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Theory and implementation

Molecules in liquids and gases do not stay in the same place, but move constantly. This process is called diffusion and it happens quite naturally in fluids at equilibrium. During this process, the motion of an individual molecule does not follow a simple path [48]. As it travels, the molecule undergoes some collisions with other molecules which prevent it from following a straight line. If the path is examined in close detail, it will be seen to be a good approximation to a random walk. Mathematically, a random walk is a series of steps where each step is taken in a completely random direction from the one before. This kind of path was famously analysed by Albert Einstein in a study of Brownian motion. He showed that the MSD of a particle following a random walk is proportional to the time elapsed. This relationship can be written as

| (4.14)

where is the MSD and t is the time. D and C are constants. The constant D defines the so-called diffusion coefficient. The figure 4.34 shows an example of a MSD analysis performed on a waterbox of 768 water molecules.

Figure 4.34: MSD calculated for a 100 ps MD simulation of 256 water molecules using NPT condition at 1 bar and 300 K.

To get the diffusion coefficient out of this plot, the slope of the linear part of the plot should be calculated.

Defining,

| (4.15)

the MSD of particle can be defined as:

| (4.16)

where and are respectively the position of particle at times and . One can introduce a MSD with respect to a given axis n:

| (4.17)

with

| (4.18)

The calculation of MSD is the standard way to obtain diffusion coefficients from MD simulations. Assuming Einstein-diffusion in the long time limit one has for isotropic systems

| (4.19)

There exists also a well-known relation between the MSD and the velocity autocorrelation function. Writing in Eq. (4.16) one can show (see e.g. [50]) that

| (4.20)

Using now the definition (4.19) of the diffusion coefficient one obtains the relation

| (4.21)

With Eq. (4.41) this can also be written as

| (4.22)

Computationally, the MSD is calculated using the Fast Correlation Algorithm (FCA) [51]. In this framework, in the discrete case, the mean-square displacement of a particle is given by

| (4.23)

where is the particle trajectory and is the the number of frames of the trajectory. We define now the auxiliary function

| (4.24)

which is splitted as follow:

|  |  | (4.25)
|  |  | (4.26)
|  |  | (4.27)

The function can be computed using the FCA method described in Section A. For the following recursion relation holds:

|  |  | (4.28)
|  |  | (4.29)

This allows one to construct the following efficient scheme for the computation of the MSD:

1. Compute .
2. Compute .
3. Compute using the FFT method.
4. Compute MSD(m) in the following loop:

It should be noted that the efficiency of this algorithm is the same as for the FCA computation of time correlation functions since the number of operations in step (1), (2), and (4) grows linearly with .

Next: Parameters Up: Mean-Square Displacement Previous: Mean-Square Displacement Contents