Ensembles

The molecular dynamics algorithm outlined in the previously makes use of the NVE ensemble (also known as the microcanonical ensemble), where the number of particles (N), volume of the system (V), and energy of the system (E) are all kept constant. This is not the only ensemble that exists, there is also other such as:

• NVT (canonical): number of particles (N), volume of system (V), temperature of the simulation (T)
• NPT (isothermal-isobaric): number of particles (N), pressure of system (P), temperature of the simulation (T)

For these ensembles, it is necessary to determine a method to modulate the temperature of the system. The temperature can be modulated using a variety of methods known as thermostating. The simplest, although not neccesarily the best, is velocity rescaling. This is where the velocities of the individual particles are changed such that the kinetic energy of the total system more accurately matches that necessary for the desired temperature. For this the instaneous temperature of the system, $T_{\text{inst}}$, is defined as,

$$T_{\text{inst}} = \frac{\sum^N_{i=0}{m_i v_i^2}}{2Nk_B},$$

where, $N$ is the number of particles, $m_i$ is the mass of particle $i$, $v_i$ is the velocity of particle $i$, and $k_B$ is the Boltzmann constant. This means that the velocities of the particles may be rescaled by the following relation,

$$v_i = v_i \sqrt{\dfrac{T_{\text{target}}}{\bar{T}}},$$

where $T_{\text{target}}$ is the target temperature for the themostat, and $\bar{T}$ is the average simulation temperature. pylj [1,2], the software that you shall use later uses this method for producing an NVT simulation, using the heat_bath function. Various other methods for thermostatting exist, such as the Anderson, Nosé-Hoover, or the Berendsen methods [3-6].

In order to achieve the NPT ensemble, it is necessary to use a barostat in addition to a thermostat. These allow the volume of the system to vary such that the pressure is constant throughout the simulation. We will not discuss barostats any further, the interested reader is directed to more detailed texts on the subject.

References

1. McCluskey, A. R.; Morgan, B. J.; Edler, K. J.; Parker, S. C. J. Open Source Educ. 2018, 1 (2), 19. 10.21105/jose.00019.
2. McCluskey, A. R.; Symington, A. R. 10.5281/zenodo.2423866.
3. Andersen, H. C. J. Chem. Phys. 1980, 72 (4), 2384–2393. 10.1063/1.439486.
4. Nosé, S. J. Chem. Phys. 1984, 81 (1), 511–519. 10.1063/1.447334.
5. Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; DiNola, A.; Haak, J. R. J. Chem. Phys. 1984, 81 (8), 3684–3690. 10.1063/1.448118.
6. Hoover, W. G. Phys. Rev. A 1985, 31 (3), 1695–1697. 10.1103/PhysRevA.31.1695.