The first law of thermodynamics says that heat is a form of energy. There is a lot of heat about! For instance, the amount of heat energy it would take to change the temperature of the world’s oceans by one degree is about \(6 \times 10^{24}\) joules. That is four orders of magnitude greater than the world’s annual energy consumption! So, if we could somehow how to figure out how to extract one degree’s worth of heat energy from the oceans, we could power the world for ten thousand years! Continue reading

# Thermodynamics II: gases and the First Law

Recall from the previous post that the First Law of Thermodynamics can be expressed

\[ \alpha+\beta = -dU \]

where \(U\) is the total energy of a thermodynamic system and \(\alpha,\beta\) are one-forms whose integrals along a transformation express the work done by the system on its environment, and the heat supplied by the system to its environment, in the course of the transformation. (In Fermi’s notation \(\alpha=dL\), \(\beta=-dQ\), where \(L,Q\) should be thought of as functions not on the state space but on the *path space* of the state space – their values depend on how you got there). If the state space is a 2-manifold parametrized by volume \(V\) and pressure \(p\), then \(\alpha=pdV\). The **thermal capacity** of the system is the derivative \(dQ/dT\), that is the marginal amount of heat absorbed for an increase in temperature. There are two versions of the heat capacity, \(C_V\) (heat capacity at constant volume) and \(C_p\) (heat capacity at constant pressure). For the situation we’re considering, \(\alpha=pdV = 0\) at constant volume, so

\[ C_V = \left(\frac{\partial U}{\partial T}\right)_V,\qquad C_p = \left(\frac{\partial U}{\partial T}\right)_p + p\left(\frac{\partial V}{\partial T}\right)_p \]

by standard calculations with partial derivatives. Continue reading

# Thermodynamics

I’ve been trying to learn a bit about classical *thermodynamics*, using Fermi’s lecture notes which are available as a low-cost Dover reprint.

That’s partly just because the subject has always been a bit mysterious to me and I would like to understand it better, but also because the Second Law of thermodynamics often gets invoked in environmental discussions – and I wonder whether it is being used accurately. (See this blog post for extended discussion about that.)

As a mathematician, I expected the discussion of thermodynamics to be *statistical*, heavily engaged with probability theory. But the main text of Fermi’s book is not about statistical mechanics at all. Instead, it is about *classical* thermodynamics; the nineteenth century theory that attempted to quantify the properties of that mysterious fluid, “heat”, and its transmission from one body to another. Continue reading

# Property A and ONL, after Kato

Hiroki Sato’s paper on the equivalence of property A and operator norm localization was recently published in *Crelle* ( “Property A and the Operator Norm Localization Property for Discrete Metric Spaces.” *Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)* 2014 (690): 207–16. doi:10.1515/crelle-2012-0065.) and I wanted to write up my understanding of this result. It completes a circle of proofs that various forms of “coarse amenability” are equivalent to one another, thus underlining the significance and naturalness of the “property A” idea that Guoliang came up with twenty years ago. Continue reading

# Dydak’s axiomatization of Euclidean geometry

Jerzy Dydak sent me an interesting paper a couple of months ago, which you can find on his website here. Entitled “A Topological Approach to the Axiomatization of Geometry”, it proposes a new way of building Euclidean plane geometry from he ground up.

Readers will know that although Euclid was the first to propose an axiomatic foundation for geometry, his axiom system is not precise enough by modern standards. In the 19th century Hilbert gave a complete system of axioms for geometry, and other mathematicians followed in his steps. In 1932 G.D.Birkhoff published a paper called *A system of axioms for plane geometry based on scale and protractor. *This appeared in the *Annals of Mathematics*! Birkhoff’s innovation was to assume the real numbers as given: his axioms stated that certain geometric quantities could be “measured” by real numbers. This was of course quite different from Euclid’s approach, where the “theory of proportion”, equivalent to what we would call today the theory of the real numbers, was developed as part of geometry (*Elements*, book 5). Birkhoff’s approach has been followed by many later writers of textbooks (such as my own *Elementary Geometry* for example).

Dydak wants to get back to the Euclidean order of business where the real numbers are developed concurrently with the foundations of geometry. The fundamental undefined notion in his theory is the ternary relation of *betweenness* (as in “C is between A and B”). This allows one to define line segments, rays, and so on. What would conventionally be called “the completeness of the real numbers” is expressed in terms of connectedness and compactness properties of line segments (themselves defined in terms of betweenness, of course); and the “arithmetic” of real numbers appears as the result of adjoining and subdividing line segments, very much in the Euclidean manner.

It would be interesting to try teaching geometry this way. You spend a lot of time thinking about one-dimensional geometry in this development; but, as a bonus, you understand the foundations of the real number system in an intuitively appealing way. Dydak also suggests that the concept of lines as defined by an abstract “betweenness” might help make models of other geometries, such as the Klein model or Poincare model of hyperbolic geometry, less mysterious: no more “we are going to call this a ‘line’ even though it doesn’t look like one.”