Three-dimensional nonequilibrium Potts systems with magnetic friction

We study the nonequilibrium steady states that emerge when two interacting three-dimensional Potts blocks slide on each other. As at equilibrium the Potts model exhibits different types of phase transitions for different numbers $q$ of spin states, we consider the following three cases: $q=2$ (i.e., the Ising case), $q=3$, and $q=9$, which at equilibrium yield, respectively, a second-order phase transition, a weak first-order transition, and a strong first-order transition. In our study we focus on the anisotropic character of the steady states that result from the relative motion and discuss the change in finite-size signatures when changing the number $q$ of spin states.

Figure 1

Schematic picture of two identical subsystems with relative motion in the $y$ direction. The two subsystems are blocks composed of $W\times L\times H$ spins, with periodic boundary conditions in all three directions. Boundaries between the two subsystems are indicated by the green (dark) areas.

Figure 2

Boundary magnetization density as a function of temperature $T$ for two Potts blocks that are either both at rest ($v=0$) or where one of the blocks is moving with respect to the other with speed $v=10$. Data for different small values of the coupling strength ratio $\kappa$ are shown. For $\kappa =0$ the two blocks are uncoupled, whereas for $\kappa =1$ the couplings at the boundary have the same strength as the couplings inside the bulk. The number of states are (a) $q=2$, (b) $q=3$, and (c) $q=9$. Every block is composed of $80\times 80\times 80$ spins. The data result from averaging over at least ten independent runs, and error bars are smaller than the symbol sizes.

Figure 3

(a)–(c) Boundary magnetization density $m_b$ and (d)–(f) boundary specific heat $C_b$ as a function of temperature $T$ for two Potts blocks that are either both at rest ($v=0$) or where one of the blocks is moving with respect to the other with speed $v=10$. Data for different large values of the coupling strength ratio $\kappa$ are shown. The number of states are (a, d) $q=2$, (b, e) $q=3$, and (c, f) $q=9$. The relative motion stabilizes the ordering of the boundary which results in an additional shift of the local transition temperature. This is clearly visible in the data for $q=2$ and $q=3$. Every block is composed of $80\times 80\times 80$ spins. The data result from averaging over at least ten independent runs, and error bars are smaller than the symbol sizes.

Figure 4

Snapshots of one of the boundaries of a system formed by two blocks with $20\times L\times 10$ spins. The speed is $v=10$ and the ratio of coupling strengths is $\kappa =9$. (a)–(d) $q=2$ and $T=3$, (e)–(h) $q=3$ and $T=2.45$, (i)–(l) $q=9$ and $T=1.55$. The length of the sample is (a) $L=20$, (b) $L=40$, (c) $L=80$, and (d) $L=160$, and similarly for the other two values of $q$. The different colors correspond to the different states of the spins.

Figure 5

Line magnetization densities in $x$ (filled squares) and $y$ (open circles) directions for a system with $q=9,\kappa =9$, and $v=10$ composed of blocks containing $20\times 160\times 10$ spins. A typical spin configuration for that system at $T=1.55$ is shown in Fig. 4.

Figure 6

Line magnetization densities in $x$ (filled squares) and $y$ (open circles) directions for $\kappa =9$. (a)–(c) $q=2$ and (d)–(f) $q=9$. The sizes of the blocks, which move with relative speed $v=10$, are $L\times L\times 20$, with (a, d) $L=20$, (b, e) $L=40$, and (c, f) $L=160$. The data result from averaging over at least ten independent runs. Error bars are only shown when the error is larger than the symbol size.

Figure 7

Boundary magnetization densities for the Ising model as a function of temperature. (a) Shift of the boundary magnetization density as a function of the speed $v$ in systems composed of blocks containing $20\times 20\times 20$ spins. (b) Boundary magnetization densities for anisotropically shaped samples with $40\times L\times 20$ spins in each block that move with relative speed $v=\infty$. Based on all our data, we estimate the critical temperature to be $T_c=2.34\left(2\right)$.