Physics Lecture Notes Statistical Mechanics

The purpose of thermodynamics is to describe the properties of various macroscopic systems at, or near, equilibrium. This is done with the help of several state variables, such as the internal energy E, the volume V , the pressure P, the number of particles N, the temperature T, the entropy S, the chemical potential µ, and others. These state variables depend only on what the state of the system is, and not on how the system was brought to that state. The variables are not all independent, and thermodynamics provides general relations among them. Some of these relations are not specific to particular systems, but are valid for all macroscopic systems.

Condensed matter systems where interactions are strong are inherently difficult to analyze theoretically. The situation is particularly interesting in low-dimensional systems, where quantum fluctuations play a crucial role. Here, the development of non-perturbative methods and the study of integrable field theory have facilitated the understanding of the behavior of many quasi one- and two-dimensional strongly correlated systems.

Although used with increasing frequency in many branches of physics, random matrix ensembles are not always sufficiently specific to account for important features of the physical system at hand. One refinement which retains the basic stochastic approach but allows for such features consists in the use of embedded ensembles.

This book demonstrates the usefulness of tools from statistical mechanics for biology. It includes the new tendencies in topics like membranes, vesicles, microtubules, molecular motors, DNA, protein folding, phase transitions in biological systems…

Over the past few decades the powerful methods of statistical physics and Euclidean quantum field theory have moved closer together, with common tools based on the use of path integrals.

This book provides a rapid overview of the basic methods and concepts in mechanics for beginning Ph.D. students and advanced undergraduates in applied mathematics or related fields. It is based on a graduate course given in 2006-07 at the Courant Institute of Mathematical Sciences. Among other topics, the book introduces Newton's law, action principles, Hamilton-Jacobi theory, geometric wave theory, analytical and numerical statistical mechanics, discrete and continuous quantum mechanics, and quantum path-integral methods.