2. HNEMD algorithm and computational technique

We start, as usual, the Green-Kubo relations (GKRS) for the hydrodynamic transport coefficients of uncharged particles [16]. It is well-known form and has been shown the standard GKRS of fluids to the YDPLS [17–22]. The typical GKRS used for the estimation of thermal conductivity of interacting dust particles for YDPLS:

$$\lambda = \frac{1}{3k\_{\rm B}V T^2} \int\_0^\infty \langle \mathbf{J}\_Q(t)\mathbf{J}\_Q(\mathbf{0})\rangle dt \tag{5}$$

where in Eq. (5), k<sup>B</sup> is the Boltzmann's constant, V is the system volume,T is the absolute temperature, and JQ is the heat flux vector. The expression for the microscopic heat flux vector JQ can be given by:

$$\mathbf{J}\_{Q}V = \sum\_{i=1}^{N} E\_{i} \frac{\mathbf{p}\_{i}}{m} - \frac{1}{2} \sum\_{i \neq j} \left(\mathbf{r}\_{i} - \mathbf{r}\_{j}\right) \left(\frac{\mathbf{p}\_{i}}{m}.\mathbf{F}\_{ij}\right) \tag{6}$$

In the above expression, Fij is the total interparticle force on particle i due to j, rij = r<sup>i</sup> � r<sup>j</sup> are the position vectors, and P<sup>i</sup> is the momentum vector of the ith particle. Ei is the energy of particle i and is given by the expression as:

$$E\_i = \frac{\mathbf{p}^2}{2m} + \frac{\mathbf{1}}{2} \sum\_{i \neq j} \phi\_{ij} \tag{7}$$

where ϕij is the Yukawa pair potential between particle i and j. Evans [23] has developed the non-Hamiltonian linear response theory (LRT) used for a moving system representing the equation of motion:

$$
\stackrel{\bullet}{\mathbf{r}}\_i = \frac{\mathbf{p}\_i}{m} \tag{8}
$$

$$\dot{\mathbf{p}}\_i = \sum\_{J=1}^{N} \mathbf{F}\_i + \mathbf{D}\_i(\mathbf{r}\_i, \mathbf{p}\_i) \mathbf{F}\_e(t) - a\mathbf{p}\_i \tag{9}$$

where <sup>F</sup><sup>i</sup> ¼ �∂ϕij=∂ri � � is the total interparticle force acting on particle <sup>i</sup> and D<sup>i</sup> ¼ D<sup>i</sup> ri; p<sup>i</sup> � � is the tensor phase variable that describes the coupling of system to the fictitious external force field Feð Þt . A mechanical work is performed through the external applied force field Feð Þt , and thus, the equilibrium cannot be maintained. In the above Eq. (9), α is the Gaussian thermostat multiplier that maintains the system temperature and it is given as [16, 23]:

$$a = \frac{\sum\_{i=1}^{N} \left[ \mathbf{F}\_i + \mathbf{D}\_i(\mathbf{r}\_i, \mathbf{p}\_i) . \mathbf{F}\_e(t) \right] . \mathbf{p}\_i}{\sum\_{i=1}^{N} p\_i^2 / m\_i} \tag{10}$$

The external force field parallel to the z-axis is of the form FeðÞ¼ t ð Þ 0; 0; FZ ; therefore, the thermal conductivity is calculated as:

$$\begin{split} \lambda &= \frac{V}{3k\_B T^2} \Bigg\int\_0^\infty \langle \mathbf{J}\_{Q\_x}(t) \mid \mathbf{J}\_{Q\_x}(\mathbf{0}) \rangle dt \\ &= \lim\_{F\_x \to 0} \lim\_{t \to \infty} \frac{-\left\langle \mathbf{J}\_{Q\_x}(t) \right\rangle}{T F\_x} \end{split} \tag{11}$$

force field strength F\* (=0.005) for the computation of the steady state values of the plasma normalized thermal conductivity is to be selected, for small varying practical system size. This feasible suitable external force field provides the steady state plasma thermal conductivity estimations, which are satisfactory over the whole

Comparison of normalized plasma thermal conductivity (λ0), computed by various numerical approaches for plasma coupling states Γ (1, 200). Results investigated by Shahzad and He for homogenous nonequilibrium MD (HNEMD) [13], Salin and Caillol for equilibrium MD (EMD) [21], Donko and Hartmann for inhomogenous NEMD [17]. Shahzad and He for homogenous perturbed MD (HPMD) [18] and Faussurier

and Murillo for variance procedure (VP) [25], (a) for N = 256, (b) N = 1372 and at κ = 1.4.

Non-Newtonian Dynamics with Heat Transport in Complex Systems

DOI: http://dx.doi.org/10.5772/intechopen.82291

Figures 1–3 display that the computed plasma thermal conductivity is in acceptable conformity with former HNEMD investigations by Shahzad and He [13], EMD calculations of Salin and Caillol [21], inhomogenous NEMD estimations of Donkó and Hartmann [17], homogenous perturbed molecular dynamics simulations

(HPMD) measurements of Shahzad and He, and theoretical predictions of Faussurier and Murillo for variance procedure (VP) [18, 25]. It can be seen from Figure 1 that our results are slightly lower as compared to earlier known numerical results based on different numerical techniques, at lower Γ. However, the present

range of the plasma state points (Γ, κ).

Figure 1.

177

where JQZ ð Þt is the z-component of the heat flux vector and the external force field FeðÞ¼ t ð Þ FZ .
