Well, one semester down, one to go. And with it, a new array of classes. I am pleased to tell you, dear readers, that I have another statistical mechanics class, and that I will be continuing the blogging my class series for this new class. So, without further ado, allow me to present the first BMC post of 2011!
With a whole year of BMC posts on statistical mechanics, I haven’t yet actually said what makes statistical mechanics different than any other aspect of physics. While this may not be the only difference, the primary tool that makes statistical mechanics unique is the use of “statistical ensembles”. These are tools used to formalize the action of complex systems by representing them as a large sum of all possible states the system can occupy. These tricks are full of mathy goodness, and are all based on a few simple principles. First, they can deal with systems that would be impossible to consider one part at a time. They also rely on the “Ergodic Principle”, the assumption that any possible state will be hit given enough time. These systems allow physicists to calculate macroscopic properties like pressure and volume for systems, simply by looking at the microscopic properties like energy states and chemical potentials.
The Holy Trinity of Statistical Ensembles
While you could invent any number of statistical ensembles, the three most useful/common are the Microcanonical Ensemble, the Canonical Ensemble, and the Grand Canonical Ensemble. These all build on each other, and each is more powerful (but at the same time more work to use) than the last. The microcanonical ensemble uses an array of states, and simplifies calculation with the assumption that no energy or matter can flow in or out of the system, and that the volume is fixed. This tool is useful for calculating the properties of thermally isolated systems like bags of snack mix (if pretzels, chips, and cheetos are the particle types, then you can treat these as thermodynamic systems. Weird, eh?). The canonical ensemble eases the restriction on the flow of energy, allowing heat to flow to or from the system. This ensemble is extremely powerful, allowing a lot more types of systems to be easily examined. The grand canonical ensemble takes the extension even further, allowing the number of particles in the system to change. This is really handy for looking at phenomena like diffusion, where particles move from areas of high concentration to low concentration. The grand canonical ensemble works by adding in additional term, the “chemical potential”. This is similar to any other energy potential, in that it pulls a system towards some equilibrium. It deals with how much the system’s energy would change with the introduction of more particles. Naturally, this new term allows the grand canonical ensemble to account for variable numbers of particles.