# Time Control of Molecular Simulation

The length of the equilibrium simulation time depends on the relaxation period of the property of interest. To obtain the statistical average of the properties of interest, it is necessary to sample 3-10 times in different relaxation periods. For example, for the vibration of the key, a period of 10 fs per vibration counts as one cycle, and you need to sample at least 50 fs. The molecular diffusion is on the order of ns, the loop-closing of the medium peptide chain takes the order of 10ns, the relaxation period to form an alpha helix is on the order of 200ns, the beta structure is 1-10 microseconds, and so on.

### The length of the relaxation period

The length of the relaxation period depends on how long a certain movement lasts and then becomes irrelevant to the previous state, or is forgotten. All motions have multiple states. The free energy barrier that this motion must cross determines the relaxation period of the motion. This is why we can simulate the average characteristic time of a process (such as protein folding), according to the Arrhenius formula, reverse the free energy barrier.

### How to know the average relaxation time of a process?

Take volume relaxation as an example. If you plot the curve of the entire volume data over time, you will see that the volume fluctuates above and below the average. It is internally driven by the interaction, vibration, and diffusion of molecules in the box, and there are fluctuations of multiple frequencies. Instantaneous fluctuations are driven by molecular vibrations, while slow, long-period fluctuations are driven by molecular aggregation and diffusion, but it is not easy to see.

### How to estimate this slow-floating period?

#### There are several ways to estimate:

• The first method is to calculate the correlation function and look at the correlation function. From any moment on, how long does it take for the volume data to be no longer correlated with each other (the correlation function value approaches 0); the length of this period of time is the characteristic relaxation time; this method is suitable for investigating the time characteristics of diffusion motion, calculating self-diffusion coefficient, viscosity, etc.
• The second method is the block average method, to see how big the block calculates the mean square deviation RMSD has nothing to do with the block size you take. The method is to divide your trajectory into many equal-length segments. Each segment is called a block. You calculate the average of each block. According to all the block averages, you can get their mean square deviation RMSD. Plot your RMSD with the block. the curve of size change.
• The third method is to calculate the fluctuation spectrum. We know that vibration has a vibration spectrum, and the wave naturally also has a wave spectrum. For example, volume fluctuation, autocorrelation calculation of volume data, and Fourier transformation, you can get a spectrum. There are a series of peaks on it, at different frequencies. The different frequencies reflect the length of time of the fluctuation, and the peak area shows the correlation with the movement of the frequency. The peak with the largest area is naturally its characteristic peak. According to its frequency, the characteristic time of its movement can be calculated. (This is how the infrared spectrum is calculated. It retains the dipole of the entire box, and then decomposes it into a lot of motions of different frequencies. The motions of different frequencies correspond to different modes of vibration. They contribute to the change of the total dipole of the box. Fourier decomposes them. Therefore, as long as a certain movement contributes to a certain property, its characteristic frequency can be decomposed by the Fourier change of the autocorrelation function of that property, and then its characteristic time can be obtained.
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