2. Mathematical model and numerical method

In this chapter the EMD Simulations are performed for a selected system, having the number of particle N = 500 with apply periodic boundary condition (PBCs) on the cubic box in three dimensions coordinates directions. These particles are placed in a cube volume V and interact with each other by pairwise Yukawa potential is given:

$$\phi(|\mathbf{r}|) = \frac{Q^2}{4\pi\varepsilon\_0} \frac{e^{-|\mathbf{r}|/\lambda\_0}}{|\mathbf{r}|},\tag{1}$$

Q is the charge, on dust particles, ε<sup>0</sup> is permittivity of free space, λ<sup>D</sup> is the Debye length which accounts for the screening of the interaction by other plasma species. The dimensionless plasma parameters have fully characterized the system under study. One is Coulomb coupling parameter and defines as Γ = Q<sup>2</sup> /4πε0awskBT (already defined in Section 1.1), where a is the Wigner-Seitz radius and is define as aws = (3/4nπ) 1/3 with n is the dust particle density,T is the temperature of the system and kB is Boltzmann constant. The screening parameter is and defines as <sup>κ</sup> � <sup>a</sup>ws/λD. In an EMD technique, Newton's motion equation is, <sup>m</sup>(d<sup>2</sup> r/dt<sup>2</sup> ) = Fi = ΣjFij, integrated numerically for N Yukawa particles with mass m positioned at ri, velocity vi and acceleration ai in the volume (V) of simulation box of particle i (i = 1, 2, 3 … ..N) is exerted a force on other particle j and it is given as Fi = Σ<sup>j</sup> Fij and i 6¼ j. The EMD is performed in the microcanonical ensemble (NVT) for constant volume and temperature [39]. In this chapter, the EMD has been used to investigate the time-dependent current correlation functions [CL(k, t) and CT(k, t)]. The dimension of the simulation box is Lx, Ly, LZ. The periodic boundary condition is used to minimize the surface size effect and applied to the simulation box. The main calculation is performed for N = 500 particles at κ = 4.5 and 5.5, plasma coupling parameters Г (temperature of the Yukawa system) varies from 1 to 100 and wavenumbers k = 0, 1, 2, and 3. The simulation time step is taken as Δt = 0.001 to allow computing the important data for sufficient 425,000 simulation run. EMD method is reported of the current correlation of SCDPs over sufficient

domain of plasma parameters of Debye screening (4.4 ≤ κ ≤ 5.5) and Coulomb coupling (1 ≤ Γ ≤ 100).
