Mean-field theory of collective transport with phase slips

The driven transport of plastic systems in various disordered backgrounds is studied within mean field theory. Plasticity is modeled using nonconvex interparticle potentials that allow for phase slips. This theory most naturally describes sliding charge density waves; other applications include flow of colloidal particles or driven magnetic flux vortices in disordered backgrounds. The phase diagrams exhibit generic phases and phase boundaries, though the shapes of the phase boundaries depend on the shape of the disorder potential. The phases are distinguished by their velocity and coherence: the moving phase generically has finite coherence, while pinned states can be coherent or incoherent. The coherent and incoherent static phases can coexist in parameter space, in contrast with previous results for exactly sinusoidal pinning potentials. Transitions between the moving and static states can also be hysteretic. The depinning transition from the static to sliding states can be determined analytically, while the repinning transition from the moving to the pinned phases is computed by direct simulation.

Figure 1

Sketches of the pinning potentials and forces studied in this paper. The pinning forces are periodic with period $2\pi$ and the pinning potential for a degree of freedom ${\theta }_i$ has minima at ${\beta }_i+2n\pi$, for integer $n$. The cases are organized primarily by the sign of $c$, with the pinning force $Y\left(x\right)=-ax-c{x}^3+O\left(x^5\right)$ for small $x={\theta }_i-{\beta }_i$. The coefficient of the harmonic part of the force satisfies $a>0$. The cases (a), (b) and (c) are for “soft” pinning forces $(c<0)$; they differ near the potential maxima, corresponding to monotonic, nonmonotonic, and continuous forces, respectively. Case (d) is a “hard” potential $(c>0)$. The “scalloped” potential, case (e), is precisely quadratic $(c=0)$ in the interval $-\pi <x<\pi$. The form of the potential especially affects the stability of the coherently pinned phase and whether “re-entrant” pinning is possible upon increasing or decreasing the force.

Figure 2

Phase diagram in the coupling-drive $(\mu –F)$ plane for a discontinuous soft cubic pinning force of the type shown in Fig. 1b. The equation of motion is Eq. (4). The corresponding $Y\left(x\right)$ is given by Eq. (42) with $a=15∕\left(8\pi \right)$ and $c=-4a^3∕27$. The strength of the pinning is $h=1$ for all degrees of freedom. The diagonally lined region indicates the IS phase, while the cross-hatched region indicates the CS phase. The light gray shaded region denotes the region of coexistence of the CM and IS phases, while the medium gray shaded region denotes the region of coexistence of the IS and CS phases. The lines $F_{\uparrow }^i$ and $F_{\uparrow }^c$ are the forces at which the system depins upon increasing the drive from the incoherent and coherent static states respectively. The line $F_{\downarrow }$ is the force at which a coherently moving system stops upon lowering the drive. The point $({\mu }_e,F_e)$ indicates where the static-moving transition goes from hysteretic to nonhysteretic. The curves ${\mu }_u\left(F\right)$ and ${\mu }_d\left(F\right)$ are the values of the coupling at which the static system makes the transition to and from finite coherence states, respectively. The inset displays the hysteresis in the coherence $r$ as the coupling strength $\mu$ is varied at $F=0$. The transitions between the IS and CS phases are first order in $r$.

Figure 3

Typical numerical results, found by integrating numerically the equations of motion [Eq. (4)], for the behavior of the mean velocity $v$ and the coherence $r$ as the driving force is slowly varied. For each pair of plots, the coupling $\mu$ is held constant, while the drive force $F$ is raised from $F=0$ to $F=1.2$ and then decreased. The pinning potential is the same as for Fig. 2. The top frames $(\mu =0.5)$ show the hysteretic behavior between the IS and the CM phases, where the coherence and velocity jump between zero and nonzero values at the same locations. The next two sets of frames $(\mu =1.14)$ are obtained by preparing the system in the IS–CS coexistence region, starting from either an initial incoherent $\left(I\right)$ or coherent $\left(C\right)$ state. When the system is prepared in an incoherent state, the velocity and coherence jump at the same value of $F(\approx 0.42)$ as $F$ is raised, but change continuously as $F$ is decreased, albeit with a change in the slope $dr∕dF$ at the repinning force $F\approx 0.32$, where $v$ goes to zero. When the system is prepared in a coherent state, there is no hysteresis and $v$ and $r$ are continuous, though $r$ again shows a singularity at depinning. The bottom frames $(\mu =1.5)$ display the behavior at the continuous depinning transition from the CS phase. The results are similar to those for $\mu =1.14$, when starting from the coherent state $\left(C\right)$. In general, depinning from the coherent state is continuous and nonhysteretic, while depinning from the incoherent state is discontinuous and hysteretic. Numerical evidence for the hysteresis does not change over the size ranges studied, strongly suggesting that these simulations accurately represent the infinite-volume limit.

Figure 4

The behavior of the coherence $r$ for couplings $\mu \approx {\mu }_u$, i.e., near the instability point of the incoherent static phase (IS) at $F=0$. The three curves show $r$ with pinning force $Y(\delta ⪡1)=-a\delta -c{\delta }^3$ for $c$ positive (hard pinning potential), negative (soft pinning potential) and zero (piecewise linear pinning force.)

Figure 5

A sample plot of $\delta$, the displacement of a degree of freedom from the minimum of the pinning potential, versus the pinning phase ${\beta }_n\left(\delta \right)$ for branch numbers $-2⩽n⩽2$. The solid portions correspond to even $n$, while the dashed portions correspond to odd $n$. The global phase $\psi$ is chosen to be zero. Here, the effective interaction is large enough, $u>a$, that $\delta \left(\beta \right)$ is multivalued. The maximum magnitude of $\delta$ is denoted by ${\delta }_{\max }$.

Figure 6

The coherence of the static state at $F=0$ as a function of the coupling strength $\mu$ for four pinning forces: (a) hard $(c>0)$ cubic pinning force, with $a=c=1∕(\pi +{\pi }^3)$; (b) piecewise linear pinning force, with $a=1∕\pi$; (c) soft $(c<0)$ cubic pinning force, with $a=1∕\pi$ and $c=-1∕{\pi }^3$; (d) sine pinning force whose maximum strength is $1∕\pi$. Also shown is the value ${\mu }_d$ where the coherence jumps from a finite value to zero upon decreasing $\mu$.

Figure 7

Upper and lower bounds for $r\left(u\right)$, plotted as the coherence $r(\mu =u∕r)$, corresponding to the maximal and minimal coherence static metastable states. The pinning force is taken to be piecewise linear with $a=1∕\pi$. A single static coherent solution is obtained only in the limit $\mu \to \infty$.

Figure 8

The region of stability of the IS phase for a piecewise linear pinning force. The single particle depinning force $F_{\mathrm{sp}}$ and the coupling strength ${\mu }_u$ for instability to the coherent or moving states are also indicated.

Figure 9

A plot of the region of stability of the incoherent static phase for a hard cubic pinning force. The nature of the instability along the $F_u\left(\mu \right)$ curve is indicated by the thickness of the bounding curve on the right. For $\mu >{\mu }_h(F>F_h)$, the transition is hysteretic, while for smaller couplings (or small, fixed driving force for varying couplings) the transition is continuous.

Figure 10

Sketches of the region of stability of the incoherent static phase for a soft cubic pinning force. (a) corresponds to a pinning force of type (a) that does not turn over (is monotonic) in each repeated interval. (b) corresponds to a pinning force of type (b) that are nonmonotonic in each period.

Figure 11

Phase diagram for the piecewise linear pinning force, $Y\left(x\right)=-x∕\pi$ [see Fig. 1b]. The lightly shaded portion is the coexistence region of the IS and CM phase $(\mu <{\mu }_u)$ and the smaller, darkly shaded region, is where the CS and CM phases coexist $({\mu }_u<\mu <{\mu }_e)$. The depinning lines $F_{\uparrow }^i=F_{\mathrm{sp}}$ and $F_{\uparrow }^c$ have been obtained analytically and confirmed by numerics. The boundary $F_{\downarrow }$ where the system repins was obtained numerically. The point $({\mu }_e,F_e)$ marks where the static-moving transition changes from hysteretic to continuous. The boundary between the IS and CS phases is $F$ independent and lies at $\mu ={\mu }_u$.

Figure 12

Mean velocity $v$ and coherence $r$ as functions of the driving force $F$ for the piecewise linear pinning force. The curves are obtained numerically by first ramping $F$ from zero to a value well within the sliding state $(F=1.2)$, and then decreasing $F$ back down to zero, while holding $\mu$ constant. The top frames show the behavior for $\mu =0.25$, where the initial static state is incoherent: this state starts sliding at the single particle depinning force $F_{\mathrm{sp}}=1$ and repins at a lower force $F\approx 0.88$. The middle frames display the results for an initially coherent static state $(\mu =0.64>{\mu }_u)$, which still displays hysteresis, both in $v$ and $r$. The bottom frames are for $\mu =1.0$, which has an initial coherent state and undergoes continuous depinning.

Figure 13

Phase diagram in the coupling-drive $(\mu –F)$ plane for a hard cubic pinning force of the type shown in Fig. 1a. The form of the pinning force $Y\left(x\right)$ is given by Eq. (34), with $a=c=1∕(\pi +{\pi }^3)$. The regions of IS–CM, CS–CM, and IS–CS–CM coexistence are shown in light, medium, and dark gray, respectively. The incoherent and coherent depinning lines are denoted by $F_{\uparrow }^i$ and $F_{\uparrow }^c$, respectively. The repinning line is denoted by $F_{\downarrow }$. The coherent depinning line and the repinning line join at $({\mu }_e,F_e)$. Beyond this point the static-moving transition is continuous. The curves ${\mu }_u\left(F\right)$ and ${\mu }_d\left(F\right)$ are the values of the coupling at which the static system makes the transition to and from finite coherence states, respectively. There curves join at $({\mu }_h,F_h)$ where the IS–CS transition becomes continuous.

Figure 14

Mean velocity and coherence versus force for the hard potential and various values of $\mu$. Solid lines are used to display the response obtained when $F$ is ramped up from zero, while dashed lines show the jumps in $v$ and $r$ when ramping $F$ back down. The top frames show the hysteretic depinning of a system prepared in the IS phase. For $\mu =0.5$ (middle frames) the system is initially in a coherent $(r\ne 0)$ static state at $F=0$. As the force is ramped up, the system first crosses the boundary from the CS to the IS phase, where $r$ jumps discontinuously from its initial finite value to zero, while the system remains pinned $(v=0)$. At a higher force the system depins by crossing the boundary from the IS to the CM phase and $r$ jumps from zero to a large finite value. The subsequent ramping down of the field goes through this sequence of phases in reverse order, but the jumps occur at distinct values of $F$. The bottom frames describes the complex response that takes place along a path that crosses the dark region of three-phase coexistence. See the text for further description.

Figure 15

Both sets of figures show the behavior of velocity and coherence for $\mu =1.25$, but for different initial states. The top frames are obtained by preparing the system in a coherent state at $\mu =1.25$ and $F=0$, and ramping $F$ up to a value above $F_{\uparrow }^i$, and then back down to zero. In this case the depinning is continuous. The bottom frames are obtained by preparing the system in an incoherent state at $\mu =1.25$ and $F=0.9$, inside the lightly shaded area of coexistence of CS and CM phases, and then ramping the force down to zero. Note the depinning upon decreasing force in this case and the subsequent repinning.

Figure 16

Phase diagram in the coupling-drive $(\mu –F)$ plane for a soft monotonic cubic pinning force of the type shown in Fig. 1a. The pinning force $Y\left(x\right)$ is given by Eq. (34) with $a=6∕\left(5\pi \right)$ and $c=-1∕5{\pi }^3$. The regions of IS–CM, CS–CM, and IS–CS–CM phase coexistence are shown in light, medium, and dark gray, respectively. The lines $F_{\uparrow }^i$ and $F_{\uparrow }^c$ are the forces at which the system depins upon increasing the drive from the incoherent and coherent static states, respectively. The line $F_{\downarrow }$ is the force at which a moving system stops upon lowering the drive. The $({\mu }_e,F_e)$ and the static-moving transition becomes continuous. The curves ${\mu }_u\left(F\right)$ and ${\mu }_d\left(F\right)$ are the values of the coupling at which the static system makes the transition to and from finite coherence states, respectively.

Figure 17

Phase diagram in the coupling-drive $(\mu –F)$ plane for a soft cubic pinning force of the type shown in Fig. 1c. The pinning force $Y\left(x\right)$ is given by Eq. (34) with $a=3\sqrt{3}∕\left(2\pi \right)$ and $c=-3\sqrt{3}∕\left(2{\pi }^3\right)$. This choice of parameters gives a nonmonotonic and continuous pinning force: the results are to be compared with $Y\left(x\right)=-\sin \left(x\right)$, another nonmonotonic and continuous force. The region of IS–CM coexistence is shown in light gray, while the IS–CS coexistence is shown in medium gray. The lines $F_{\uparrow }^i$ and $F_{\uparrow }^c$ are the forces at which the system depins upon increasing the drive from the incoherent and coherent static states, respectively. The line $F_{\downarrow }$ is the force at which a moving system stops upon lowering the drive. The CS region is very small for these values of parameters, corresponding to ${\mu }_T\approx 1.85$ and ${\mu }_d\left(0\right)\approx 1.44$, and it has been magnified in the inset. The CS phase does not exist at finite $F$ for coupling larger than ${\mu }_T$. Shown within this inset is the point $({\mu }_e,F_e)$ where the $F_{\uparrow }^c$ and $F_{\downarrow }$ lines join and the static-moving transition ceases to be hysteretic. The curves ${\mu }_u\left(F\right)$ (only visible within the inset) and ${\mu }_d\left(F\right)$ are the values of the coupling at which the static system makes the transition to and from finite coherence states, respectively.

Figure 18

Mean velocity and coherence, obtained from numerical calculations, for a continuous cubic pinning potential and parameter values given in Fig. 17. The top frames record the hysteretic response of a system prepared in the incoherent state at $\mu =0.8$ and $F=0$, while the bottom frames show the continuous $F=0$ depinning of system prepared in the coherent state at $\mu =1.67$.

Figure 19

Phase diagram for a sinusoidal pinning force $Y\left(x\right)=-\sin \left(x\right)$. The IS–CM coexistence region is shaded gray. The $F=0$ region in which the system can only exist in the CS phase is denoted by a series of ×’s. The region of IS–CS coexistence is denoted by medium on-axis gray shading. The $\mathrm{IS}\to \mathrm{CS}$ and $\mathrm{IS}\leftarrow$ phase boundaries are the points $(\mu ={\mu }_u=2,F=0)$ and $(\mu ={\mu }_d\approx 1.49,F=0)$, respectively. The line $F_{\uparrow }^i$ is the forces at which the system depins from the incoherent state. The line $F_{\downarrow }$ is the force at which a moving system stops upon lowering the drive.

Figure 20

Phase diagram in the $\mu –F$ plane for a piecewise linear pinning force, with $a=1∕\pi$ and $c=0$, and $\rho \left(h\right)=e^{-h}$. The depinning curve has been obtained numerically for a system with $N=1024$ and a ramp rate of $dF∕dt={10}^{-6}$.

Figure 21

Phase diagram, redrawn in the disorder-drive plane, for a discontinuous soft cubic pinning force of the type shown in Fig. 1b and $\rho \left(h\right)=\delta (h-1)$. The disorder $h$ and drive $F$ are normalized by the strength of the phase-slip interaction, $\mu$. The parameter values and the symbols are the same as in Fig. 2.

Figure 22

The figure shows the behavior of the phase $\delta \left(\beta \right)$ for three values of $u$. The $n=\pm 1$ half-sections are dashed, while the $n=0$ section is solid. Curve (a) corresponds to $u⩽a$ and is single valued. Curves (b) and (c) are both multivalued and correspond to (b) $a<u⩽Y(\pi ∕2)$ and (c) $u>Y(\pi ∕2)$. The section ${\beta }_0$ ends at the points $\pm {\delta }_{\max }$, where the half sections $\beta =\mp 1$ begin. For curve (b) these points lie within the portion of the curve that must be included in the integral to determine $f\left(u\right)$. For curve (c) they lie outside. ${\delta }_{*}$ denotes the nonzero value of the phase at $\beta =-\pi$.

Figure 23

The figure shows the phase $\delta$ versus ${\beta }_n$ at $u=0.8$, for $n=0,\pm 1,\pm 2$, for the continuous pinning force of Fig. 1c, with $a=3\sqrt{3}∕\left(2\pi \right)$ and $c=-3\sqrt{3}∕\left(2{\pi }^3\right)$. The upper and lower branches, lying outside the range $[-{\delta }_{\max },{\delta }_{\max }]$ are unstable, while the central branch is stable.

Figure 24

Plot of $F_{\mathbf{tot}}$ versus $\delta$ for $\beta =\pi ∕2$, $u=0.8$ for the continuous pinning force of Fig. 1c, with $a=3\sqrt{3}∕\left(2\pi \right)$ and $c=-3\sqrt{3}∕\left(2{\pi }^3\right)$. The equation $F_{\mathbf{tot}}=0$ has two solutions. The left-hand solution, with negative slope is stable, while the right-hand solution, with positive slope, is unstable.

Figure 25

The phase $\delta \left(\beta \right)$ as a function of ${\beta }_n$ for $-2⩽n⩽2$ for two sets of values of $(u,F)$, corresponding to single-valued [for $(u=0.1,F=0.2)$] and multivalued [for $(u=0.35,F=0.2)$] solutions. Even $n$ branches are drawn as solid lines and odd $n$ branches are dashed.

Figure 26

The phase $\delta$ as a function of $\beta$ for various values of $u$ and $F<F_{\uparrow }^c\left(u\right)$. Also shown in each frame are the values of ${\delta }_{\min }$ and ${\delta }_{\max }$ defined in Eq. (B4) and the boundary points ${\delta }_R$ and ${\delta }_L$ of the connected portion of the function $\delta \left(\beta \right)$ that is used to evaluate the integrals determining the coherence in each case. The four curves corresponds to the four cases discussed in the text: (a) $u<u_{sv}\left(F\right)$, where $\delta \left(\beta \right)$ is single valued. In this case we can choose ${\delta }_L={\delta }_{\max }$, which requires ${\delta }_R={\delta }_{\max }$. As $F$ is increased, ${\delta }_{\max }$ grows until ${\delta }_{\max }={\delta }_L=\pi$ at $F=F_{\uparrow }^c$. (b) $u_{sv}\left(F\right)<u⩽u_1\left(F\right)$; (c) $u_1\left(F\right)<u⩽u_2\left(F\right)$; and (d) $u>u_2\left(F\right)$.

Figure 27

The force $F_{\uparrow }^c\left(u\right)$, as a function of $u$ for the hard pinning force, $Y\left(x\right)=-(x+x^3)∕(\pi +{\pi }^3)$. The arrows indicate the values $u_{sv}^{*}$, $u_1^{*}$, and $u_2^{*}$ separating the four different regions discussed in the text. These values are defined by the relations $u_{sv}\left(F_{\uparrow }^c\left(u_{sv}^{*}\right)\right)=u_{sv}^{*}$, $u_1\left(F_{\uparrow }^c\left(u_1^{*}\right)\right)=u_1^{*}$, and $u_2\left(F_{\uparrow }^c\left(u_2^{*}\right)\right)=u_2^{*}$. The plot becomes nonlinear beyond $u_2^{*}$ where the threshold goes from being determined by ${\delta }_{\max }=\pi$ to being determined by ${\delta }_L=\pi$.

Figure 28

(a) Area of the hysteresis loop in $v\left(F\right)$ as $F$ is cycled from large values to zero and then back up again near ${\mu }_e\approx 0.75$ for the scalloped potential. Different system sizes and ramp rates $dF∕dt$ are shown. Plot (b) is just a blowup of (a) very near ${\mu }_e$.

Figure 29

Area of the hysteresis loop in $v\left(F\right)$ near ${\mu }_e$ for the hard case, where ${\mu }_e$ is the intersection of $F_{\uparrow }^i$ and $F_{\uparrow }^c$. Different system sizes and ramp rates are shown.