μ0

5.727 657 506(58) — 10’4

µ0 1/(4π)

fel 1(ex) 138.935 4574(14)

c 137.035 99911(46) 299 792.458(ex)

4.222 18(63) — 10’32 1.108 28(17) — 10’34

G

h 2π 0.399 031 2716(27)

1(ex) 0.063 507 799 32(43)

5.485 799 0945(24) — 10’4

me 1(ex)

e 1(ex) 1(ex)

u 1 822.888 484 93(80) 1(ex)

mp 1 836.152 672 61(85) 1.007 276 46688(13)

mn 1 838.683 6598(13) 1.008 664 915 60(55)

md 3 670.482 9652(18) 2.013 553 212 70(35)

mμ 206.768 2838(54) 0.113 428 9264(30)

mH 1 837.152 645 89(85) 1.007 825 032 13(13)

7.297 352 568(24) — 10’3 7.297 352 568(24) — 10’3

±

±’1 137.035 999 11(46) 137.035 999 11(46)

5.291 772 108(18) — 10’2

a0 1 (ex)

R∞ 0.5(ex) 0.010 973 731 568 525(73)

μB 0.5(ex) 57.883 818 04(39)

3.166 8154(55) — 10’6

kB 0.008 314 472(15)

0

vm 254 496.34(44) 37.712 467(66)

Part I

A Modeling Hierarchy for Simulations

1

Introduction

1.1 What is this book about?

1.1.1 Simulation of real systems

Computer simulations of real systems require a model of that reality. A

model consists of both a representation of the system and a set of rules that

describe the behavior of the system. For dynamical descriptions one needs in

addition a speci¬cation of the initial state of the system, and if the response

to external in¬‚uences is required, a speci¬cation ofthe external in¬‚uences.

Both the model and the method of solution depend on the purpose of

the simulation: they should be accurate and e¬cient. The model should be

chosen accordingly. For example, an accurate quantum-mechanical descrip-

tion of the behavior of a many-particle system is not e¬cient for studying

the ¬‚ow of air around a moving wing; on the other hand, the Navier“Stokes

equations “ e¬cient for ¬‚uid motion “ cannot give an accurate description of

the chemical reaction in an explosion motor. Accurate means that the sim-

ulation will reliably (within a required accuracy) predict the real behavior

of the real system, and e¬cient means “feasible with the available technical

means.” This combination of requirements rules out a number of questions;

whether a question is answerable by simulation depends on:

• the state of theoretical development (models and methods of solution);

• the computational capabilities;

• the possibilities to implement the methods of solution in algorithms;

• the possibilities to validate the model.

Validation means the assessment of the accuracy of the model (compared to

physical reality) by critical experimental tests. Validation is a crucial part

of modeling.

3

4 Introduction

1.1.2 System limitation

We limit ourselves to models of the real world around us. This is the realm

of chemistry, biology and material sciences, and includes all industrial and

practical applications. We do not include the formation of stars and galax-

ies (stellar dynamics) or the physical processes in hot plasma on the sun™s

surface (astrophysics); neither do we include the properties and interactions

of elementary particles (quantum chromodynamics) or processes in atomic

nuclei or neutron stars. And, except for the purposes of validation and

demonstration, we shall not consider unrealistic models that are only meant

to test a theory. To summarize: we shall look at literally “down-to-earth”

systems consisting of atoms and molecules under non-extreme conditions of

pressure and temperature.

This limits our discussion in practice to systems that are made up of

interacting atomic nuclei, which are speci¬ed by their mass, charge and spin,

electrons, and photons that carry the electromagnetic interactions between

the nuclei and electrons. Occasionally we may wish to add gravitational

interactions to the electromagnetic ones. The internal structure of atomic

nuclei is of no consequence for the behavior of atoms and molecules (if we

disregard radioactive decay): nuclei are so small with respect to the spatial

spread of electrons that only their monopole properties as total charge and

total mass are important. Nuclear excited states are so high in energy

that they are not populated at reasonable temperatures. Only the spin

degeneracy of the nuclear ground state plays a role when nuclear magnetic

resonance is considered; in that case the nuclear magnetic dipole and electric

quadrupole moment are important as well.

In the normal range of temperatures this limitation implies a practical

division between electrons on the one hand and nuclei on the other: while

all particles obey the rules of quantum mechanics, the quantum character

of electrons is essential but the behavior of nuclei approaches the classical

limit. This distinction has far-reaching consequences, but it is rough and

inaccurate. For example, protons are light enough to violate the classical

rules. The validity of the classical limit will be discussed in detail in this

book.

1.1.3 Sophistication versus brute force

Our interest in real systems rather than simpli¬ed model systems is con-

sequential for the kind of methods that can be used. Most real systems

concern some kind of condensed phase: they (almost) never consist of iso-

lated molecules and can (almost) never be simpli¬ed because of inherent

1.1 What is this book about? 5

symmetry. Interactions between particles can (almost) never be described

by mathematically simple forms and often require numerical or tabulated

descriptions. Realistic systems usually consist of a very large number of in-

teracting particles, embedded in some kind of environment. Their behavior

is (almost) always determined by statistical averages over ensembles con-

sisting of elements with random character, as the random distribution of

thermal kinetic energy over the available degrees of freedom. That is why

statistical mechanics plays a crucial role in this book.

The complexity of real systems prescribes the use of methods that are

easily extendable to large systems with many degrees of freedom. Physical

theories that apply to simple models only, will (almost) always be useless.

Good examples are the very sophisticated statistical-mechanical theories for

atomic and molecular ¬‚uids, relating ¬‚uid structural and dynamic behav-

ior to interatomic interactions. Such theories work for atomic ¬‚uids with

simpli¬ed interactions, but become inaccurate and intractable for ¬‚uids of

polyatomic molecules or for interactions that have a complex form. While

such theories thrived in the 1950s to 1970s, they have been superseded by ac-

curate simulation methods, which are faster and easier to understand, while

they predict liquid properties from interatomic interactions much more ac-

curately. Thus sophistication has been superseded by brute force, much to

the dismay of the sincere basic scientist.

Many mathematical tricks that employ the simplicity of a toy model sys-

tem cannot be used for large systems with realistic properties. In the exam-

ple below the brute-force approach is applied to a problem that has a simple

and elegant solution. To apply such a brute-force method to a simple prob-

lem seems outrageous and intellectually very dissatisfying. Nevertheless, the

elegant solution cannot be readily extended to many particles or complicated

interactions, while the brute-force method can. Thus not only sophistication

in physics, but also in mathematics, is often replaced by brute force methods.

There is an understandable resistance against this trend among well-trained

mathematicians and physicists, while scientists with a less elaborate train-

ing in mathematics and physics welcome the opportunity to study complex

systems in their ¬eld of application. The ¬eld of simulation has made theory

much more widely applicable and has become accessible to a much wider

range of scientists than before the “computer age.” Simulation has become

a “third way” of doing science, not instead of, but in addition to theory and

experimentation.

There is a danger, however, that applied scientists will use “standard”

simulation methods, or even worse use “black-box” software, without real-