In statistical mechanics the thermodynamic quantities are defined in terms
of ensemble averages. For example, the temperature is proportional to the
value of the kinetic energy averaged over every state in the ensemble. On
the other hand, a molecular dynamics simulation produces a time sequence of
events and physical properties are averaged in time over the duration of the
simulation. The ergodic hypothesis asserts that the ensemble average is
equal to the time average (averaged over all times). So, for example, using
the equipartition function to relate the temperature to the average of
the kinetic energy for N particles, we have
where the brackets < ... > denotes the time average. The right-hand
side of is the quantity computed in the molecular dynamics simulation.
Because the simulation will be performed over a finite time, it is required
that the simulation sample phase space sufficiently well.
The number of "atoms" included in the model is not always equal to the number of atoms in the true system. To reduce the amount of computer resources needed, it is quite common to combine hydrogens with the heavy atoms to which they are attached. The only hydrogens that are treated explicitly are those capable of potentially forming hydrogen bonds. The parameters of the heavy atom are altered to compensate for this "fusion" process and the resulting parameter set is termed a "united atom" or "extended atom" force field. Explicit treatment of all hydrogens in the problem involves the use of the "all atom" force field.
To decrease the computational costs associated with the evaluation of order N2 nonbonded terms it is customary to ignore interaction pairs which are separated by more than a certain cutoff distance which is denoted as Rc. This is usually implemented by first generating a list of nonbonded partners which meet the criteria of not being excluded based on either connectivity or distance. This list is then used by the energy routine a number of times before being updated to correct for the drift of the system. The generation of the list is still an order N2 procedure but only simple distance calculations and logical operations are involved.
As an example of the utility of the list procedure, consider that a 3000 atom system has a total of approximately 4.5 million N2/2 nonbonded partners. In the simplest form of a list routine the only numerical computation is a determination of the square of the distance between two atoms. This involves five operations: three multiplications and two additions. The nonbonded energy evaluation requires roughly twice as many mathematical operations as does the list routine. If no list were employed, this would require 450 million operations for 10 nonbonded energy evaluations. However, generation of a list (22.5 million operations), using a cutoff of 8 Å typically reduces the average number of nonbonded partners from 1500 (N/2) to about 50, giving a total of only 15 million operations in the energy routine for 10 evaluations. This corresponds to a total savings of 412.5 million operations or about 92% that of not employing a list.
The use of nonbonded lists does lead to two complications; namely, the introduction of discontinuities in the potential energy surface, and the splitting of dipoles. These problems must be appropriately resolved in order to maintain accuracy in the simulation or minimization process.
In cases where forces fall off rapidly with distance, the contribution
of many far-off particles may be approximated by a single forces term.
The multipole expansion of the force in A approximates the total
forces on a particle in region B due to all particles in region
A with an error bounded by R/(R+D).
For particles involved in long range interactions like electrostatic or gravitational forces, this approach may significantly speed up the calculation by avoiding all or the particle-particle interactions. The fast algorithm relies upon replacing portions of direct sums with series expansions.