molecules, one is often better able to understand, interpret, and simulate such condensed media systems. This is where the power of statistical mechanics comes into play A. The Distribution of Energy Among levels One of the most important concepts of statistical mechanics involves how a specified amount of total energy e can be shared among a collection of molecules and among the internal(translational, rotational, vibrational, electronic)degrees of freedom of these molecules. The primary outcome of asking what is the most probable distribution of energy among a large number n of molecules within a container of volume v that is maintained in equilibrium at a specified temperature t is the most important equation in statistical mechanics, the Boltzmann population formula E; /kT)Q This equation expresses the probability Pi of finding the system(which, in the case introduced above, is the whole collection of N interacting molecules)in its quantum state, where E, is the energy of this quantum state, T is the temperature in K, Q2, is the degeneracy of the jth state, and the denominator Q is the so-called partition function Q=2,Q, exp(-E,/kT) The classical mechanical equivalent of the above quantum Boltzmann population formula for a system with M coordinates(collectively denoted q) and M momenta( denoted p)is PAGE 2PAGE 2 molecules, one is often better able to understand, interpret, and simulate such condensed￾media systems. This is where the power of statistical mechanics comes into play. A. The Distribution of Energy Among Levels One of the most important concepts of statistical mechanics involves how a specified amount of total energy E can be shared among a collection of molecules and among the internal (translational, rotational, vibrational, electronic) degrees of freedom of these molecules. The primary outcome of asking what is the most probable distribution of energy among a large number N of molecules within a container of volume V that is maintained in equilibrium at a specified temperature T is the most important equation in statistical mechanics, the Boltzmann population formula: Pj = Wj exp(- Ej /kT)/Q. This equation expresses the probability Pj of finding the system (which, in the case introduced above, is the whole collection of N interacting molecules) in its jth quantum state, where Ej is the energy of this quantum state, T is the temperature in K, Wj is the degeneracy of the jth state, and the denominator Q is the so-called partition function: Q = Sj Wj exp(- Ej /kT). The classical mechanical equivalent of the above quantum Boltzmann population formula for a system with M coordinates (collectively denoted q) and M momenta (denoted p) is: