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Statistical Mechanics  Calculating Equilibrium Averages
According to statistical mechanics, the probability that a given state with energy E is occupied in equilibrium at
constant particle number N, volume , and temperature T (constant
, the `canonical' ensemble) is proportional to
, the `Boltzmann factor.'

(1) 
The equilibrium value of any observable O is therefore obtained by averaging over all states accessible to the system,
weighting each state by this factor.
Quantum mechanically, this averaging is performed simply by summing over the discrete set of microstates (Figure 1):

(2) 
where Z is the partition function:

(3) 
and is the expectation value of the quantity O in the energy eigenstate:

(4) 
Classically, a microstate is specified by the positions and velocities (momenta) of all particles,
each of which can take on any value. The averaging over states in the classical limit is done by
integrating over these continuous variables:
where the integrals are over all phase space (positions and momenta ) for the N particles in 3 dimensions.
When all forces (the potential energy V) and the observable O are velocityindependent,
the momentum integrals can be factored and canceled:

(5) 
where
is the total kinetic energy, and . As a result,
Monte Carlo simulations compare V's, not E's.
Next: Classical vs. Quantum Mechanics:
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Steinbach
20190201