In the R.I.N.G.S. code the existence or the absence of a bond between two atoms

Precisely the program will consider that a bond exists if the interatomic distance

In such a network a series of nodes and links connected sequentially without overlap is called a path. Following this definition a ring is therefore simply a closed path.

If we study thoroughly a specific node of this network we see that this node can be involved in numerous rings.

Each of these rings is characterized by its size and can be classified based upon the relations between the nodes and the links which constitute it.

However, the wide variety of sequences and structural defects that may be encountered in these materials make this type of analysis particularly challenging.

Some scenarios should therefore be reviewed, to lay the groundwork to look for rings in these compounds. In particular we need to evaluate the influence of different bond defects in the amorphous network.

However, in each case both the impact on the connectivity, and the impact on the physical properties which can be deduced from the analysis, have to be studied.

Furthermore it is necessary to determine if the influence of the bond defects depends on the criteria used to define a ring. ]]>

The main goal is to allow to link this analysis method to other physical properties of the material that can be measured experimentally.

The following article illustrates features implemented in the R.I.N.G.S. code in order to improve the results of the ring statistics.

]]>S(q) allows to study the organization of the particles in the material and can be directly compared to neutrons and X-rays scattering experiments.

Thereafter you will find the theoretical elements that will help you to understand theses techniques, and to obtain an overview of the formalisms implemented in the R.I.N.G.S. code.

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Indeed even considering the calculation power of the actual computers it is not possible to run molecular dynamics simulations on systems with a number of atoms close to the Avogadro number ( 10^{23}).

It appears that the path follow by this atom can be described by a random pedestrian walk. Mathematically this represents a sequence of steps done one after another where each step follows a random direction which does not depend on the one of the previous step (Markovien process).

In the case of a one-dimensional system (straight line) the displacement of the atom will therefore be either a forward step (+) or a backward step (-). Furthermore it will be impossible to predict one or the other direction (forward or backward) since they have an equal probability to be chosen.

One can conclude that the distance this atom may have traveled is closed to zero. Nevertheless if we choose not to sum the displacements them-self (+/-) but the square of these displacement then we will add a positive quantity to the total distance which therefore increases at each step.

Consequently this allows to obtain a better evaluation of the real (square) distance traveled by this atom.

The **M**ean **S**quare **D**isplacement **MSD **is defined by the relation:

where (*t*) is the position of the atom *i* at the time *t*, and the 〈 〉 represent an average on the time steps and/or the particles.

However during the analysis of the results of molecular dynamics simulations it is important to subtract the probable drift of the center of mass of the simulation box:

where (*t*) represents the position of the center of mass of the system at the time *t*.

The MSD also contains information on the diffusion of atoms. If the system is solid (frozen) then the DCM towards to a value (saturation), the kinetic energy is not sufficient enough to reach a diffusive behavior. Nevertheless if the system is not frozen (ex: liquid) then the DCM will grow linearly in time. In such case it is interesting to characterized the behavior of the system compared to the slope of the MSD. The slope of the MSD or so called diffusion constant D is defined by:

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