The Cluster's Statistical Mechanics
A page about statistical mechanics — connecting microscopic laws to macroscopic behavior.
The statistical mechanics
Statistical mechanics is the branch of physics that connects the microscopic laws (quantum mechanics, classical mechanics) to the macroscopic laws (thermodynamics). It does this by counting the number of microstates compatible with a given macrostate. The entropy is S = k_B log W, where W is the number of microstates. In the cluster, statistical mechanics connects the microscopic edit laws (quantum edit theory, classical edit dynamics) to the macroscopic edit laws (cluster thermodynamics).
The microcanonical ensemble
The microcanonical ensemble describes an isolated system with fixed energy E, volume V, and particle number N. All microstates with energy E are equally probable. The entropy is S(E,V,N) = k_B log Omega(E,V,N), where Omega is the number of microstates with energy E. In the cluster, the microcanonical ensemble describes an isolated cluster section with fixed edit energy E, volume V, and edit number N.
The canonical ensemble
The canonical ensemble describes a system in thermal contact with a heat bath at temperature T. The probability of a microstate with energy E_i is P_i = exp(-beta E_i) / Z, where Z is the partition function. The free energy is F = -k_B T log Z. In the cluster, the canonical ensemble describes a cluster section in thermal contact with an edit bath at temperature T. The probability of edit microstate i is P_i = exp(-beta E_i) / Z.
The ergodic hypothesis
The ergodic hypothesis states that the time average of a physical quantity equals its ensemble average. This is the foundation of statistical mechanics. If the system is ergodic, then a single long trajectory samples all accessible microstates. In the cluster, the ergodic hypothesis states that the time average of an edit quantity equals its ensemble average. A single long edit trajectory samples all accessible edit configurations.
This mechanics
This page is about statistical mechanics. S = k_B log W. Omega(E,V,N) is the number of microstates. Z = sum exp(-beta E_i). P_i = exp(-beta E_i) / Z. F = -k_B T log Z. The ergodic hypothesis equates time and ensemble averages. The statistics are real.