Box-size scaling and system-size scaling for different thermodynamics

Using Monte-Carlo simulation for N particles in a cubic box of length L and with the periodic boundary conditions, we generate the configurations of the Lennard-Jones fluid with a linear term and with different cutoff distances [10]. Given in table 3.2 for different thermodynamic system we have done and in table 4.1 for the particle number N and the box length L, the simulations of five system sizes are performed.

Table 4.1: The cube of the length is calculates by L＝ ,

(a)

N 3000 6000 12000 24000 48000

L 14.56 18.38 23.12 29.12 36.69

(b)

N 3000 6000 12000 24000 48000

L 14.42 18.17 22.89 28.85 36.34

And we used the configurations to construct the 3N3N Hessian matrices. By using the JADAMILU package to solve the Hessian matrices, we achieve the eigenvectors and

the imaginary-frequency branch of INMs spectrum. At first, we roughly locate the ME by the property of and reveal the system-size invariance with the eigenvectors and eigenvalues receive from Hessina matrices. And, we are going on further analysis to check the universality of different thermodynamic states. Under the box-counting measuring, there are two different finite size scaling analysis methods: system-size scaling and box-size scaling.

But the scaling behavior will breakdown for small box l near lattice constant a, the choice of small box size should be .

By average INMs of N=3000,6000,12000,24000 and 48000 at the ME in the imaginary-frequency branch and taking the ratio in the box-size scaling method as an integer varied from 2 to 8, we have calculated the and for q between -5 and 5. And in the system-size scaling, we defined l=2.427 in =0.972 and l=2.403 in =1.0, it means in the simulated system of N=3000 was exactly partitioned into 216 boxes. For other larger simulated systems and with this l, L/l is not exactly an integer so that we partition each realization into small boxes of size l as many as possible, with some remains not enough to be a small box. Therefore, the five system sizes that we have simulated, the values of are 6,7,9,12 and 15. is maximum integer which is smaller than or equal to Lennard-Jones fluid [37]. The results of SSP curve provide an evidence to confirm the location of the ME in the INMs spectrum. The reference figures all obtain from [40], the purpose is to compare with my data and confirm the mobility edges at different thermodynamics states still have the same properties. And, the ME at different

thermodynamic states are shown in Table.(4.3).

In conclusion, the system-size scaling is better than the box-size scaling in theory. Because the system-size scaling is exactly partitioned all the simulated system into equal size of small box. The box-size scaling is partitioned one simulated system into several unequal sizes of small boxes. Therefore, the fluctuation of the particle number between each small box for system-size scaling is much smaller than box-size scaling. From the view of our results, the system-size scaling is much sensitive to the precision of the ME frequency spectrum. Because it is hard to tell the different between the SSP curve at the frequency spectrum A and the frequency spectrum B. A and B frequency are all nearby the ME. Finally, we locate the exactly frequency of ME based on the system-size scaling result.

(All the reference data in Anderson model and INMs of TLJ fluid come from [38][39][40])

Table.(4.2) The ME at different thermodynamic states.

c ME( )

2.5 0.972 0.836 -69.60.5

2.5 1.0 0.836 -70.40.5

2.5 1.0 0.7 -59.60.5

2.5 1.0 0.5 -43.30.5

3.5 0.972 0.836 -72.10.5

Table.(4.3) The position of the singularity strength (q=0) under box-size scaling and

Fig.(4.1-a) At thermodynamic state , the box-size scaling of the singularity spectrum of the INMs LJ simple fluids at a ME. The INMs of

imaginary-frequency is calculated with (blue line with error bar). And, the circles, squares and red dashed line are all reference data come from [38][39][40].

Fig.(4.1-b) At thermodynamic state , the system-size scaling of the singularity spectrum of the INMs LJ simple fluids at a ME. The INMs of imaginary-frequency is calculated with (green line with error bar) for five different system sizes from N=3000 to 48000. In each panel, is generated with the data of and with a step of .

Fig.(4.2-a) At thermodynamic state , the box-size scaling of the singularity spectrum of the INMs LJ simple fluids at a ME. The INMs of

imaginary-frequency is calculated with (blue line with error bar). And, the circles, squares and red dashed line are all reference data come from [38][39][40].

Fig.(4.2-b) At thermodynamic state , the system-size scaling of the singularity spectrum of the INMs LJ simple fluids at a ME. The INMs of imaginary-frequency is calculated with (green line with error bar) for five different system sizes from N=3000 to 48000. In each panel, is generated with the data of and with a step of .

Fig.(4.3-a) At thermodynamic state , the box-size scaling of the singularity spectrum of the INMs LJ simple fluids at a ME. The INMs of imaginary-frequency is calculated with (blue line with error bar). And, the circles, squares and red dashed line are all reference data come from [38][39][40].

Fig.(4.3-b) At thermodynamic state , the system-size scaling of the singularity spectrum of the INMs LJ simple fluids at a ME. The INMs of imaginary-frequency is calculated with (green line with error bar) for five different system sizes from N=3000 to 48000. In each panel, is generated with the data of and with a step of .

Fig.(4.4-a) At thermodynamic state , the box-size scaling of the singularity spectrum of the INMs LJ simple fluids at a ME. The INMs of

imaginary-frequency is calculated with (blue line with error bar). And, the circles, squares and red dashed line are all reference data come from [38][39][40].

Fig.(4.4-b) At thermodynamic state , the system-size scaling of the singularity spectrum of the INMs LJ simple fluids at a ME. The INMs of imaginary-frequency is calculated with (green line with error bar) for five different system sizes from N=3000 to 48000. In each panel, is generated with the data of and with a step of .

Fig.(4.5-a) At thermodynamic state , the box-size scaling of the singularity spectrum of the INMs LJ simple fluids at a ME. The INMs of

imaginary-frequency is calculated with (blue line with error bar). And, the circles, squares and red dashed line are all reference data come from [38][39][40].

Fig.(4.5-b) At thermodynamic state , the system-size scaling of the singularity spectrum of the INMs LJ simple fluids at a ME. The INMs of imaginary-frequency is calculated with (green line with error bar) for five different system sizes from N=3000 to 48000. In each panel, is generated with the data of and with a step of .