The Second Law of Thermodynamics has been called the most important law of nature: It is the law that gives a direction to processes that is not inherent in the laws of motion, that says the state of the universe is driven to thermal equilibrium. Its mathematical formulation is simple: The entropy of a closed system cannot decrease. Since the recognition that macroscopic phenomena have an atomic basis, it has remained a fundamental problem to reconcile the increase of entropy with the known reversibility of all the laws of microscopic physics. Professor Michael Mackey of McGill University here explores the dynamical basis for the Second Law, that is, he seeks to illuminate the fundamental dynamical properties required for the construction of a successful statistical mechanics. Aimed at physicists and applied mathematicians with an interest in the foundations of statistical mechanics, the book includes such new material as: a demonstration that the black body radiation law can be deduced from maximal entropy principles; a discussion of sufficient conditions for the existence of at least one state of thermodynamic equilibrium; a description of the behavior of entropy in asymptotically periodic systems; a necessary and sufficient condition for the evolution of entropy to a global maximum; and a presen- tation of the three main types of ergodic theorems and their proofs. He also explores the potential role of incomplete knowledge of dynamical variables, measurement imprecision, and the effects of noise in giving rise to entropy increases.

The Schrodinger equation is the basic equation of quantum theory. The study of this equation plays an exceptionally important role in modern physics. From a mathematician's point of view the Schrodinger equation is as inexhaustible as mathematics itself.

Newton's polyhedron of a polynomial in several variables is the convex hull of the set of exponents of its monomials completed in some way. The boundary of Newton's polyhedron can be interpreted as one of the possible generalizations of the degree of a polynomial in one variable to the case of several variables. This is a much more informative notion than that of the ordinary degree. At the same time, it is not invariant with respect to linear transformations of coordinates and thus is related to a fixed coordinate system. An intermediate notion between the degree and Newton's polyhedron is the notion of a weighed degree in which the calculation of the "degree" of a monomial is performed by means of summation of the degrees of the various variables with different weights. Newton's polyhedron accunmlates information about the various weighted degrees and principal parts which correspond to its faces.