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
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.
The Feynman Lectures on Physics (3 Vols)
(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.
(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
(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
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 and an Introduction to
(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.
(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.
(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
Equilibrium Statistical Physics
(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
Non-Equilibrium Statistical Thermodynamics
(Cambridge University Press, 2004)
Statics, Dynamics and Renormalization
(World Scientific, 2000)
Renormalization Group Theory
Scaling and Renormalization
in Statistical Physics
(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.
Renormalization Group Methods in Physics
(John Wiley, 1991)
- Out of print.
A Guide for Beginners
(Oxford University Press, 2004)
Lectures on Phase Transitions
and the Renormalization Group
(Westview Press, 1992)
Quantum and Statistical Field Theory
(Oxford University Press, 1992)
Liquids and Soft Condensed Matter
for Simple and Complex Liquids
(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
(Oxford University Press, 2002))
A nice phenomenological introduction to the Physics of
polymers, gels, colloids, and liquid crystals.
Soft Matter Physics
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.
Energy, Information, Life
(W. H. Freeman, 2003)
A book on the undergraduate level on the physical foundations
of microbiology, mostly probability and thermodynamics of small
Quantum Mechanics (2 Vols.)
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
(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
(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 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
(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
( 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.
A Primer of Analytic Number Theory:
From Pythagoras to Riemann
(Cambridge University Press, 2003)
An introduction to analytic number theory,
requiring nothing more than basic calculus. A remarkable
Making Transcendence Transparent:
An intuitive approach to classical transcendental
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
Linear Algebra Done Right
(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.
Ordinary Differential Equations
(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
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
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 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
Nonlinear Dynamics and Chaos: With Applications
to Physics, Biology, Chemistry and Engineering
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
(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
(Cambridge University Press, 2nd ed, 2002)
Functional Analysis, Fourier Theory, and Distributions
Fourier Analysis and Its Applications
(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
(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 Made Easy
A Primer on Wavelets
and Their Scientific Applications
Probability Theory and Statistics
An Introduction to Probability Theory
and Its Applications (Vols 1 and 2)
(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
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
Time Series Analysis
The Analysis of Time Series:
(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.
The Fractal Geometry of Nature
(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
Mathematical Foundations and Applications
(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!
for Scientists and Engineers
(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
(The Mathematical Association of America,
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
Numerical Recipes in C:
The Art of Scientific Computing
(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
(Network Theory Ltd., 2nd ed, 2003)
Computation and Simulation
An Introduction to Computer Simulation Methods:
Applications to Physical System
(Addison Wesley, 2nd ed, 1995)
Computer Simulation of Liquids
(Oxford University Press, 1989)
(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.
Non-Linear Systems (Phenomenology)
Non-Linear Physics for Beginners:
Fractals, Chaos, Pattern Formation, Solutions,
Cellular Automata and Complex Systems
(World Scientific, 1990)
Introduction to Nonlinear Physics
Fractals and Disordered Systems
(Springer, 2nd ed, 1995)
"Chaos and Fractals"
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
Modeling Complex Systems
Critical Phenomena in Natural Sciences:
Chaos, Fractals, Selforganization and Disorder:
Concepts and Tools
(Springer, 2nd ed, 2006)
The purpose of computing is insight, not pictures.
The Computational Beauty of Nature:
Computer Explorations of Fractals, Chaos, Complex Systems,
(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.
An Exploration of Computer Worlds
(W.H. Freeman and Company, 1988)
A Handbook of Computer Sorcery
(W.H. Freeman and Company, 1990)
The Tinkertoy Computer and Other Machinations:
Computer Recreations from the Pages of Scientific American and
(W.H. Freeman and Company, 1993)
- All out of print.