What is Statistical Mechanics? To answer, we need to have a little detour and start to talk about another topic, namely Thermodynamics.
Thermodynamics is interested in the relationship among a small number of macroscopic quantities of a body. e.g. pressure, volume, temperature, magnetisation, etc. As such, the basic concepts in thermodynamics rely directly on empirical, experimental confirmation. The power and appeal of thermodynamics is that its predictions are general and robust, and without reference to the microscopic nature of a system: Einstein, once asked about a theory of matter that would change very little or not at all over the next two hundred years, stated that he would make his bet on Thermodynamics.
In a mathematical perspective, thermodynamics has to do with mathematical identities involving derivatives of well defined function. In a physical sense, it has to do first and foremost with the equilibrium properties of a systems, but also transformations such as heating, cooling, melting, volume compression, (de)-magnetisation, etc, chemical reactions, conversion of mechanical work into heat, conversion of heat into mechanical work, etc.
Statistical mechanics is concerned with explaining how such properties and transformation occur as a consequence of the microscopic properties of a system. At small scales, e.g. at the quantum scale, things change incessantly in time, due to electronic, atomic and molecular motion and collisions. How to reconcile the micro- and macro- pictures? For example, how to understand that, while both Newton’s and Schrodinger’s equations appear to be reversible in time at the microscopic level (i.e. without preferred direction of time), on the macroscopic scale we often observe irreversible changes, like at the breaking of a glass, or the mixing of coffee with milk in a cup ? Why does a broken glass not recompose spontaneously? Why do not coffee and milk in a cappuccino mutually separate spontaneously? Or, how is it possible that in thermodynamical equilibrium, the temperature, the pressure, etc. of a body stay constant, given the never-ending motion of the micro-constituents of the body?
Statistical mechanics provides answers and explanations to these fascinating and deep questions, reconciling microscopic world and macroscopic observations. But the number of microscopic degrees of freedom of a macroscopic body is enormous. So one has to abandon a deterministic approach to the dynamics of each individual particle, and adopt a statistical perspective. Thus concepts like probability, averages, fluctuation become centerstage, and yet in this way a clear connection with the thermodynamical level of description can be established. This program can be accomplished both at the classical and quantum level.
What has been just described constitutes the essence of the course in statistical mechanics, where starting from a microscopic description of matter,
- the central concept of entropy is introduced,
- a connection with thermodynamical quantities is established, and
- paradigmatic systems such as the ideal (quantum an classical) gases are studied.
The overall topic is truly fascinating, since it shows in a beautiful way how what we experience in everyday life naturally emerges from the behavior of matter down at the fundamental space, time, and energy scales.
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