Books

This is a very subjective list of books which I like and which cover topics that fit well with Toy Problems. Their level of difficulty varies greatly, but I have placed emphasis on books which should be accessible to most people with some undergraduate Math and Sciences.

In no way is this a comprehensive list of relevant books, but more an invitation to a wonderful world - many books on this list share the excitement of understanding this world with their readers.

Some books have been included because they are such particular favorites of the editor, although they have no direct bearing on the topics discussed on ToyProblems. In such cases, a smaller font has been used.



Physics

Basic Physics

The Feynman Lectures on Physics (3 Vols) by Richard P. Feynman, Robert B. Leighton, Matthew Sands (Addison-Wesley, repeatedly re-issued)
The legendary classic. An introduction into basic physics from Feynman's unique view point. "Elementary" treatment, but not an easy read. Rather self-contained, including introductions to mathematical topics such as Algebra, Complex Numbers, and Vector Calculus on the same "elementary" (but not easy) level. Stresses deep, conceptual insights, but not phenomenological details and niche topics. The second and third volumes (on Electromagnetism and Quantum Mechanics, resp.) are particularly recommended. Only the choice of topics shows the age - it would have been fascinating to hear Feynman's opinion on modern topics such Chaos Theory, Soft Condensed Matter, and Biological Physics.
Physik by Christian Gerthsen, et. al. (Springer - repeatedly re-issued, under varying titles and with different co-authors)
A treasure trove of phenomenological facts from all parts of physics. Includes many interesting exercises. Not the place to go for deep, conceptual explanations or insightful derivations, but a tremendous compendium of basic Physics. In German


Classical Mechanics

Klassische Mechanik by Friedhelm Kuypers (Wiley-VCH, 7th ed, 2005)
An exceptionally clear exposition, which brings out the important concepts with particular clarity. Several hundred fully worked problems are an essential part of the book. In German.
Classical Dynamics: A Contemporary Approach by Jorge V. José, Eugene J. Saletan (Cambridge, 1998)
A substantial re-work of an earlier text by Saletan and Cromer. Modern treatment throughout. Combines broad coverage of traditional topics in Classical Mechanics, with substantial chapters on modern topics such as Symplectic Transformations, Non-linear Dynamics, and Solitons. May provide more breadth than depth.


Thermodynamics

Thermodynamics and an Introduction to Thermostatistics by Herbert B. Callen (Wiley, 2nd ed, 1985)
Thermodynamics ("Gas-bottle physics") is developed as an independent topic, based on a set of very general axioms. It is remarkable how far-reaching predictions can be made based on nothing but very weak assumptions. A model for the development of theories. The classic in the field.


Statistical Mechanics

Statistical Mechanics by Kerson Huang (Wiley Interscience, 2nd ed. 1987)
Probably the quickest and most elementary way to Ensemble Theory in Statistical Mechanics. Does not emphasize deep understanding, but gets you doing homework problems quickly.
Statistical Mechanics by R. K. Pathria (Butterworth-Heinemann 2nd ed, 1996)
Rigorous and comprehensive. The current edition is a significant re-write of a pre-Renormalization-Group classic. As with many such deep revisions, I am not entirely sure that the revised version retains all the qualities that made the original a classic.
Equilibrium Statistical Physics by Michael Plischke, Birger Bergersen (World Scientific, 2nd ed, 1994)
One of the first books to emphasize model-oriented statistical mechanics at in favor of ideal quantum systems. Rather comprehensive, but brief. Expects previous knowledge on the level of Huang's book (cf. above), but serves as a very handy reference for the more advanced reader.
Equilibrium and Non-Equilibrium Statistical Thermodynamics by Michel Le Bellac, Fabrice Mortessagne, G. George Batrouni (Cambridge University Press, 2004)
Statistical Physics: Statics, Dynamics and Renormalization by Leo P. Kadanoff (World Scientific, 2000)

Renormalization Group Theory

Scaling and Renormalization in Statistical Physics by John Cardy (Cambridge University Press, 1996)
A small volume of lecture notes, stressing the conceptual meaning of Renormalization Group Theory, instead of the technical formalism. Not an easy read and pre-supposes at least some prior knowledge of Statistical Mechanics.
Introduction to Renormalization Group Methods in Physics by Richard J. Creswick, Horacio A. Farach, Charles P. Poole (John Wiley, 1991)
Out of print.
Renormalization Methods: A Guide for Beginners by W. D. McComb (Oxford University Press, 2004)
Lectures on Phase Transitions and the Renormalization Group by Nigel Goldenfeld (Westview Press, 1992)
Quantum and Statistical Field Theory by Michel Le Bellac (Oxford University Press, 1992)

Liquids and Soft Condensed Matter

Basic Concepts for Simple and Complex Liquids by Jean-Louis Barrat, Jean-Pierre Hansen (Cambridge University Press, 2003)
This rather short book provides a very pleasant introduction to the Physics of classical (i.e. non-quantum) liquids, including their thermodynamics, structure, and dynamics.
Soft Condensed Matter by Richard A.L. Jones (Oxford University Press, 2002))
A nice phenomenological introduction to the Physics of polymers, gels, colloids, and liquid crystals.
Soft Matter Physics by Mohamed Daoud (Editor), Claudine E. Williams (Editor) (Springer, 1999)
A collection of fascinating, but rather advanced reviews by researchers mostly from the "French School" on topics in soft condensed matter, including capillarity and wetting, colloids and surfactants, polymers and liquid crystals.
Biological Physics: Energy, Information, Life by Philip Nelson (W. H. Freeman, 2003)
A book on the undergraduate level on the physical foundations of microbiology, mostly probability and thermodynamics of small systems. Expensive.


Quantum Mechanics

Quantum Mechanics (2 Vols.) by Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe (Wiley-Interscience, 1992)
Big, fat, comprehensive. Relatively easy to read. Employs a unique format, in which applications and advanced topics are delegated to chapter Appendices, so as not to break up the main flow.
Modern Quantum Mechanics by J. J. Sakurai (Addison Wesley, 2nd ed, 1994)
Short, concise, succinct. Rather terse in places. Expects (but does not rigorously require) at least some previous knowledge of the subject matter. The exposition is particularly clear and strives towards a "canonical" treatment.

Quantum Mechanics of Condensed Matter Systems

A Quantum Approach to Condensed Matter Physics by Philip L. Taylor, Olle Heinonen (Cambridge University Press, 2002)
A book on the quantum theory of (mostly) the solid state - i.e. phonons and electrons. Employs second quantization throughout, but spends no time on field-theoretic formalism. The treatment and choice of topics are up-to-date.


Path Integrals

Path integrals are a fascinating topic: they often seem like a particularly intricate prank. Clearly, they must be a tremendous solution - but what is the problem?

Quantum Mechanics and Path Integrals by Richard P. Feynman, A. R. Hibbs (McGraw-Hill Companies, 1965)
The absolute classic. The early chapters provide the most lucid introduction to path integrals to date. Currently out of print (but watch for a re-print edition).
Techniques and Applications of Path Integration by L. S. Schulman ( Wiley-Interscience, 1981; re-published by Dover, 2005)
This book consists of two parts: a very approachable introduction to path integrals, and a collection of miscellaneous applications, some of which quite whimsical. A very entertaining book.


Mathematics

Number Theory

A Primer of Analytic Number Theory: From Pythagoras to Riemann by Jeffrey Stopple (Cambridge University Press, 2003)
An introduction to analytic number theory, requiring nothing more than basic calculus. A remarkable achievement.
Making Transcendence Transparent: An intuitive approach to classical transcendental number theory by Edward B. Burger, Robert Tubbs (Springer, 2004)
An introduction to the theory of transcendental numbers, succeeding in making this apparently esoteric subject surprisingly accessible. Despite its fanciful title, this is not a "recreational" mathematics book - it requires real work, but few other formal prerequisites.


Linear Algebra

Linear Algebra Done Right by Sheldon Axler (Springer, 2nd ed, 2004)
The book lives up to its (pompous) title. Linear Algebra is treated strictly as the theory or linear mappings. Determinants are avoided and connections with systems of linear equations are de-emphasized. This approach achieves a tremendously clear and coherent presentation, and allows concentration on the important underlying concepts. Avoidance of determinants (and, therefore, the tedious calculation of exact eigenvalues and the associated difficulties that the characteristic polynomial may fail to factor) opens the road for more general theorems regarding the existence of at least some eigenvalues. For completeness sake, determinants are introduced in the last chapter. Easily accessible to an undergraduate.


Differential Equations

Ordinary Differential Equations by Garrett Birkhoff, Gian-Carlo Rota (Wiley, 4th ed, 1989)
A relatively compact and practical presentation of the standard material on Ordinary Differential Equations. Modern presentation despite its age (originally published in 1959, the current edition is a re-print of the 1978 edition), including a discussion of stability theory for 2-dimensional systems. Beware of reported misprints, in particular in the exercises!
Ordinary Differential Equations: A Brief Eclectic Tour by David A. Sánchez (MAA, 2002)
This is not actually a book for students, but for instructors. The author tries to point out those topics which are essential to Ordinary Differential Equations, and which are mostly auxiliary to certain solution approaches (such as Linear Algebra or Numerical Analysis). A refreshing road map for the confused.
Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory by Carl M. Bender, Steven A. Orszag (McGraw-Hill, 1978; re-published by Springer in 1999)
If this book were a movie, it would be X-rated: It is strictly for grown-ups! The authors provide an in-depth discussion of methods to obtain accurate approximations to differential equations. It is striking to see what methods exist to go beyond the usual "leading order behavior" typically employed in theoretical physics and how good (numerically!) even low-order approximations can be made. More striking still is how the common confusion regarding ordinary differential equations is overcome in this volume by placing the usual text-book results in a much wider context. Amazing, and thoroughly funny in places.


Dynamical Systems

Dynamical Systems are properly a subset of the theory of differential equations, but with a focus on non-linear equations and therefore an emphasis on approximations and qualitative behavior.

Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering by Steven H. Strogatz (Addison-Wesley, 1994; re-published by Perseus Books Group, 2001)
Advanced undergraduate level. The book was originally written during the hey-day of dynamical systems theory and breathes the excitement of that time. It contains a wealth of fascinating problems, but runs the danger of leaving the reader with the same sensation as a Chinese banquet: full, but quite confused as to what has just been consumed.
Differential Equations, Dynamical Systems, and an Introduction to Chaos by Morris W. Hirsch, Stephen Smale, Robert Devaney (Academic Press, 2nd ed, 2003)
Beginning graduate level. To achieve a more "canonical" treatment, the authors intentionally do without many of the exciting applications. The book tries to be self-contained (carefully introducing all required linear algebra and eigentheory in the early chapters), but at the price of getting of to a slow start for more advanced readers.
Chaos in Dynamical Systems by Edward Ott (Cambridge University Press, 2nd ed, 2002)


Functional Analysis, Fourier Theory, and Distributions

Fourier Analysis and Its Applications by Gerald B. Folland (Brooks Cole, 1992)
A wonderful book covering Fourier Series (and other orthogonal function systems) and their applications to partial differential equations in the first part, and Fourier Transforms and the theory of distributions in the latter part. A very successful mixture of theory and application and striking a level of mathematical sophistication well-matched to the needs of the theoretical physicist. Highly recommended. May unfortunately be hard to find.
An Introduction to Fourier Analysis and Generalised Functions by M. J. Lighthill (Cambridge University Press, 1958)
A short, theoretical classic. May be too concise to serve as a first introduction to the overall topic, but heads and shoulders above many of the "application" centered texts on Fourier Theory.

Wavelets

Wavelets Made Easy by Yves Nievergelt (Birkhauser, 2000)
A Primer on Wavelets and Their Scientific Applications by James S. Walker (CRC, 1999)


Probability Theory and Statistics

An Introduction to Probability Theory and Its Applications (Vols 1 and 2) by William Feller (Wiley, 3rd ed, 1968 and 2nd ed, 1971)
A timeless classic. The first volume (on discrete sample spaces) is particularly accessible.
Data Analysis: A Bayesian Tutorial by Devinderjit Sivia, John Skilling Oxford University Press, 2nd ed, 2006)
A short and pragmatic introduction to the Bayesian interpretation of Statistics, with many examples from Physics experiments. Succeeds in making the Bayesian viewpoint intuitively plausible. Does not lead far, but serves as excellent primer for further study.


Time Series Analysis

The Analysis of Time Series: An Introduction by Chris Chatfield (Chapman and Hall/CRC, 6th ed, 2003)
Relatively short and pragmatic. Written for the mathematically educated, but assumes no previous knowledge of either time series or stochastic processes.


Fractals (Mathematics)

The Fractal Geometry of Nature by Benoit B. Mandelbrot (W. H. Freeman, 1982)
The classic in the field, written by the man who single-handedly invented the fractal concept. In large parts accessible to the non-specialist.
Fractal Geometry: Mathematical Foundations and Applications by Kenneth Falconer (John Wiley, 2nd ed, 2003)
This is one of the very few books out there which attempts to get away from the naively-intuitive approach and to put the theory of fractals on a more formal mathematical foundation. The formality is there, but it is not easy to see how this helps our understanding of the topic.


Numerics, Computation and Simulation

The purpose of computation in science is insight, not numbers!

Richard Hamming

Numerics

Numerical Methods for Scientists and Engineers by Richard Hamming (Dover, 2nd ed, 1987)
Technically obsolete, this book is still valuable because of its motto (cf. above) and because of its last chapter, which tries to bring some much needed perspective to the use of computation in Science.
Numerical Methods that (usually) Work by Forman S. Acton (The Mathematical Association of America, 1997)
Originally published in 1970 and re-printed (without changes!) in 1990 and again in 1997, this book is probably the best introduction to the Numerical Analyst's mindset available. The emphasis is on understanding and intuition, and not on the details of any particular algorithm. Particular attention is drawn to the ways numerical algorithms can lead the unwary astray.
Numerical Recipes in C: The Art of Scientific Computing by William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling (Cambridge University Press, 2nd ed, 1992)
The ubiquitous "Numerical Recipes" is the worst book on Numerical Analysis, except for all the others that have been written (paraphrasing Winston Churchill). The book's strengths are its comprehensive coverage of topics, the authors' thoroughly practical attitude, and the fact that it comes with full implementations of all algorithms. It's weaknesses are fact that the authors' recommendations are often questionable, the references and algorithms are outdated, and that many of the implementations are awfull. Nevertheless, it is an incredibly empowering book, and, since in the end it is more important to solve the right problem the wrong way, than solving the wrong problem the right way, one of the most valuable books in the field.
Gnu Scientific Library: Reference Manual by Mark Galassi, Jim Davies, James Theiler, Brian Gough, Gerard Jungman, Michael Booth, Fabrice Rossi (Network Theory Ltd., 2nd ed, 2003)


Computation and Simulation

An Introduction to Computer Simulation Methods: Applications to Physical System by Harvey Gould, Jan Tobochnik (Addison Wesley, 2nd ed, 1995)
Computer Simulation of Liquids by M. P. Allen, D. J. Tildesley (Oxford University Press, 1989)
Computational Physics by J. M. Thijssen (Cambridge University Press, 1999)
A rather advanced (graduate level) text. The usual suspects MC and MD make an appearance, but most of the book deals with methods to solve the Schroedinger Equation (i.e. Quantum problems) in a variety of situations.


Interdisciplinary Topics

Non-Linear Systems (Phenomenology)

Non-Linear Physics for Beginners: Fractals, Chaos, Pattern Formation, Solutions, Cellular Automata and Complex Systems by Lui Lam (Editor) (World Scientific, 1990)
Introduction to Nonlinear Physics by Lui Lam (Editor) (Springer, 2003)
Fractals and Disordered Systems by Shlomo Havlin (Editor), Armin Bunde (Editor) (Springer, 2nd ed, 1995)


"Chaos and Fractals"



Grand Synthesis

We seem to be standing on the cusp of a grand synthesis of ideas, linking fractals with scaling laws and phase transitions and renormalization group ideas. This synthesis has not yet happened, but there is a large collection of individual pieces which somehow seem to be aspects of a bigger whole.

The books below are fascinating catalogues of models and results. The book by Boccara is more accessible, the one by Sornette more comprehensive.

Modeling Complex Systems by Nino Boccara (Springer, 2003)
Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools by Didier Sornette (Springer, 2nd ed, 2006)


Virtual Worlds

The purpose of computing is insight, not pictures.

Nick Trefethen

The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation by Gary William Flake (The MIT Press, 2000)
An interesting and beautifully produced book with many fascinating models suitable for computational study. Unfortunately, the discussion remains strictly descriptive, and fails to provide a deeper understanding of the systems presented.
Armchair Universe: An Exploration of Computer Worlds by A. K. Dewdney (W.H. Freeman and Company, 1988)
Magic Machine: A Handbook of Computer Sorcery by A. K. Dewdney (W.H. Freeman and Company, 1990)
The Tinkertoy Computer and Other Machinations: Computer Recreations from the Pages of Scientific American and Algorithm by A. K. Dewdney (W.H. Freeman and Company, 1993)
All out of print.