3 IV. NUMERICAL RESULTS AND DISCUSSION In this section, we present our numerical results for the velocity response of the PFC model in presence of an external pinning potential under a uniform applied force. For the numerical calculations, the phase field ψ ( ⃗ x ) and momentum density field ⃗ g ( ⃗ x ) are defined on a space square grid with dx = dy = π/ 4 with periodic boundary conditions. System sizes L × L with L = 64 to 128 where used. The Laplacians and gradients were evaluated using finite differences. We consider a pinning potential V ( ⃗ x ) representing a substrate with square symmetry the particle number density ρ ( ⃗ x ) from a uniform reference value ρ$_{0}$ , such that ψ ( ⃗ x ) = ( ρ ( ⃗ x ) − ρ$_{0}$ ) /ρ$_{0}$ . It is a con- served field and its average value, ¯ ψ , represents another parameter in the model. The intrinsic wave vector of the model ⃗ k$_{i}$ has no preferred directions and its magnitude is set to unity in the present work. In the absence of a pinning potential, the Hamiltonian of Eq. (12) is minimized by a configuration of the field ψ ( ⃗ x ) forming a hexagonal pattern of peaks with recipro- cal lattice vectors of magnitude | ⃗ k$_{h}$ | ≈ 1, in an appro- priate range of values for the parameters r and ¯ ψ in the model. This periodic structures of peaks in ψ can be regarded as a simple model of an atomic layer. To study the nonlinear dynamical behavior we take the form of H$_{int}$ given by Eq.(12) together with the ki- netic energy and the external force terms for the total effective Hamiltonian H$_{t}$ in the dynamical equations Eq. (11) and in Eq. (9). We make the additional simplify- ing approximation that ρ ( ⃗ x ) ≈ ρ$_{0}$ in Eq. (9) and in the coefficient of the first term in Eq. (11). This approxima- tion ensures that the effective diffusion coefficient in the model is positive definite for any temperature and driv- ing force. Another motivation for this approximation is to show that the dynamical equations used in the previ- ous works 13,15 follows from the more general Eqs. (10) and (11). Setting ρ$_{0}$ = 1, we obtain, where k$_{0}$ defines the period of the pinning potential for both the x and y directions. The lattice misfit between the phase-field crystal and the pinning potential can be defined as δ$_{m}$ = (1 − k$_{0}$ ). We choose the parameters of the PFC model as r = − 0 . 25, ¯ ψ = − 0 . 25, a lattice mismatch δ = − 0 . 5 and a pinning strength V$_{0}$ = 0 . 15. For these parameters, the ground-state configuration in the absence of an external driving force is a c (2 × 2) phase $^{14,15}$, where every second site of the lattice of the pinning potential corresponds to a peak in the phase field ψ ( ⃗ r ). This commensurate configuration is stable below a commensurate melting temperature T$_{c}$ ≈ 0 . 055. This is determined from the temperature dependence of the structure factor as well as that of the specific heat. The structure factor S ( ⃗ k ) is calculated from the the positions ⃗ R$_{j}$ of the peaks in the phase field as Here we have redefined ⃗ g → ⃗ g + ρ$_{0}$ ⃗ f /η to remove a uni- form term on right hand side of the equation for ⃗ g . The above coupled equations can be combined in a single equation by applying the operator ∇· to both sides of the second equation and using the first one to eliminate ∇ · ⃗ g , giving Here, 〈 ... 〉 denotes a time average which is equivalent to a thermal average at equilibrium. For increasing tem- peratures thermal fluctuations disorder the layer and the scaled structure factor S ( ⃗ k$_{c}$ ) /N$_{p}$ evaluated at the pri- mary reciprocal lattice vector ⃗ k$_{c}$ of the commensurate phase decreases from a value of unity at T = 0 rapidly through T$_{c}$ to zero at higher temperatures. The transi- tion is broadened due to finite size effects as shown in Fig. 1(a). Another signature of this commensurate melt- ing transition is that the specific heat develops a broad peak near T$_{c}$ corresponding to an increase in the fluctu- ations in ψ at the transition as shown in Fig. 1(b). Fi- nally, the mobility µ defined as lim$_{f}$ → $_{0}$( v/f ) where v is the drift velocity, also changes qualitatively through the transition. In the commensurate phase, a finite threshold for the sliding of the overlayer exists and hence the mo- bility is vanishingly small. As the system melts above T$_{c}$ the mobility rapidly rises and reaches a plateau at higher temperatures. This behavior is shown in Fig. 1c. For the study of the nonlinear response and sliding friction of the system, we focus on an initial state which corresponds to When the driving force ⃗ f and the external pinning po- tential are set to zero, the dynamical equation above is identical to the one used in Ref. 22 to study propagat- ing density modes in the PFC model. It can also be obtained by introducing inertial effects in the dynamical equation of the PFC model through a memory function of exponential form $^{23}$. In the limit of large η when ∂g$_{i}$/∂t in Eqs. (13) or, equivalently, ∂ $^{2}$ψ/∂t2 in Eqs. (14), can be neglected, these equations reduce to the familiar overdamped dynamical equations used in the previous works 13,15 without inertial effects at zero temperature. ∂ $^{2}$ψ ∂t 2 + η ∂ψ ∂t = ∇ $_{2}$δH$_{pfc}$ δψ − ⃗ f · ∇ ψ + ξ ( ⃗ x, t ); (14) 〈 ξ ( ⃗ x, t ) ξ ( ⃗ x $^{′}$, t$^{′}$) 〉 = 2 k$_{B}$T η ∇ $^{2}$δ ( ⃗ x − ⃗ x $^{′}$) δ ( t − t $^{′}$) . S ( ⃗ k ) = 〈 N$_{P}$ ∑ j,j ′ =1 1 N$_{P}$ e − i ⃗ k · ( ⃗ R$_{j}$ $_{−}$⃗ R$_{j}$$_{′}$ $^{)}$〉 . (16) ∂ψ ∂t = −∇ · ⃗ g ; (13) ∂g$_{i}$ ∂t = −∇$_{i}$ δH$_{pfc}$ δψ + ψf$_{i}$ − ηg$_{i}$ + ν$_{i}$ ( ⃗ x, t ); 〈 ν$_{i}$ ( ⃗ x, t ) ν$_{j}$ ( ⃗ x $^{′}$, t$^{′}$) 〉 = 2 k$_{B}$T ηδ ( ⃗ x − ⃗ x $^{′}$) δ ( t − t $^{′}$) δ$_{i,j}$. V ( ⃗ x ) = − V$_{0}$ [cos( k$_{0}$x ) + cos( k$_{0}$y )] , (15)