# Molecular Dynamics¶

In this chapter performing molecular dynamics simulations using MAPP will be discussed.

## Supported Ensembles¶

`mapp.md.nvt` (T,dt) |
ensemble |

`mapp.md.nst` (S,T,dt) |
ensemble |

`mapp.md.muvt` (mu,T,dt,gas_element,seed) |
ensemble |

## NHC Thermostat¶

The thermostat that is employed in MAPP is the well-known Nose-Hoover chains (NHC). Here we will touvh on the equations of motions of NHC breifly and its implementation in MAPP in order to clarify the meaning of some of the variables that users have control over, namely , , and . Interested reader might refer to [1] and [2] for a full discription and mathematical formulation of of NHC governing equations. With that being said the main idea behind NHC is as follows: In order to simulate the coupling of a system with an external thermal bath, we couple its kinetic energy with kinetic enery of a series of fictitious object with fictitious masses, namely links in NHC chain. With index denoting th link in the chain.

Schematic representation of NHC of length .

Suppose we have a general dimeensional system with deggrees of freedom (they can be positions of atoms or anything else), namely . The equations of motion are as follows:

where is the potential energy, and is the mass associated th degree of freedom; and are degree of freedom of and mass of th link in Nose-Hoover chain, respectively. Here denotes derivative with respect to time. is the external (bath) temperature; is the internal (current) temperature of system, calculated using

As it is apparent from the equations above is uinitless and is of unit of . Let us intorduce the following change of variable

where is a unitless parameter and is a positive time scale, which in turn simplify thermostat equations of motion:

in MAPP as a matter of convention are chosen as follows:

which gives us the final form of equations of motion implemented in MAPP:

For completeness here are the equations of motions for :

and :

The two parameters and can be adjusted by user through the following attributes

:

:

Another variable that can be adjusted is , which is related to the numerical time integration. In order to increase the percision of numerical integration instead of eveolving the system through one step of , it can be performed through steps of . can be adjusted by user through the following attributes

:

## Isothermal-Isostress¶

Isothermal-isostress ensemble that is used in mapp is taken from [3]. Aside from external stress, temperature and thermostat related parameters (see NHC Thermostat), there is one other matrix parameter ( ) that user has control over. Before going over the equations of motion it should be noted that our formultion is different from orginal presentation in three major ways:

- stress is coupled to a separate thermostat rather than particle thermostat
- unitcell matrix ( ) is the transpose of the one suggested in the original formulation ( )
- unlike the original formulation unitcell mass is not scalar i.e. each of components have their own mass, and are defined using a time scale matrix

Cosider a dimensional unitcell defined by tensor, containing atoms, interacting via a given potential energy, where denotes the position of atom . The particles in the system have degrees of freedom (which is not necessarily equal to ) and are coupled with NHC thermostat. The system is subjected to an external temperature of and an extrenal stress tensor . The equations of motions regarding the particles are as follows

here represents derivative with respect to time. Please note that here is not the actual particle trajectory but rather is. is the force exerted on atom due to potential energy and is its mass. is a tensor representing the equivalent of velocity for ; is the particle thermostat related parameter, see NHC Thermostat. The equations of motion pertaining to are:

where is a given length scale matrix; is the stress thermostat related parameter (see NHC Thermostat), and

where is the current volume (determinant of ); and describe the reference configuration and is its volume, respectively; and

:

## References¶

[1] | Mark E. Tuckerman. Statistical mechanics : theory and molecular simulation. Oxford graduate texts. Oxford ; New York : Oxford University Press, 2010., 2010. ISBN 978-0-19-852526-4. |

[2] | Daan Frenkel and Berend Smit. Understanding molecular simulation : from algorithms to applications. Computational science: v. 1. San Diego, Calif. ; London : Academic, c2002., 2002. ISBN 978-0-12-267351-1. |

[3] | W. Shinoda, M. Shiga, and M. Mikami. Rapid estimation of elastic constants by molecular dynamics simulation under constant stress. Physical Review B (Condensed Matter and Materials Physics), 69(13):134103–1–8, April 2004. doi:10.1103/PhysRevB.69.134103. |