### Ground states and critical points for generalized frenkel-kontorova models in Z^d

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As described by Eq. From Fig. In the continuum limit 29 , the dependence is known to decay exponentially with the coupling strength,. In this limiting case, fronts are smooth and the discrete FK model, Eqs. The value of D in the stochastic simulation effectively accounts for both thermal fluctuations and structural disorder, leading to rounding of the transition close to the depinning point and converging to the deterministic result away from it.

Black continuous and dashed lines represent the predictions in the discrete, Eq. In the continuum reaction-diffusion formalism we obtain an estimate,.

We emphasize that neither of the asymptotic predictions, Eqs. The only discrepancy occurs close to the depinning point, where the experimental data deviate from the theoretical prediction. On one hand, this distinction can be attributed to the presence of thermal fluctuations and structural disorder. As known from the literature 42 , 43 , these factors accounted as effective thermal noise result in rounding of the transition in the vicinity of depinning.

As confirmed by our simulations see Fig. On the other hand, a reliable estimate for the front speed close to the depinning point, where front speeds are small, requires accumulation of large enough statistics. These amounts of data can be obtained within the framework of the numerical model but are not always available in the experiment. From the comparison of the results of the full FK model, Eqs.

Points correspond to experimental data, and lines are fits according to the theoretical model, Eq. Experimental conditions correspond to the two circles in panel a. As follows from Eqs. Note that the equilibria positions of the inner particles remain unaffected, which particularly indicates that the mechanism is essentially independent of the chain length, ensuring its universality. Their approximate evaluation yields the following expressions:. Equation 13 can be considered as a function of the H x field, , where c is a fit parameter. We have presented an experimental system showing discrete fronts which propagate along chains of paramagnetic colloidal particles held together and driven by an external magnetic field.

We develop a reduced analytically tractable model which exquisitely matches and explains the observed features in a wide range of field strengths ranging from the strongly discrete to continuum limits.

## Ground States and Critical Points for Aubry–Mather Theory in Statistical Mechanics

The finite size of chains is used to polarize the emerging front via a symmetry breaking mechanism. We note that the actual number of particles in the chain is unimportant and the mechanism is effectively universal, being valid for both short and long chains. The ability to control the front dynamics, including its propagation direction, in driven colloidal systems is appealing for potential applications such as e. Finally, the generic form of the developed model presents a firm implication for a greater variety of systems exhibiting similar phenomena.

Particular details of our experimental system present no restrictions for our findings, which can expectedly be extended to other nonlinear systems in biological and condensed matter contexts. The particles were diluted in highly deionized water and deposited above the FGF surface. The applied magnetic field was provided via custom-made Helmholtz coils perpendicular to each other. The dynamics of particles can be well described by two-dimensional 2D overdamped Langevin equations. To derive an efficient rigorously reduced one-dimensional 1D description in terms of a generalized Frenkel-Kontorova FK model, we perform two consecutive steps reducing the complexity of the full 2D time-dependent model, Eqs.

Note that while obtaining Eq. We also note that the time dependence enters Eq. Thus, expressions 18 and 19 are time independent and the corresponding slow-timescale equations of motion are fully autonomous in the co-moving reference frame. While the evaluation of the force due to individual interaction with the field of substrate yields a simple expression. Note that although dipolar forces are long ranged and formally require to account for further neighbors 48 , the nearest neighbor approximation is often successfully applied to simplify the theoretical analysis 41 and is known to work particularly well for systems of paramagnetic colloidal particles coupled via dipolar interactions both in and out of equilibrium Regulating wave front dynamics from the strongly discrete to the continuum limit in magnetically driven colloidal systems.

We thank T. Fischer, A. Pikovsky, L. Schimansky-Geier and I. Sokolov for valuable discussions. Author Contributions P. All authors analyzed and interpreted the data. National Center for Biotechnology Information , U. Sci Rep. Published online Feb 3. Johansen , 3, 4 and Arthur V.

## Read Ground States And Critical Points For Generalized Frenkel Kontorova Models In Zd

Straube b, 5. Tom H. Arthur V. Author information Article notes Copyright and License information Disclaimer. Received Sep 8; Accepted Dec This work is licensed under a Creative Commons Attribution 4. Supplementary Information. Abstract The emergence of wave fronts in dissipative driven systems is a fascinating phenomenon which can be found in a broad range of physical and biological disciplines.

Open in a separate window. Figure 1. Propagation of fronts along propelling colloidal chains. Coarse-grained model capable of front propagation To obtain insight into the basic physics and quantify the dynamics of fronts, we apply a reduced one-dimensional 1D model capable of front propagation along propelling chains. Dynamic state diagram.

### Associated Data

Figure 2. Existence of fronts and dynamics of individual colloidal particles. Figure 3. Magnetic control of front speed switching from the discrete to continuum limits. Figure 4.

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Magnetic control of the direction of front propagation. Conclusions We have presented an experimental system showing discrete fronts which propagate along chains of paramagnetic colloidal particles held together and driven by an external magnetic field. Derivation of the reduced theoretical model To derive an efficient rigorously reduced one-dimensional 1D description in terms of a generalized Frenkel-Kontorova FK model, we perform two consecutive steps reducing the complexity of the full 2D time-dependent model, Eqs.

Supplementary Movie S2: Click here to view. Supplementary Information: Click here to view. Acknowledgments We thank T.