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a well pinned c (2 × 2) phase initially, corresponding to T = 0 . 01 ≪ T$_{c}$ , and a damping parameter η = 0 . 4. 

In our numerical calculations, a driving force along the x direction is increased from zero to a maximum value above a critical depinning force and then decreased back to zero. For each value of the force, the coupled equa- tions (13) are solved using an Euler algorithm with time step dt = 0 . 001 − 0 . 005. The implicit trapezoidal method was also used to check numerical stability. For each value of the force, 10 6 time steps were used to allow the sys- tem to reach a velocity steady state and an additional period with the same number of time steps were used to calculate the average velocity and other time averaged quantities. 

To study the velocity response of the PFC model, we need to determine the velocity of the peaks 14 in the phase field ψ ( ⃗ x ). This is done by determining the time depen- dence of the peak positions ( ⃗ R$_{i}$ ( t )) in ψ . The steady state drift velocity ⃗ v for the system is obtained from the peak velocities ⃗ v$_{i}$ ( d ⃗ R$_{i}$/dt ) as 

where N$_{P}$ is the number of peaks and 〈 ... 〉 denotes time average. The steady state structure factor S ( ⃗ k ) which is a measure of translational order, is also calculated from the the peak positions ⃗ R$_{j}$ , as in Eq.(16). 

The most notable qualitative feature of the velocity response to the driving force ⃗ f (as shown in Fig. 2) is that it shows hysteresis behavior with two different crit- ical force thresholds f$_{a}$ ≈ 0 . 075 for increasing forces and f$_{b}$ ≈ 0 . 045 for decreasing forces. These two threshold val- ues correspond to the static frictional force and kinetic frictional force respectively. As the force is increased be- yond f$_{a}$ , the velocity jumps abruptly from zero to a finite value whereas when the force is decreased below f$_{b}$ the velocity of the sliding layer drops abruptly and become pinned by the external potential to form an immobile commensurate state again. Below, we present the micro- scopic details of the configurations for the system in the neighborhood of the two thresholds. This allows us to characterize the change in velocity response at f$_{a}$ as a dynamical force induced melting transition of the initial commensurate state, and the second transition at f$_{b}$ as a dynamical force induced freezing of the sliding phase. 

To study these transitions we first examine the behav- ior of the steady state structure factor, as shown in Fig. 3. We focus on the dependence of S ( Q ) on the driving force, where ⃗ Q is the dominant reciprocal lattice vector of the layer. ⃗ Q = ⃗ k$_{c}$ corresponds to the primary recip- rocal lattice vector for the c (2 × 2) phase and ⃗ Q = ⃗ k$_{h}$ to the reciprocal lattice vector of the hexagonal phase in absence of the driving force. Consistent with the veloc- ity response behavior in Fig. 2, on increasing the force beyond f$_{a}$ , S ( k$_{c}$ ) drops abruptly to zero. This is the onset of the dynamical force induced melting transition 

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of the initial c (2 x 2) commensurate state. The behavior of S ( k$_{c}$ ) is analogous to the temperature induced disor- dering transition shown in Fig. 1(a). On decreasing the force, the value of S ( k$_{c}$ ) stays vanishingly small until the force drops below the threshold f$_{b}$ , at which point S ( k$_{c}$ ) rapidly increases to a value corresponding to the com- mensurate pinned state. The other interesting feature is that for f > f$_{c}$ ≈ 0 . 12, the structure factor shows clear 

