Luiza Angheluta
<aside> 📖 Statistical mechanics provides us with microscopic descriptions of thermodynamic systems as ensembles of microscopic entities (e.g. atoms, spins, colloidal particles, etc.) interacting with each other and their surroundings according to the laws of classical/quantum mechanics. It is a “statistical” description of a thermodynamic state in the sense that thermodynamic variables - such as pressure, temperature, entropy, etc - can be determined by ensemble (statistical) averages of microscopic quantities - e.g. forces, velocities, collision rates, etc. In classical statistical mechanics, the microscopic particles follow the Newtonian dynamics. Classical fluids (liquid or gases) are described as ensembles of weakly-interacting atoms obeying Newton’s law of motion in the presence of internal forces generated by the mutual interaction potential (van der Waals-like potential) and external forces due to externally imposed conditions. This classical statistical mechanics description of fluids breaks down at very low temperatures where quantum effects become important, i.e. when atoms become quantum objects with an associated wavefunction. Luckily, concepts and methods from classical can be readily extended into the quantum realm with few, but very important modifications. One obvious modification is that classical mechanics needs to be replaced by quantum mechanics to describe the evolution of the particle wavefunctions. A more subtle, yet groundbreaking modification is at the statistical level and has to do with a distinct way in which we count quantum states to compute statistical averages. This quantum statistics is intrinsically related to the quantum nature of particles, i.e. bosons and fermions. Thus, we distinguish Bose-Einstein statistics obeyed by bosons from Fermi-Dirac statistics satisfied by fermionic systems. These two types of quantum statistics lead to emergent phenomena. The degeneracy pressure in Fermi gases at 0K which keeps matter from collapsing under its own gravitational pull. Bose gases below a critical temperature tend to collective crowd the ground state and form an emergent state of matter called a Bose Einstein condensate with distinct exotic flow properties like superfluidity. Yet, at high enough temperatures, both fermions and bosons behave as classical Newtonian particles obeying the Maxwell-Boltzmann statistics.
As it turns out concepts from statistics are fundamental for counting microscopic configurations and computing average properties. This lecture is intended to briefly review key concepts from probability and combinatorics, and apply them to describe thermodynamic states (also called macrostates) as ensembles of microscopic configurations (also called microstates). For more on details on microstates and macrostates check out this short video
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