Statistical mechanics

Statistical mechanics is the mathematical investigation of the mass behavior of very large systems of "particles" based on the microscopic laws that govern these particles. Such laws might include the way individual particles interact with their nearest-neighbors, for example. Statistical mechanics provides the formal (i.e. theoretical) foundation for the area of physics known as thermodynamics.

An important part of statistical mechanics is developing theorems about phase transitions. When the micro-level behavior of the system depends on some parameter, such as (for a physical example) the temperature, the system may spontaneously undergo a significant change in its bulk behavior as this parameter increases past a critical point.

Another example of such a parameter is the probability that vertices in a randomly-generated graph will have an edge between them. It has been shown (that is, formally proven) that when this probability is low, the random graph thus generated will be exemplified by having a large number of disconnected subgraphs (i.e. components), but as the probability of attachment increases, there is a phase-transition in the kinds of graph that result. Once the critical probability has been passed, the random graphs will suddenly be "dominated" by a single, very large component.

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