2.1 Current correlation functions

The SCDPs support both longitudinal and transverse waves. The experimental importance of time-dependent correlation function is that the spectroscopic technique an example of this technique is neutron scattering. Investigate microscopic dynamical quantities through the MD approach and then comparison by Fourier analysis of the simulation result. The local density gives information about the atom's distribution. There is also possible to analyze the motion of atoms. The Fourier component of Particle current or momentum current for a single atomic particle in MD unit is given as.

$$
\pi(r, t) = \sum\_{j} \nu\_{j} \delta(r - r\_{j}(t)) \tag{2}
$$

where vj and rj are the velocity and position of a jth particle, by using the Fourier transformation of particle current becomes as for a given wavenumber vector (k).

$$
\pi(k, t) = \sum\_{j} \nu\_{j} e^{-ik \cdot r\_{j}(t)} \tag{3}
$$

The correlation function of the current vector component is defined as

$$\mathcal{C}\_{a\beta}(k,t) = \frac{k^2}{N\_m} \left( \pi\_a(k,t)\pi\_\beta(-k,0) \right) \tag{4}$$

For the isotropic fluid under consideration of symmetry above equation can be expressed in term of longitudinal current correlation and transverse current correlation in the relative direction of k, where k is the wave vector and equal to multiple of integers k = 2π/L and L is the size of the simulation box. Wave vector k becomes equal to k = 2π/L (k0, k1, k2, k3), kj ϵ Z, j = 0, 1, 2, 3; L is the length of simulation box and V = L<sup>3</sup> .

$$k = |k| = \frac{2\pi}{L} |(\mathfrak{x}, \mathfrak{y}, \mathfrak{z}| \tag{5}$$

Here x, y, and z are integers.

$$\mathbf{C}\_{a\beta}(k,t) = \frac{k\_a k\_\beta}{k^2} \mathbf{C}\_L(k,t) + \left(\delta\_{a\beta} - \frac{k\_{a\beta}}{k^2}\right) \mathbf{C}\_T(k,t) \tag{6}$$

By putting k = k z the time-dependent longitudinal current correlation becomes as.

$$\mathbf{C}\_{L}(k,t) = \frac{k^2}{N\_m} \left\langle \Sigma \boldsymbol{\nu}\_i \boldsymbol{e}^{-ik \, \boldsymbol{x}\_j(t)}(k,t) \Sigma \boldsymbol{\nu}\_j \boldsymbol{e}^{-ik \, \boldsymbol{x}\_j}(k,t) \right\rangle \tag{7}$$

Where Nm represents the number of particles, vi and vj are the velocity of the ith and jth particles, <....> gives the statistical average of particle current. Longitudinal current correlation function explains the direction of the waves along the wave vector (wavenumber) and a transverse direction perpendicular to k.

$$\mathbf{C}\_{T}(\mathbf{k},t) = \frac{\mathbf{k}^{2}}{2\mathbf{N}\_{m}} \left< \pi\_{\mathbf{x}}(\mathbf{k},t)\pi\_{\mathbf{x}}\left(-\mathbf{k},\mathbf{0}\right) + \pi\_{\mathbf{y}}(\mathbf{k},t)\pi\_{\mathbf{y}}(-\mathbf{k},t) \right> \tag{8}$$

The longitudinal current correlation also related to the dynamical structure factor.

$$\mathcal{S}(k, a) = \frac{1}{a^2} \mathcal{C}\_L(k, a) \tag{9}$$

In Eq. (9), the dynamical structure factor and longitudinal current correlation contain the same physical information of the systems. Transverse current and longitudinal current also explain the wave spectra in 3D SCDPs. In our EMD simulation model, the current correlation function is the only function of wavenumber and time (k, t). Through this mathematical model of current correlation, we checked out variation in frequency and peak amplitude of transverse and longitudinal waves in SCDPs for at Г, κ, N, and k [14, 19, 32, 35].
