Best Book On Solid Mechanics
Nayra Waddles
This course is your starting point towards really understanding solid mechanics. You need to understand the basics to enable you to apply good practice in your finite element analysis, and this course will give you the knowledge you really need for a good understanding of the principles that every engineer or designer should know.
The course covers these topics in a concise and practical manner. Unlike traditional university courses, this course avoids lengthy mathematical derivations and highlights many practical examples to illustrate the application of solid mechanic theories in modelling and analysing engineering structures. Where possible, exercises that can be done by hand are included so that attendees can test their knowledge. Full solutions will be provided within a few weeks of completing the course.
Personal passwords are provided to allow you to access e-learning backup material via our discussion forum. Reading lists, homework submissions and supplementary information are all available via the discussion forum.
To get the most out of the course, participation in forum discussions is very much encouraged. Typically the forum remains open for 4 weeks after the last live session, giving you plenty of time to catch up with homework, review and ask questions.
Mechanics depends upon an understanding of statics where motionless bodies of any shape, size and material are considered to be in a state of equilibrium under the action of external forces (session 1).
However, each material behaves in a different way so that building a component from steel is very different than building it from plastic. Decisions on material usage requires an understanding of the constitutive or material relationships (session 3) so that the developing stresses when the body undergoes straining can be calculated.
In our desire to determine stresses in a certain component made of a certain material many hand calculations have been developed historically based on laboratory studies and in-field readings but understanding their presentation and their limitations is important (session 4).
Many components are part of larger systems and a very important part of understanding the statics is the understanding of the externally applied loads and boundary conditions or external restrictions that may exist (session 5).
The more math I read, the more I see concepts from statistical mechanics popping up -- all over the place in combinatorics and dynamical systems, but also in geometric situations. So naturally I've been trying to get a grasp on statistical mechanics for a while, but I haven't been very successful. I've skimmed through a couple of textbooks, but they tended to be heavy on the physical consequences and light on the mathematical underpinnings (and even to an extent light on the physical/mathematical intuition, which is inexcusable!)
I suspect that part of the problem is that, unlike the analogous situation with quantum mechanics, I'm not sure what mathematics I can fall back on if I don't "get" some statistical model. So, is there a good resource for statistical mechanics for the mathematically-minded?
A classic book on solvable two-dimensional models is Baxter's "Exactly Solved Models in Statistical Mechanics" now available in a new edition from Dover. The Yang-Baxter equation, of course, has many connections with important branches of mathematics. This book explains its origins and use in solving certain physically motivated models.
If you are looking for a book, the real answer is "not really". As a mathematician masquerading as a physicist (more often than not of a statistical-physical flavor) I have looked long, hard, and often for such a thing. The books cited above are some of the best for what you want (I own or have read at least parts of many of them), but I would not say that any are really good for your purposes.
Many bemoan the lack of The Great Statistical Physics text (and many cite Landau and Lifshitz, or Feynman, or a few other standard references while wishing there was something better), and when it comes to mathematical versions people naturally look to Ruelle. But I would agree that the Minlos book (which I own) is better for an introduction than Ruelle (which I have looked at, but never wanted to buy).
Other useful books not mentioned above are Thompson's Mathematical Statistical Mechanics, Yeomans' Statistical Mechanics of Phase Transitions and Goldenfeld's Lectures On Phase Transitions And The Renormalization Group. None of them are really special, though if I had to recommend one book to you it would be one of these or maybe Minlos.
You might do better in relative terms with quantum statistical mechanics, where some operator algebraists have made some respectable stabs at mathematical treatments that still convey physics. But really that stuff is at a pretty high level (and deriving the KMS condition from the Gibbs postulate in the Heisenberg picture can be done in a few lines) so the benefit is probably marginal at best.
It sounds like your goal is primarily to get a quick overview of how mathematicians use statistical physics, with lots of intuition, and lots of applications to nearby areas, but not necessarily much physics or any 'best possible' results. If that is the case, the Montanari/Mezard book `Information, Physics and Computation' is excellent. The emphasis is on teaching many different techniques, rather than on the statistical physics itself, and it is really written as a textbook rather than a reference book (i.e. the theorems are the easiest ones to understand, not the most powerful ones to use). In other words, it has exactly the mathematical underpinnings, but doesn't cover what many of the other books treat as central material.
We have finished writing an introductory book on (some aspects of) equilibrium statistical mechanics, with mathematicians (and mathematically-inclined physicists) in mind. A draft of the version of the complete book, as it was sent to its publisher (Cambridge University Press), can be downloaded from this page. It should more or less coincide with the final version (modulo corrections that will be made on the galley proofs), that we hope will be published mid-2017.
"Statistical Mechanics: Entropy, Order Parameters, and Complexity" by James Sethna (my favorite) and "Statistical Mechanics" by Kerson Huang are both really good books. Sethna's book is very readable and engaging; Huang's is more of a syatematic textbook.
Ruelle's book is mathematically rigorous but is aimed at people who already know something about the field. Baxter's book is incredibly valuable for what it covers, but it is highly specialized and provides no motivation for people who aren't already comfortable with the basic formalism.
And, there are many surveys and currently appears new reviews about some topics, some examples(I included pages of mathematicians that work on the subject, in their pages you can get some introductory texts):
There are apparently only a few books on rigorous results in statistical mechanics. David Ruelle's books are apparently standard, though I found them difficult to digest when I picked them up. One which I found more accessible is "Introduction to Mathematical Statistical Physics" by Minlos.
A personal note is that I found statistical mechanics very unintuitive and difficult to learn at first. I felt that the formalism didn't come together for me until I was familiar with a multitude of physical systems.
Two other books which are worthwhile I find R.B. Israel: Convexity in the Theory of Lattice gases. It has wonderful introduction by Wightman , which is like book in itself. it is limited in scope but is excellent in what it treats.T.C. Dorlas Statistical mechanics, is written by a mathematical physicist and covers many topics in a more rigorous way than most physics textbooks do.
It's certainly not a basic book, but Itzykson and Drouffe's "Statistical Field Theory" gives a good overview of the use of quantum field theory techniques in statistical mechanics. Some of the chapters are quite readable.
Start here: An Introduction to Thermal Physics by Dan Schroeder is an excellent introduction that provides a single, consistent mathematical underpinning providing the insight necessary to truly understand things like entropy and the differences between the classical and quantum cases of statistical mechanics. I can't emphasize enough how important the core ideas of that book are to understanding the foundations of statistical mechanics. (As an aside, Schroeder also happens to be the coauthor, with Michael Peskin, of An Introduction to Quantum Field Theory which has purportedly replaced Bjorken and Drell as the standard in that field, though this is only hearsay.)
This topic is old, but I'll still add my 2. I usually don't really like statistical mechanics books aimed at physicists, as they are often much more focused on computational techniques than on concepts. There are however very good lecture notes by Yoshi Oono, available on his page:
This book is aimed at physicists, but contains a very unusual amount of mathematical content, esp. in footnotes, with many references to the math. phys. literature. This book assumes that the reader already has some knowledge of this field. There are other lecture notes, aimed at undergraduates, on his page as well (I haven't looked closely at those, so I cannot comment on their quality):
Maybe I didn't read the replies well enough, but apparently no one mentions A. Khinchin's book Mathematical Foundations of Statistical Mechanics. I just started reading this book. It's definitely mathematical and specifically says that it's written for a mathematician. The notation is a bit old, but the book is very readable, and far, far better than any book of physics I've seen lately.
A really good first textbook for statistical mechanics is David Chandler's Introduction to Modern Statistical Mechanics. It's written by a physical chemist for senior undergraduates and does an excellent job distilling down the very fundamental material into a one-semester course at Berkeley. It's not at all math heavy.
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