Contents

From Statistical Mechanics: Algorithms and Computations

Statistical Mechanics: Algorithms and Computations is organized in seven chapters which treat different subjects of classical and quantum statistical physics. On this page, you can read excerpts of SMAC-book, and find links to pages in SMAC-wiki that illustrate the contents of the book, and the wiki.

Contents 1 Preface 2 Chapter 1: Monte Carlo methods 3 Chapter 2: Hard disks and spheres 4 Chapter 3: Density matrices and path integrals 5 Chapter 4: Bosons 6 Chapter 5: Order and disorder in spin systems 7 Chapter 6: Entropic forces 8 Chapter 7: Dynamic Monte Carlo methods

[edit] Preface

This is how SMAC-book begins:

Statistical Mechanics: Algorithms and Computations is meant for students and researchers ready to plunge into statistical physics, or into computing, or both. It has grown out of my research experience, and out of courses that I have had the good fortune to give, over the years, to beginning graduate students at the Ecole Normale Supérieure and the Universities of Paris VI and VII, and also to summer school students in Drakensberg, South Africa, undergraduates in Salem, Germany, theorists and experimentalists in Lausanne, Switzerland, young physicists in Shanghai, China, among others. Hundreds of students from many different walks of life, with quite different backgrounds, listened to lectures and tried to understand, made comments, corrected me, and in short helped shape what has now been written up, for their benefit, and for the benefit of new readers that I hope to attract to this exciting, interdisciplinary field...

Read the entire Preface (pdf)

Have a look at the complete Table of contents (pdf).

Check out the Index (pdf)

[edit] Chapter 1: Monte Carlo methods

Opening page:

Children randomly throwing pebbles into a square, as in this figure, illustrate a very simple direct-sampling Monte Carlo algorithm that can be adapted to a wide range of problems in science and engineering, most of them quite difficult, some of them discussed in this book.

The basic principles of Monte Carlo computing are nowhere clearer than where it all started: on the beach, computing π....

Read the first three pages of Chapter 1 (pdf)

The first algorithm in this book, Direct pi, implements the above children's game on a computer, or a pocket calculator. This is incredibly easy (see Pages for all ages for other simple problems). During the course of the first chapter, we will visit an adults' version of the game,Markov pi, our first Markov-chain Monte Carlo program. Twenty-seven programs later, direct_gamma will illustrate the famed Levy distributions. Characteristically, direct_gamma is still a five-line program, but the theory behind it is advanced University-level mathematics.

The sections of Chapter 1 are as follows: (complete Table of contents, including subsections (pdf)

1.1 Popular games in Monaco

1.2 Basic sampling

1.3 Data analysis

1.4 Computing

Exercises

References

[edit] Chapter 2: Hard disks and spheres

Opening page:

Four hard disks move about a square box much like billiard balls. The rules for wall and pair collisions are quickly programmed on a computer, allowing us to follow the time evolution of the hard-disk system (see the figure on the left). Given the initial positions and velocities at time t = 0, a simple algorithm allows us to determine the state of the system at t = 10.37, but the unavoidable numerical imprecision quickly explodes. This manifestation of chaos is closely related to the statistical description of hard disks and other systems, as we shall discuss in this chapter.....

Read the first three pages of Chapter 2 (pdf)

The sections of Chapter 2 are as follows: (complete Table of contents, including subsections (pdf)

2.1 Newtonian deterministic mechanics

2.2 Boltzmann's statistical mechanics

2.3 Pressure and the Boltzmann distribution

2.4 Large hard-sphere systems

2.5 Cluster algorithms

Exercises

References

[edit] Chapter 3: Density matrices and path integrals

Opening page:

A quantum particle in a harmonic potential is described by energies and wave functions that we know exactly (see the figure). At zero temperature, the particle is in the ground state ; it can be found with high probability only where the ground-state wave function differs markedly from zero. At finite temperatures, the particle is spread over more states, and over a wider range of x - values. In this chapter, we discuss exactly how this works for a particle in a harmonic potential and for more difficult systems. We also learn how to do quantum statistics if we ignore everything about energies and wave functions.

Read the first three pages of Chapter 3 (pdf)

The sections of Chapter 3 are as follows: (complete Table of contents, including subsections (pdf)

3.1 Density matrices

3.2 Matrix squaring

3.3 The Feynman path integral

3.4 Pair density matrices

3.5 Geometry of paths

Exercises

References

[edit] Chapter 4: Bosons

Opening page:

We suppose that a three-dimensional harmonic trap, with harmonic potentials in all three space dimensions, is filled with bosons (see figure on the left). Well below a critical temperature, most particles populate the state with the lowest energy. Above this temperature, they are spread out into many states, and over a wide range of positions in space. In the harmonic trap, the Bose--Einstein condensation temperature increases as the number of particles grows. We shall discuss bosonic statistics and calculate condensation temperatures, but also simulate thousands of (ideal) bosons in the trap, mimicking atomic gases, where Bose--Einstein condensation was first observed, in 1995, at microkelvin temperatures.

Read the first three pages of Chapter 4 (pdf)

The sections of Chapter 4 are as follows: (complete Table of contents, including subsections (pdf)

4.1 Ideal bosons (energy levels)

4.2 The ideal Bose gas (density matrices)

Exercises

References

[edit] Chapter 5: Order and disorder in spin systems

Opening page:

In a ferromagnet, up spins want to be next to up spins and down spins want to be next to down spins. Likewise, colloidal particles on a liquid surface want to be surrounded by other particles (see the figure). At high temperature, up and down spins are equally likely across the system and, likewise, the colloidal particles are spread out all over the surface. At low temperature, the spin system is magnetized, either mostly up or mostly down; likewise, most of the colloidal particles are in one big lump. This, in a nutshell, is the statistical physics of the Ising model, which describes magnets and lattice gases, and which we shall study in the present chapter.

Read the first three pages of Chapter 5 (pdf)

The sections of Chapter 5 are as follows: (complete Table of contents, including subsections (pdf)

5.1 The Ising model-exact computations

5.2 The Ising model-Monte Carlo algorithms

5.3 Generalized Ising models

Exercises

References

[edit] Chapter 6: Entropic forces

Opening page:

Clothes-pins are randomly distributed on a line (see the figure): any possible arrangements of pins is equally likely. What is the probability of a pin being at position x? Most of us would guess that this probability is independent of position, but this is not the case: pins are much more likely to be close to a boundary, as if attracted by it. They are also more likely to be close to each other. In this chapter, we study clothes-pin attractions and other entropic interactions, which exist even though there are no charges, currents, springs, etc. These interactions play a major role in soft condensed matter, the science of colloids, membranes, polymers, etc, but also in solid state physics...

Read the first three pages of Chapter 6 (pdf)

The sections of Chapter 6 are as follows: (complete Table of contents, including subsections (pdf)

6.1 Entropic continuum models and mixtures

6.2 Entropic lattice models: dimers

Exercises

References

[edit] Chapter 7: Dynamic Monte Carlo methods

Opening page:

Disks are dropped randomly into a box (see figure), but they stay put only if they fall into a free spot. Most of the time, this is not the case, and the last disk must be removed again. It thus takes a long time to fill the box. In this chapter, we study algorithms that do this much faster: they go from time t = 4262 to t = 20332 in one step, and also find out that the box is then full and that no more disks can be added. All problems considered in this chapter treat dynamic problems, where time dependence plays an essential role.

Read the first three pages of Chapter 7 (pdf)

The sections of Chapter 7 are as follows: (complete Table of contents, including subsections (pdf).

7.1 Random sequential deposition

7.2 Dynamic spin algorithms

7.3 Disks on the unit sphere

Exercises

References