Supersonic kinks

The velocity of the single kink versus the force (β=1/π,  g=1,  η=0.05) enlarged view	The critical kink velocity  vc(β)  at the fixed force  f = 0.5

 Anharmonic interaction: (multiple) kinks  ≈  Toda solitons:

	vToda c = sinh(πβp) πβp (enlarged view)
	p = 3:  vToda/c ≈ 3.34 p = 2:  vToda/c ≈ 1.81 p = 1:  vToda/c ≈ 1.18   (enlarged view)

Collisions of driven kinks

	if the stability intervals overlap: single kink + double kink → triple kink (β=0.01,  f=0.15,  g=1,  η=0.012)
	if the stability intervals do not overlap: single kink + double kink → running state (β=1/π,  f=0.25,  g=1,  η=0.012)

Traffic jams

Atomic trajectories for  β = 1/π,  f = 0.33	Zoom of the rectangle on the left figure

Schematically:

Phase diagram (N=256,  g=0.1,  f=0.5,  T=0.1)	Section of the left figure (enlarged view)

Traffic jams do appear in the underdamped FK model with anharmonic interaction just before the transition to the running state, if the following conditions are satisfied:

See the paper by O.M. Braun, B. Hu, A. Filippov, and A. Zeltser, Phys. Rev. E 58 (1998) 1311 "Traffic jams and hysteresis in driven one-dimensional systems" (pdf files may be found here) and also the web page  "Driven Lattice-Gas Model"

The soft FK model

In the soft FK model, the particles have a complex structure treated in a mean-field fashion: particle collisions are inelastic and also each particle is considered as having its own thermostat. The model has a truly equilibrium ground state. When an external dc force is applied to the atoms, the model exhibits a hysteresis even at high temperatures due to the clustering of atoms with the same velocity. Another effect of clustering is phase separation in the steady state when the system splits into regions of immobile atoms ("traffic jams") and regions of running atoms.

Idea: Consider a system consisting of complex particles, which have their own structure with internal degrees of freedom. The internal modes may be excited due to inter-particle collisions, that take away the kinetic energy of the translational motion, so that the collisions are inelastic. This is a typical situation in soft-matter physics, for example, in physics of granular gases. The kinetic energy of atomic translational motion that is lost in a collision is stored as the energy of excitation of internal degrees of freedom and may be released later as the kinetic energy. In a simple case, when the number of internal degrees of freedom is “large” and their coupling is nonlinear, the energy lost in collisions is transformed into the “heating” of particles. We propose a new type of stochastic models, a model with “multiple” thermostats, where, in addition to the standard “substrate” thermostat, each particle is considered as having its own “thermostat”. A natural description of such a model is one with a specific type of Langevin equations (or the corresponding Fokker-Planck equation).

Motion equations: The equation of motion for the lth particle has the form

where the dot (prime) indicates the time (spatial) derivative. The substrate thermostat is modeled by the Gaussian stochastic force  δF_l(t) which has zero average and the standard correlation function

The mutual damping  h_l  was chosen to depend on the distance between the NN atoms in the same way as the potential, h_l = h^*exp[-g (x_l- x_l-1- a_A)],  where h^* is a parameter which describes the inelasticity: the interaction is elastic in the case of  h^*=0, while in the limit h^*→ ∞ the collisions are completely damped.

This set of Langevin equations is equivalent to the Fokker-Planck-Kramers equation for the distribution function W,

It is easy to check that in the undriven case,  f = 0,  the Maxwell-Boltzmann distribution is a solution of the FPK equation. Thus, the model has the truly thermodynamically equilibrium state.

Simulation results:

Figure: Dependence of the normalized mobility B on the force  f  for three values of the intrinsic damping: h^*=0 (blue up triangles, the "elastic" model), h^*=e^-a_A » 0.0393 (red down triangles), and h^*=10 e^-a_A » 0.393 (black diamonds) for an increasing force (solid curves and symbols) and a decreasing force (dotted curves and open symbols). Other parameters are the following:  N/M = 144/233 = golden mean,  g  = 1/p,  g = 1, h = 0.01, and  T = 1.  Inset: B(f)  for  h^* » 0.0393 for three values of the rate of force changing:  R » 10^-6 (blue up triangles),  R » 2·10^-7 (red down triangles), and  R » 4·10^-8 (black diamonds).

Figure: Atomic coordinates as functions of time in the "traffic-jam" regime for  f = 0.095,  h^*» 0.0393,  g = 1/p,  g = 1,  h = 0.01,  and  T = 1.

Figure: Dependence of the normalized correlation of coordinates K_x/K_x0 (black diamonds) and velocities K_v/K_v0 (red triangles) on the driving force  f  for  h^*» 0.0393,  g = 1/p,  g = 1,  h = 0.01,  and  T = 1  for an increasing force (solid curves and symbols) and a decreasing force (dotted curves and open symbols). Note that for a spatially homogeneous state we should have  K_x » K_x0 = T/g  and  K_v » K_v0 = 2T.

Conclusion: The dynamics of the soft model with inelastic interaction drastically differs from the classical (elastic) one. First, the system exhibits hysteresis even at huge temperatures. The reason why the 1D model exhibits hysteresis is that the soft model is effectively infinite-dimensional. The particles have an infinite number of internal degrees of freedom treated in a mean-field fashion. Second, the soft model allows the coexistence of two phases (the TJ regime) for a much wider range of model parameters. Both effects are due to the clustering of atoms in the soft model.

Two-dimensional driven FK model

Half-filled layer (θ=1/2)

small damping
larger damping
enlarged view

(exponential repulsion,  β = 0.37,  g_eff = 0.15)

Dynamical phase diagram:

enlarged view

 

Partially-filled layer (θ=3/4)

We also studied the nonlinear dc response of a two-dimensional underdamped system of interacting atoms subject to an isotropic periodic external potential with triangular symmetry. When driving force increases, the system transfers from a disorder locked state to an ordered sliding state corresponded to a moving crystal. By varying the values of the effective elastic constant, damping and temperature, we found different scenarios and intermediate phases during the ordering transition. For a soft atomic layer, the system passes through a plastic-channel regime that appears as a steady-state regime at higher values of the damping coefficient. For high values of the effective elastic constant, when the atomic layer is stiff, the intermediate plastic phase corresponds to a traffic-jam regime with immobile islands in the sea of running atoms. At a high driving of the stiff layer, a soliton-like elastic flow of atoms has been observed.

Movies:

A. Locked-to-sliding transition for  g=0.1, η=0.1, T=0.1,  F changes from F=0.355 to F=0.4:  avi 21 Mb

 

B. Steady states for:

1. g=0.1, η=0.3, T=0.001,  F=0.645:  animated gif 5.5 Mb (channel plastic flow)

2. g=0.1, η=0.3, T=0.1,  F=0.57:  animated gif 3.9 Mb (traffic jams)

3. g=0.3, η=0.3, T=0.001,  F=0.6:  animated gif 3.1 Mb (traffic jams)

4. g=0.3, η=0.3, T=0.001,  F=0.615:  animated gif 3.8 Mb (channel plastic flow)

5. g=1, η=0.1, T=0.001,  F=0.165:  animated gif 3.4 Mb

6. g=1, η=0.1, T=0.001,  F=0.35:  animated gif 3.1 Mb (elastic flow)

7. g=1, η=0.3, T=0.001,  F=0.4:  animated gif 3.4 Mb

8. g=1, η=0.3, T=0.001,  F=0.65:  animated gif 2.6 Mb (solitonic, or DW motion)

 

Closely-packed layer (θ=1): islands of moving atoms

A. Stiff layer: g_eff ~ 1 or > 1  (example for LJ interaction with  g_eff = 0.9,  η = 0.141,  T = 0.05,  and  f = 0.9933)

B. Weak layer: g_eff << 1  (example for exponential interaction with  g_eff  = 0.1,  η = 0.1,  T = 0.01,  and  f = 0.94)

Figures: The positions of the particles are indicated by circles. The x component of the particle velocity is shown in a grey scale by the color of the circle: black corresponding to zero velocity and the lightest grey to velocities over a certain velocity cutoff.

 

For details, see also (pdf files may be found here)

• O.M. Braun, M.V. Paliy, J. Röder, and A.R. Bishop, Phys. Rev. E 63 (2001) 036129 "Locked-to-running transition in the two-dimensional underdamped driven Frenkel-Kontorova model"

• J. Tekic, O.M. Braun, and Bambi Hu, Phys. Rev. E 71 (2005) 026104 "Dynamic phases in the two-dimensional underdamped driven Frenkel-Kontorova model"

Fuse-safety device

A simplified model:  V_sub(x,y) = 1 – cos x + ½ ω²y²

Figure: small dots & solid curves:  η = 0.1; bold dots and dashed curves:  η = 0.2;

labels: (1) θ = 64/56;   (2) θ = 64/60

	enlarged view

The fuse-safety device was introduced and studied in the paper by Oleg Braun, Maxim Paliy, and Bambi Hu, Phys. Rev. Lett. 83 (1999) 5206 "Fuse safety device on an atomic scale"

 

Some open questions

 

For the 2D FK model,

• Ø θ < 1: a TJ state ?

• Ø role of defects in the 2D model ?

• Ø quasi-periodic / random substrate potential ?

•  a.c. driven underdamped models (1D & 2D) ?

•  ratchet scenarios?

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Last updated on October 5, 2008 by Oleg Braun.                                   Translated from L^AT_EX by T_TH