## 4 Computational Results for the Dispersion Force on Nanoparticles from Graphene  
Here, we present the computational results for the nonequilibrium dispersion force \(F_{neq}\) acting on nanoparticles of radius \(R\) on the source side of a graphene sheet characterized by the mass-gap parameter \(\Delta\), which takes the typical values of 0.1 and 0.2 eV [82,83]. The temperature of nanoparticles is assumed to be the same as that of the environment, i.e., \(T_{E}=300\) K, whereas the temperature of a graphene sheet can be either cooler, \(T_{g}=77\) K, or hotter, \(T_{g}=500\) K, than the environmental temperature. These temperatures are chosen as a representative example. The first is the temperature of liquid nitrogen, whereas the second is close to that employed in the experiment measuring the nonequilibrium Casimir-Polder force [85]. The developed formalism allows for computation of the nonequilibrium dispersion interaction for any experimental temperatures. In line with the assumptions made in Sections 2 and 3, computations were performed in the separation range from 200 nm to 2 \(\upmu\)m and it is assumed that \(R\) is sufficiently small. The polarizabilities of dielectric and metallic nanoparticles are presented in (1). Numerical computations were performed by Equations (2)-(4), (6) and (7) using the expressions for the polarization tensor presented in Section 3. For this purpose, we worked out a program written in the C++ programming language. The program utilizes the Gauss-Kronrod and double-exponential quadrature methods from the GNU Scientific Library [86] and Boost C++ Libraries [87] for numerical integration. High-precision computation is achieved with the help of the Boost Multiprecision Library [88]. The program also employs the OpenMP Library [89] for parallelization. The results presented below were obtained using computational resources of the Supercomputer Center of the Peter the Great Saint Petersburg Polytechnic University. The computational results for \(F_{\text{neq}}\) are normalized to the classical limit of the equilibrium dispersion force acting on a nanoparticle on the source side of an ideal metal plane [71]  
\[F_{c}(a,T_{E})=-\frac{3k_{B}T}{4a^{4}}\,\alpha(0). \tag{29}\]  
The normalized values do not depend on the static polarizability of a nanoparticle \(\alpha(0)\). The absolute values of \(F_{\text{neq}}\) for the nanoparticles made of some specific material can be obtained using Equations (1) and (29) by fixing the values of \(R\) and \(\varepsilon(0)\). In doing so, the value of \(R\) is restricted by the conditions \(R\ll a\) and \(R\ll\hbar c/(k_{B}T)\), considered in the beginning of Section 2. In Figure 1, the \(F_{\text{neq}}/F_{c}\) ratio is shown as a function of the nanoparticle-graphene separation by the two bottom (blue) lines, which are plotted for graphene sheets with the mass-gap parameter \(\Delta\) equal to 0.2 and 0.1 eV, kept at temperature \(T_{g}=77\) K.  
For comparison, the two top (black) lines in Figure 1 show the ratio \(F_{\text{eq}}/F_{c}\) for the same nanoparticles and graphene sheets computed at a state of thermal equilibrium, i.e., when the temperatures of graphene and nanoparticles are equal to the environmental temperature, \(T_{g}=T_{E}=300\) K. In this case, the quantity \(F_{eq}\) is computed as  
\[F_{\text{eq}}(a,\Delta,T_{E})=-\frac{2k_{B}T_{E}\alpha(0)}{c}\sum_{l=0}^{ \infty}{}^{\prime}\int\limits_{0}^{\infty}k\,dke^{-2 aq_{l}(k)}\Big{\{}\Big{[} 2q_{l}^{2}(k)c^{2}-\xi_{E,l}^{2}\Big{]}\,R_{\text{TM}}(i\xi_{E,l},k;\Delta,T_{ E})-\xi_{E,l}^{2}R_{\text{TE}}(i\xi_{E,l},k;\Delta,T_{E})\Big{\}}. \tag{30}\]  
Equation (30) is obtained from (3) by putting \(T_{g}=T_{E}\). As shown in Figure 1, for a cooled graphene sheet the change in the value of \(\Delta\) has a lesser impact on \(F_{\text{neq}}\) than on \(F_{\text{eq}}\) at short separations but has a greater impact on \(F_{\text{neq}}\) than on \(F_{\text{eq}}\) at large separations.