this vast body of data; even a small system of 103 structureless atoms followed over a mere 104 timesteps can produce up to 6 × 107 numbers, corresponding to a full chronological listing of the atomic coordinates and velocities. A great deal of data averaging and filtration of various kinds is required to reduce this to a manageable and meaningful level; how this is achieved depends on the questions that the simulation is supposed to answer. Much of this processing will be carried out while the simulation is in progress, but some kinds of analysis are best done subsequently, using data saved in the course of the simulation run; the choice of approach is determined by the amount of work and data involved, as well as the need for active user participation in the analysis.

Software availability

Readers interested in downloading the software described in this book in a computer-readable form for personal, noncommercial use should visit the Cambridge University Press web site at http://uk.cambridge.org, where the home page for this book and the software can be found; a listing of the programs included in the software package appears in the Appendix. Additional material related to the book, as well as contact information, can be found at the author's website – http://www.ph.biu.ac.il/~rapaport.

Legal matters

The programs appearing in this book are provided for educational purposes only. Neither the author nor the publisher warrants that these programs are free from error or suitable for particular applications, and both disclaim all liability from any consequences arising out of their use.

Introduction

This chapter provides the introductory appetizer and aims to leave the reader new to MD with a feeling for what the subject is all about. Later chapters will address the techniques in detail; here the goal is to demonstrate a working example with a minimum of fuss and so convince the beginner that MD is not only straightforward but also that it works successfully. Of course, the technique for evaluating the forces discussed here is not particularly efficient from a computational point of view and the model is about the simplest there is. Such matters will be rectified later. The general program organization and stylistic conventions used in case studies throughout the book are also introduced.

Soft-disk fluid

Interactions and equations of motion

The most rudimentary microscopic model for a substance capable of existing in any of the three most familiar states of matter – solid, liquid and gas – is based on spherical particles that interact with one another; in the interest of brevity such particles will be referred to as atoms (albeit without hint of their quantum origins). The interactions, again at the simplest level, occur between pairs of atoms and are responsible for providing the two principal features of an interatomic force. The first is a resistance to compression, hence the interaction repels at close range. The second is to bind the atoms together in the solid and liquid states, and for this the atoms must attract each other over a range of separations.

The second edition of The Art of Molecular Dynamics Simulation is an enlarged and updated version of the first. The principal differences between the two editions are the inclusion of a substantial amount of new material, both as additional chapters and within existing chapters, and a complete revision of all the software used in the case studies to reflect a more modern programming style. This style change is a consequence of the population shift in the research community. At the time the first edition was written older versions of the Fortran language were still in widespread use; despite this fact, C was chosen as the programming language for the book in preference to Fortran, but in a form that would appear familiar to Fortran programmers of the era. Now that C – and related languages – are in widespread use, and Fortran has even evolved to become more like C, the expressive capabilities of C can be employed to the full, resulting in software that is easier to follow. The power of desktop computers has also increased by a large factor since the case studies of the first edition were developed; in recognition of this fact some of the studies consider larger systems, reflecting a shifting view of what is considered a ‘short’ computation. Other minor changes and corrections have been incorporated throughout the text. The exhortation to employ this volume as a cookbook remains unchanged.

Role of simulation

Computer simulation in general, and molecular dynamics in particular, represent a new scientific methodology. Instead of adopting the traditional theoretical practice of constructing layer upon layer of assumption and approximation, this modern alternative attacks the original problem in all its detail. Unfortunately, phenomena that are primarily quantum mechanical in nature still present conceptual and technical obstacles, but, insofar as classical problems are concerned, the simulational approach is advancing as rapidly as computer technology permits. For this class of problem, the limits of what can be achieved remain well beyond the horizon.

Theoretical breakthroughs involve both new concepts and the mathematical tools with which to develop them. Most of the major theoretical advances of the just-finished twentieth century rest upon mathematical foundations developed during the preceding century, if not earlier. Whether still undeveloped mathematical tools and new concepts will ever replace the information presently only obtainable by computer simulation, or whether the simulation is the solution, is something only the future can tell. Whether computer modeling will become an integral part of theoretical science, or whether it will continue to exist independently, is also a big unknown. After all, theory, as we know it, has not been around for very long.

To what extent can simulation replace experiment? In the more general sense, this is already happening in engineering fields, where models are routinely constructed from well-established foundations.

Introduction

The range of problems amenable to study using MD knows few bounds and as computers become more powerful the range continues to expand. Because of the enormous breadth of the subject, we have chosen to concentrate on the simplest of systems and avoid overly specialized models. Most of the case studies up to this point have been based on short-ranged, two-body interactions; within this framework a considerable variety of problems can be studied, but a few conspicuous gaps remain. Pair potentials have their limitations, and while certain kinds of intermolecular interaction can be imitated by the appropriate combinations of pair potentials, it is sometimes essential to introduce many-body interactions to capture specific features of the ‘real’ intermolecular force [mai81].

In this chapter we present two different approaches to the introduction of many-body interactions, namely, three-body interactions and the embedded-atom method, each in the form of a case study. We cannot do justice to the range of applications to which these and other enhancements of the MD method, such as those discussed in Chapter 13, have contributed, but in the prevailing culinary atmosphere we hope the reader will gain at least a taste of what is involved.

Three-body forces

The problem

Even when regarded simply as effective potentials, the capacity of the pair potential to reproduce known behavior has its limitations.

Introduction

The elementary constituents of most substances are structured molecules, rather than the spherically symmetric atoms treated in previous chapters. The emphasis on simple monatomic models is justified for a number of reasons: the dynamics are simpler, thereby making life easier for newcomers; it reflects the historical development of the field, since the original work establishing the viability of the MD approach as a quantitative tool dealt with liquid argon [rah64]; and once the basic techniques have been mastered they can be extended to a variety of more complex situations. In this chapter we discuss the first of these excursions – to molecules constructed from a rigidly linked atomic framework. This approach is suitable for small, relatively compact molecules, where rigidity seems a reasonable assumption, but if this is not true then motion within the molecule must also be taken into account, as we will see in later chapters. There is really no such thing as a rigid molecule, but from the practical point of view it is a very effective simplification of the underlying quantum problem; the model also does not account for chemical processes – no mechanism is provided for molecular formation and dissociation.

The chapter begins with a summary of rigid-body dynamics, but with a slightly unfamiliar emphasis. In treatises on classical mechanics Euler angles play a central part [go180]; while they provide the most intuitive means for describing the orientation of a rigid body and are helpful for analyzing certain exactly soluble problems, in numerical applications they actually represent a very poor choice.

Introduction

Practically all the simulations described so far have involved systems that are either in equilibrium or in some time-independent stationary state; while individual results are subject to fluctuation, it is the well-defined averages over sufficiently long time intervals that are of interest. In this chapter we extend the MD approach to a class of problem in which the behavior is not only time dependent, but the properties themselves are also spatially dependent in ways that are not always predictable. The analysis of the behavior of such systems cannot be carried out following the methods described earlier, which were generally based on the evaluation of system-wide averages or correlations, and one is therefore compelled to resort to graphical methods. Here we focus on MD applications in fluid dynamics, a subject in which atomic matter is conventionally replaced by a continuous medium for practical purposes; recovering the atomic basis is part of trying to understand more complex fluid behavior of the type studied in rheology. For more on the microscopic approach to hydrodynamics see [mar92].

Open systems

Most current MD applications deal with closed systems; this implies either total isolation from the outside world, or coupling to the environment in a way described by one of the ensembles of statistical mechanics. The coupling can occur, for example, with the aid of a thermostat (§6.3), in which case the equilibrium properties are those of the canonical ensemble.

Introduction

Some internal degrees of freedom are important to molecular motion, while others can be regarded as frozen. Classical mechanics allows geometrical relations between coordinates to be included as holonomic constraints. We have already encountered constraints in connection with non-Newtonian modifications of the dynamical equations (Chapter 6); here the constraints occur in a Newtonian context, so that there is little doubt as to the physical nature of the trajectories.

In this chapter we focus on a class of model where constraints play an important role, namely, the polymer models used for studying alkane chains and more complex molecules, in which a combination of geometrical constraints and internal motion is required. The treatment of constraints is not the only new feature of such models; the interactions responsible for bond bending and torsion are essentially three- and four-body potentials, and some rather intricate vector algebra is required to determine the forces. The particular alkane model described here incorporates one further simplification, namely, the use of the often encountered ‘united atom’ approximation – the hydrogen atoms attached to each carbon atom in the backbone are absorbed into the backbone atoms and are thereby eliminated from the problem.

Geometric constraints

Role of constraints

The notion of a constraint acting at the molecular level is merely an attempt at simplification; the justification for assuming that certain bond lengths and angles are constant is that, at the prevailing temperature, there is insufficient energy to excite the associated vibrational degrees of freedom (or modes) out of their quantum ground states.

Historical background

The origins of molecular dynamics – MD – are rooted in the atomism of antiquity. The ingredients, while of more recent vintage, are not exactly new. The theoretical underpinnings amount to little more than Newton's laws of motion. The significance of the solution to the many-body problem was appreciated by Laplace [del51]: ‘Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it – an intelligence sufficiently vast to submit these data to analysis – it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes’. And the concept of the computer, without which there would be no MD, dates back at least as far as Babbage, even though the more spectacular hardware developments continue to this day. Thus MD is a methodology whose appearance was a foregone conclusion, and indeed not many years passed after digital computers first appeared before the first cautious MD steps were taken [ald57, gib60, rah64].

The N-body problem originated in the dynamics of the solar system, and the general problem turns out to be insoluble for three or more bodies. Once the atomic nature of matter became firmly established, quantum mechanics took charge of the microscopic world, and the situation became even more complicated because even the constituent particles seemed endowed with a rather ill-defined existence.