Physics
Statistical mechanics: collective phenomena and solvable models
Instructor: Mauricio Romo and Leonardo Santilli
Course Introduction: At the early 20th century the atomic theory of matter became well established. A few grams of a gas (such as Oxygen) will contain about 1023 atoms. Other basic constituents of matter, such as molecules, will also come in similar orders of magnitude even if we consider relatively small volumes.
This leads to the question, if we want to study the properties of matter, do we have to solve a coupled system of differential equations on ∼ 1023 variables? Besides being a technical and computationally formidable problem, the experimental observations seem to point out this solution is not the right approach.
Many body systems (many ∼ 1023, in nature) must require a completely new perspective: just using classical or quantum mechanics, based on deterministic equations (such as Newton's, Maxwell's or Schrodinger's equations, for example) is not enough. Hence, statistical mechanics, which comes in classical and quantum versions.
In this course, we will start from basic concepts of probability theory and show how they can be applied to understand the dynamics of systems with many constituents. By introducing essential concepts such as the ergodic hypothesis, we will motivate the theory of ensembles that constitutes the foundations of statistical mehcanics. These concepts, even though motivated by physics, can be connected with several branches of mathematics such as stochastic processes and measure theory. However, our main focus will be, through examples, to show why statistical mechanics is the key to unlock new phenomena that are not accessible purely from the properties of the individual constituents (microscopic properties). Such phenomena are known as 'collective phenomena' and present some of the most suprising applications of statistical mechanics. For instance, the existence of phase transitions in many body systems.
We plan to present several solvable models that will help us to understand deep concepts such as the Ising model, ideal gases and other lattice models. Last but not least, it should be emphasized that statistical mechanics, and its mathematics, not only have applications to the study of matter but also to quantum computing, random processes and economics.