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To better understand the dynamic melting transition at f$_{a}$ and the freezing transition at f$_{b}$ , we inspect the steady state configurations near the two thresholds as well as the region between the two thresholds in real space, which yield more direct and detailed information than the structure factor at the peak values. Treating the peaks in the phase field as ”particles”, the coordina- tion number of each particle in the commensurate c (2 × 2) phase is 4 while for the incommensurate hexagonal phase it is 6. Furthermore, the core of dislocations in a hexago- nal lattice correspond to particles with 5-fold and 7-fold coordination numbers. Thus, further insight into the mi- croscopic nature of the steady state configurations can be obtained by inspecting the fraction of particles with co- ordination numbers p$_{4}$ , p$_{5}$ , p$_{6}$ , p$_{7}$ . The results are shown in Figs. 4 and Fig. 5 for increasing and decreasing ap- plied forces respectively. Fig. 4 shows that when the force f is increased beyond f$_{a}$ , besides the rapid drop in p$_{4}$ consistent with the structure factor data, the fraction of other coordination numbers p$_{5}$ , p$_{6}$ , p$_{7}$ also increases significantly untill f reaches f$_{c}$ , beyond which p$_{5}$ and p$_{7}$ start to decrease and we have a continuous transition to an incommensurate hexagonal phase. Thus the nature of steady state above f$_{a}$ is a strongly disordered state analogous to the high temperature phase in the absence of driving force above the commensurate melting tem- perature. On decreasing the force below f$_{a}$ , the data in Fig. 5 shows that the system remains in a melted state with large disorder until the threshold f$_{b}$ is reached, be- low which we have only 4-fold coordination number and the system returns to a pinned c (2 × 2) phase. Taken together, the qualitative behavior of the structure factor and the coordination number strongly suggest that the transitions at f$_{a}$ and f$_{b}$ can be regarded as a force in- duced dynamical melting and freezing transition respec- tively. 

Finally, we look at a snapshot of the phase field ψ ( x ) obtained in the steady state for a driving force just above the dynamical freezing threshold f$_{b}$ . This is shown in Fig. 6. It consists of stripes of commensurate c (2 × 2) phase separated by disordered domain walls. These do- main walls are mobile liquid like regions as confirmed by a count of the fraction of various coordination numbers shown in Fig. 7. The commensurate stripes have only 4- fold coordination numbers, while in the liquid like domain walls, there is a mixture of 5-fold, 6-fold and 7-fold coor- dination numbers. Starting from this steady state which still has a non-zero average sliding velocity, we can watch the dynamical freezing in real time as the driving force is reduced to a value just below the freezing threshold f$_{b}$ . The time sequence of the freezing is shown in Fig. 8. In returning to the pinned state, the domain wall regions gradually shrink and eventually disappear. 

The main features of the dynamical melting and freez- ing and hysteresis effects in PFC model described above are similar to those found earlier by molecular dynamics simulations of particle models with interacting Lennard- Jones potentials 3 and the uniaxial Frenkel-Kontorova 

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model 5 under a driving force starting from a commensu- rate state. The similarity of results from these very differ- ent models demonstrates the universality of the hysteresis loop for models with inertia effects and the macroscopic consequence of stick and slip motion. When compared with the Lennard-Jones particle model one notable dif- ference is the mechanism of the dynamical freezing at f$_{b}$ . In the present PFC model, it involves parallel liquid-like domain walls similar to the uniaxial Frenkel-Kontorova model, although in the absence of the driving force there is no easy direction. In the Lennard-Jones model, nu- cleation and growth occur via closed pinned domains $^{3}$. The origin of this intriguing difference requires further investigation of both atomistic and PFC models. One possibility is that the nucleation of stripes is related to the absence of a fixed constraint on the number of peaks in the PFC model. In this case, the mechanism of the dy- namical freezing found in the present PFC model should be compared to the results of particle models with a con- stant chemical potential rather than with a fixed particle number. Unfortunately, such results are currently un- available. 

In this paper, we have derived general stochastic dy- namical equations for the particle and momentum den- sity fields including both thermal fluctuations and iner- tial effects. The new equations are applied to the study of the nonlinear response to an external driving force 

## V. CONCLUSIONS 

