Computer simulation 
microscopic information (i.e. atomic positions, velocities, etc.)
Goal of statistical mechanics:
understand and predict macroscopic phenomena from microscopic or molecular information
microscopic information macroscopic properties
(i.e. pressure, internal energy, etc.)
A. Definitions
Assume we have a system with N particles
phase space: 6N-dimensional space consisting of the momenta and spatial coordinates of the N particles
momentum: 
(3N-dimensional vector)
where 
, 
,
and 
are mass, velocity, and momentum of particle i
spatial coordinates: 
(3N-dimensional vector)
represents a point in phase space: 
ensemble
-a collection of a very large number Z of systems
-each system is a replica (on a macroscopic level) of the system of interest
-each system has the same chosen fixed macroscopic parameters
(i.e. NPT,NVT,NVE, etc.)
-each system evolves independently in time
-a collection of points in phase space, each one representing one of the Z systems
at a particular instant of time
probability density 
-points in an ensemble are distributed according to probability density
- determined by the chosen fixed macroscopic parameters (NPT, NVT, etc.)
: a weight function describing the relative probabilities for points
partition function: sum over states
probability density:
is normalized: 
is an unnormalized version of
Partition function 
is connected to a thermodynamic quantity
:
B. Ensembles
1.Microcanonical ensemble
Constant N,V,E
The density is:
The delta function selects out those states of an N-particle system in a container of volume V that have energy E
The partition function is:
The appropriate thermodynamic potential is the negative of the entropy:
Recall the equation from thermodynamics: 
where N is
the number of microscopic states.
The classical mechanical equations of motion conserve energy and thus provide a
useful way to sample states in this ensemble. From a practical standpoint,
numerical errors (roundoff and turncation) during the integration process cause a drift in energy.
2. Canonical ensemble
Constant N,V,T
The density is:
The partition function is:
The appropriate thermodynamic potential is the Helmholtz free energy:
For this ensemble, all values of the energy are allowed. Thus, the classical
mechanical equations of motion (where energy is conserved) are not an adequate way to sample states in this ensemble.
However, for very large N it can be shown that the canonical and microcanonical
ensembles are practically equivalent (i.e. the density for the canonical ensemble
as a function of energy has a very sharp maximum for some particular value of the
energy E).
See Statistical and Thermal Physics by Reif, p.94-99
Thus, this ensemble can be obtained using the classical mechanical equations of motion with temperature control devices.
3. Other ensembles
Isothermal-isobaric ensemble
-constant N,P,T
-volume V varies
-thermodynamic potential is Gibbs free energy
Grand canonical ensemble
-constant 
,V,T
-number of particles N varies
-thermodynamic potential is -PV
C. Ensemble averages and ergodicity
Ensemble average:
average value of a mechanical property A over all members of the ensemble
General assumption:
the experimentally observable macroscopic property 
= the time average of 
over a long time interval:
Ergodic:
A system is ergodic if there is at least one trajectory that passes through all
points in phase space (for which 
is non-zero) so each system
will visit all these points
the average properties could be deduced from a snapshot rather than following the complete trajectory of one system
replace the time average by an average taken over all members of the ensemble frozen at a particular time:
Ensembles are equivalent
For commonly used ensembles, ensemble averages produce consistent average properties
D. Fundamentals of molecular dynamics and Monte Carlo
Molecular dynamics
1. is just the time average of .
The time evolution of the system is governed by Newton's classical mechanical
equations of motion, which can be solved on a computer using a large finite number 
of time steps of length
: total time; should be infinite but for practical purposes is a long finite time
2. Newton's equations generate a succession of states in accordance with the probability function 
for the microcanonical ensemble, and
is the ensemble average:
Ergodic system: time average and ensemble average are equivalent
-Must ensure adequate sampling
-system must be ergodic so there aren't independent closed circuits where a trajectory could become trapped and not fully sample phase space
-must also be sure that bottlenecks (due to high barriers) do not cause poor sampling even if system is ergodic
Monte Carlo
-Generate a set of states in phase space that are sampled from the complete set in
accordance with the probability density , but not necessarily
for microcanonical ensemble (i.e. not using true equations of motion)
-Invent a means of generating from one state point 
another point
(need not have a physical interpretation)
-This prescription should satisfy the following conditions
1. probability density should not change as system evolves
2. any reasonable starting distribution 
should tend to this stationary
solution as the simulation proceeds
3. ergodicity should hold (although it can't be proven, there should be a reasonabe argument)
where here 
runs over succession of states generated by this prescription
-Must ensure adequate sampling
Both molecular dynamics and Monte Carlo
-must have satisfactory phase space exploration in feasible computational time
-must be able to reproduce results given identical macroscopic parameters
(density, energy, etc.) but different initial conditions (atomic positions and momenta)