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Register a new userEmergent friction in two-dimensional Frenkel-Kontorova models
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Simple models for friction are typically one-dimensional, but real interfaces are two-dimensional. We investigate the effects of the second dimension on static and dynamic friction by using the Frenkel-Kontorova (FK) model. We study the two most straightforward extensions of the FK model to two dimensions and simulate both the static and dynamic properties. We show that the behavior of the static friction is robust and remains similar in two dimensions for physically reasonable parameter values. The dynamic friction, however, is strongly influenced by the second dimension and the accompanying additional dynamics and parameters introduced into the models. We discuss our results in terms of the thermal equilibration and phonon dispersion relations of the lattices, establishing a physically realistic and suitable two-dimensional extension of the FK model. We find that the presence of additional dissipation channels can increase the friction and produces significantly different temperature-dependence when compared to the one-dimensional case. We also briefly study the anisotropy of the dynamic friction and show highly nontrivial effects, including that the friction anisotropy can lead to motion in different directions depending on the value of the initial velocity.
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A 1D model of interacting particles moving over a periodic substrate and in a position dependent temperature profile is considered. When the substrate and the temperature profile are spatially asymmetric a center-of-mass velocity develops, corresponding to a directed transport of the chain. This autonomous system can thus transform heath currents into motion. The model parameters can be tuned such that the particles exhibit a crossover from an ordered configuration on the substrate to a disordered one, the maximal motor effect being reached in such a disordered phase. In this case the manybody motor outperforms the single motor system, showing the great importance of collective effects in microscopic thermal devices. Such collective effects represent thus a free resource that can be exploited to enhance the dynamic and thermodynamic performances in microscopic machines.
A two-dimensional Frenkel-Kontorova model is set up. Its application to the tribology is considered. The materials and the commensurability between two layers strongly affect the static friction force. It is found that the static friction force is larger between two layer of same materials than that for different materials. For two-dimensional case the averaged static friction force is larger for the uncommensurate case than that for the commensurate case, which is completely different from one-dimensional case. The directions of the propagation of the center of mass and the external driving force are usually different except at some special symmetric directions. The possibility to obtain superlubricity is suggested.
The Frenkel Kontorova (FK) model is known to exhibit the so called Aubrys transition which is a jamming or frictional transition at zero temperature. Recently we found similar transition at zero and finite temperatures in a super-conducting Josephson junction array (JJA) on a square lattice under external magnetic field. In the present paper we discuss how these problems are related.
We study the mechanism of decay of a topological (winding-number) excitation due to finite-size effects in a two-dimensional valence-bond solid state, realized in an $S=1/2$ spin model ($J$-$Q$ model) and studied using projector Monte Carlo simulations in the valence bond basis. A topological excitation with winding number $|W|>0$ contains domain walls, which are unstable due to the emergence of long valence bonds in the wave function, unlike in effective descriptions with the quantum dimer model. We find that the life time of the winding number in imaginary time diverges as a power of the system length $L$. The energy can be computed within this time (i.e., it converges toward a quasi-eigenvalue before the winding number decays) and agrees for large $L$ with the domain-wall energy computed in an open lattice with boundary modifications enforcing a domain wall. Constructing a simplified two-state model and using the imaginary-time behavior from the simulations as input, we find that the real-time decay rate out of the initial winding sector is exponentially small in $L$. Thus, the winding number rapidly becomes a well-defined conserved quantum number for large systems, supporting the conclusions reached by computing the energy quasi-eigenvalues. Including Heisenberg exchange interactions which brings the system to a quantum-critical point separating the valence-bond solid from an antiferromagnetic ground state (the putative deconfined quantum-critical point), we can also converge the domain wall energy here and find that it decays as a power-law of the system size. Thus, the winding number is an emergent quantum number also at the critical point, with all winding number sectors becoming degenerate in the thermodynamic limit. This supports the description of the critical point in terms of a U(1) gauge-field theory.
We solved the Frenkel-Kontorova model with the potential $V(u)= -frac{1}{2} |lambda|(u-{rm Int}[u]-frac{1}{2})^2$ exactly. For given $|lambda|$, there exists a positive integer $q_c$ such that for almost all values of the tensile force $sigma$, the winding number $omega$ of the ground state configuration is a rational number in the $q_c$-th level Farey tree. For fixed $omega=p/q$, there is a critical $lambda_c$ when a first order phase transition occurs. This phase transition can be understood as the dissociation of a large molecule into two smaller ones in a manner dictated by the Farey tree. A kind of ``commensurate-incommensurate transition occurs at critical values of $sigma$ when two sizes of molecules co-exist. ``Soliton in the usual sense does not exist but induces a transformation of one size of molecules into the other.
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