European Physical Journal E 34, 117 (2011)

Characterizing Plastic Depinning Dynamics with the Fluctuation Theorem

J. A. Drocco1, C. J. Olson Reichhardt2, and C. Reichhardt2,a

1 Department of Physics, Princeton University, Princeton, NJ 08544, USA
2 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Received: 8 February 2011 and Received in final form 14 June 2011
Published online: 28 October 2011 - ©EDP Sciences/ Societa Italiana di Fisica / Springer-Verlag 2011

Abstract. We demonstrate that the fluctuation theorem can be used to characterize plastic flow phases in collectively interacting particle assemblies driven over quenched disorder when strong fluctuations and crackling noise with 1/fα character occur. By measuring the frequency of entropy-destroying trajectories and the diffusivity near the threshold for motion, we map out the different dynamic phases and demonstrate that the fluctuation theorem holds in the strongly fluctuating plastic flow regime which was previously shown to be chaotic. For different driving rates and disorder strength, we find that it is possible to define an effective temperature which decreases with increasing drive, as expected for this type of system. When the size of the pinning sites is large, we identify specific regimes where the fluctuation theorem holds only at long times due to an excess of negative entropy events that occur when particles undergo circular motions within the traps. We discuss how the fluctuation theorem could be applied to plastic flow in other driven nonthermal systems with quenched disorder such as superconducting vortices, magnetic domain walls, Coulomb glasses, and earthquake models.
1 Introduction
2 Method and Simulation
3 Nonthermal Fluctuations
4 Application of the Fluctuation Theorem
5 Nonequilibrium Temperature
6 Large Pinning Trap Regimes and Negative Events
7 Relation to Experiments
8 Summary
References

1  Introduction

There are a number of systems that can be modeled as a collection of interacting particles moving over random quenched disorder. Examples include vortices driven through strongly-pinned type-II superconductors [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16], electron crystals in solid-state systems [17,18], colloidal particles moving over random substrates [19,20], and charge transport through metallic dot arrays [21]. Such systems exhibit strongly non-thermal fluctuations in a disordered plastic flow regime. Similar fluctuating dynamics are also observed in the motion of magnetic domains [22,23] and assemblies of dislocations [24]. All of these show a transition from a pinned phase to a disordered moving phase, in which plastic flow occurs and the particles tear past their neighbors [1,2,8,9,11,12,15,19,20]. Plastic flow is associated with non-Gaussian velocity distributions [7,8,9], 1/f velocity noise [8,10,14], and rapidly changing one-dimensional flow channels [12,15]. Moreover, the particle lattice is strongly disordered, contains numerous topological defects, and has a liquidlike structure factor. At higher applied drive, however, the system ceases to flow plastically [3,4,5,6,7,8,9,13] and may show partial ordering into a smectic flow of partially coupled one-dimensional channels [4,5,6,8,9] or fuller ordering into an anisotropic crystal on shorter length scales with a smectic ordering on longer length scales [4,5,6,8].
Since the plastic flow is associated with nonequilibrium fluctuations, a novel means of characterizing the disordered flow regimes is with the recently developed fluctuation theorems (FTs) [28,29,30,31,32,33,34,35,36,37]. The FT has been described as a generalization of the second law of thermodynamics in systems outside the thermodynamic limit [30,31,32], and, accordingly, has attracted enormous attention in recent years. The Gallavotti-Cohen FT was demonstrated to hold analytically in the long-time limit for a class of time-reversible dynamical systems [28], and there are also extensions of the FT to stochastic or non-reversible systems that have been developed by several groups [33,35,36,37,41]. The FT has mainly been studied in systems exhibiting thermal fluctuations; however, recent studies show that the FT can be applied to intrinsically nonthermal driven systems [38] such as sheared colloids in the jammed state [39] and fluidized granular media [40]. This suggests that a similar application of fluctuation theorems could also provide a useful way to characterize the dynamics of plastic systems.
In this work we show how the FT can be used as a means to characterize plastic depinning in an overdamped system lacking thermal fluctuations. The FT has been shown to apply to the power dissipated by a thermal bath for particles obeying Langevin dynamics [35,36], but it is not guaranteed to apply in the zero temperature limit [42]. The noise which produces fluctuations in our system is strictly athermal and arises as a result of driving the system. Although the FT is generally thought to apply in systems which obey the chaotic hypothesis, the system we treat does not strictly obey the chaotic hypothesis. In previous work on plastic depinning it was shown that certain plastic flow regimes are chaotic while others are not, which implies that it may be possible to distinguish the chaotic plastic flow from nonchaotic plastic flow by determining whether the FT is obeyed by the fluctuations in the flow. The time window over which the FT appears to be valid can provide insight into how the flow dynamics change with parameters such as drive or disorder strength.

2  Method and Simulation

There are many FTs and we specifically examine the FT described in Ref. [34]. One of the main predictions of the FT is that the probability density function (p.d.f.) of the injected power p(Jτ) obeys the following relation:
p(Jτ)
p(−Jτ)
 =eJττ/βτ ,
(1)
where Jτ is the injected power, τ is the duration of the trajectory over which the power is integrated, and βτ approaches a constant value β∞ as τ→ ∞. p(Jτ) represents the probability of a positive injected power of a given magnitude, and p(−Jτ) represents the probability of an injected power of the same magnitude but negative, sometimes called a "second law violation". The value β∞ is referred to as a "nonequilibrium temperature" [33,40,43,44,45]. It is not related to an ambient temperature, which is zero for our system. We note that numerous other definitions of effective temperature have been proposed for nonequilibrium systems such as glasses [46]. Wang et al. [38] experimentally measured the quantity in Eq. 1 from the trajectories of a colloid driven through a thermal system. In the system we consider, there is no thermal bath and T = 0; instead, the particles experience only an external drive, a random quenched background, and interactions with other particles. Here, the fluctuations are generated via the plastic motion of the particles.
We consider colloidal spheres confined to two dimensions and driven with an electric field in the presence of randomly distributed pinning sites. This particular model system has been shown to exhibit the same general dynamical features, including plastic flow and moving crystalline phases [8], observed in other collectively interacting particle systems driven over random disorder such as vortices in type-II superconductors [1,2,7]; thus, we believe the behavior we observe will be generic to other systems of this type. Additionally, experimental realizations of the colloid system would permit the direct measurement of the particle trajectories [20].
We simulate a system of Nc=792 colloids at density ρ = 0.5 with periodic boundary conditions in the x and y directions, and employ overdamped dynamics such that the equation of motion for a single colloid i is
η dri
dt
 =fYi+fpi+fd .
(2)
All quantities are rescaled to dimensionless units, and the damping constant η is set to unity. There is no thermalization. The colloid interaction force fYi is given by the following screened Coulomb repulsion: fYi=∑j ≠ iNcAc([4/(rij)]+[1/(rij2)])e−4rij∧rij. Here Ac is an adjustable coefficient, ri(j) is the position of vortex i(j), rij=|ri−rj| and ∧rij=(ri−rj)/rij. The quenched disorder introduces a force fpi which is modeled by Np nonoverlapping randomly placed attractive parabolic pinning sites of strength Ap=0.5 and radius rp=0.45, fpi=∑k=1Np(−Aprik/rp)Θ(rp−rik)∧rik, where Θ is the Heaviside step function. Np varies slightly based on the number of sites which can be placed given the random seed used in a particular realization, but is approximately normally distributed with μN=1000 and σN=17. The driving force fd=fd∧x is a constant unidirectional force applied equally to all colloids. We initialize the system using simulated annealing [47] in order to eliminate undesirable transient effects due to relaxation, and apply a driving force. The equations of motion are then integrated by the velocity Verlet method [48] for 105−107 simulation time steps, depending on Nc. The time step dt=0.002.
We compute the longitudinal and transverse diffusivities Dα with α = x,y by fitting
〈[(ri(t+∆t)−ri(t)) ·
^
α
 
 ]2 〉 = 2Dα ∆t.
(3)
The injected power computed for a single colloid i over a time period of length τ is given by:
Jτ=⌠
⌡ 		t+τ

t 
 fd·vi(s) ds
(4)
where vi represents the instantaneous velocity of colloid i. A particle which moves in a retrograde fashion, or opposite to the direction of the driving force, makes a negative contribution to the injected power. We measure Jτ for each individual particle in a single run and combine this data to obtain p(Jτ). We resample the same data set to identify Jτ for a variety of τ ranging from a minimum of 10 simulation time steps to roughly one tenth the duration of the entire simulation.
Fig1.png
Figure 1: Illustration of noise properties. (a) Velocity time series v(t) for a single colloid in a system with fd=0.345, showing strong intermittency. Occasional motion against the driving force, indicated by negative values, appears. (b) Noise power spectrum S(f) for the v(t) shown in (a) averaged over an ensemble of 100 colloids. For f > 0.1, we find S(f) ∝ f−2.

3  Nonthermal Fluctuations

We first demonstrate that our colloidal system exhibits strong nonthermal fluctuations under T = 0 dynamics, as is characteristic of the class of systems that exhibit crackling noise. This system has previously been shown to exhibit disordered plastic flow in both simulations and experiments [19,20]. We focus on the plastic flow phase in which a portion of the particles are temporarily trapped for a period of time while the remaining particles are mobile. The distribution of particle velocities in the plastic flow regime is bimodal. The noise induced in collectively interacting systems with quenched disorder differs significantly from that found in simulations of thermal noise. In Fig. 1(a), we plot a velocity time series v(t) for a single colloid at fd=0.345, near the depinning transition. Here, v = vi ·∧x. Frequent, aperiodic starts and stops appear, as well as intermittent motion in the direction opposite to the driving force (indicated when the signal drops below zero). The velocity noise is not Gaussian, unlike the noise found for colloids in a thermal bath. In addition, the power spectrum of the velocity noise is not white, as shown by the plot of
S(f) = ⎢
⎢ 		⌠
⌡ 		e−i2πf v(t)dt⎢
⎢ 		2
 
(5)
in Fig. 1(b). We find a 1/f2 scaling of the noise power at high frequencies, and a 1/f noise signature at lower frequencies. Power spectra of this form are characteristic of nonthermal systems that exhibit crackling noise, such as dislocation dynamics, plastically deforming superconducting vortex matter, and magnetic Barkhausen noise.
Fig2.png
Figure 2: Colloid trajectories (lines) for all 792 colloids during 4 ×105 simulation time steps in a T=0 system with quenched disorder at (a) fd=0.27; (b) fd=0.34. In the lower drive system (a), the only motion is transient and limited to a small subset of the colloids. In the higher drive system (b), motion is persistent and trajectory traces are longer and more dense.
Fig3.png
Figure 3: Demonstration of FT in a nonthermal system driven over quenched disorder. (a) p(Jτ) for all observed trajectories with fd=0.345 at τ = 0.02, 4.02, 8.02, 12.02, and 16.02 (from upper right to lower right). (b) Identical to (a) with fd=0.375. (c) A blow-up of panel (a) highlighting the discontinuous behavior near Jτ=0. Dashed line is a least-squares fit to a non-centered Gaussian for p(Jτ) with τ = 12.02, indicating that the power p.d.f.'s deviate from a normal distribution. (d) Fit to Eq. 1 of the driven system with fd=0.345 for all 10.02 ≤ τ ≤ 30.02, with darker color indicating larger τ. The slope of the fit gives the inverse of the nonequilibrium temperature β∞.

4  Application of the Fluctuation Theorem

We inspect the distribution of power dissipated by individual colloids to test whether Eq. 1 is satisfied in the interacting colloid system. For fd < 0.33 the colloids are pinned and there is no nontransient motion, as illustrated in Fig. 2(a). Just above the depinning transition at fd = 0.34, the colloid motion persists with time and the trajectories are highly disordered as shown in Fig. 2(b). Approximately one third of the colloids are pinned at any given time; however, all of the colloids take part in the motion. In Fig. 3(a) we plot the strongly non-Gaussian p(Jτ) that appear in the absence of thermalization for τ = 0.02, 4.02, 8.02, 12.02, and 16.02 at fd = 0.345. The τ = 0.02 curve, most closely representative of the instantaneous distribution, peaks at Jτ=0 and is skewed in the positive direction by the applied drive. The power p.d.f. at a slightly larger drive of fd = 0.375, shown in Fig. 3(b), shares this feature, though for larger τ a clear peak emerges at Jτ > 0. At fd=0.345 near the depinning transition, however, each p.d.f. always has its largest peak at Jτ=0, where cusp-like discontinuous derivative occurs [Fig. 3(c)] due to the motionless particles trapped in pinning sites. As a result, there is an effective jump discontinuity at the sampled resolution of Jτ=0, violating the implicit assumption of continuity in Eq. 1. We expect that this will be a very general feature of systems with pinned states. While there is a positive correlation between log(p(Jτ)/p(−Jτ)) and Jτ, the fluctuation theorem is not satisfied for small τ as these curves do not pass through the origin. Taking τ larger effectively smooths the p.d.f. and lessens the discontinuity, making a fit to Eq. 1 possible at large τ, as shown in Fig. 3(d). We note that the fit to Eq. 1 is possible over a range of Jτ that is much larger than the range over which it is possible to fit a non-centered Gaussian to the power p.d.f. [Fig. 3(c), dashed line].
The quality of the fit to Eq. 1 depends on the value of fd. Earlier studies showed that systems with depinning transitions can exhibit different dynamical regimes as a function of external drive, including a completely pinned phase where there is no motion, a stable filamentary channel phase at depinning where a small number of particles move in periodic orbits [1], chaotic flow at higher drives when the filaments change rapidly with time [1,7], and a dynamically recrystallized phase at even higher drives where the particle paths are mostly ordered [7]. To quantify the quality of the fits to Eq. 1 we calculate the Pearson product-moment correlation coefficient r [49], which is a measure of the linear correlation between two variables, and the coefficient of determination R2, which measures the fraction of variance explained by the fitted model [50]. In Fig. 4(a) we plot the mean dissipation η〈v〉/fd versus fd for the system in Fig. 2(b), along with the corresponding longitudinal and transverse diffusivities Dx and Dy.
In Fig. 4(b) we show the value of R2 for varied fd and for all τ < τc(fd), where
τc(fd)= sup
{t | r(fd,τ) ≥ 0.5  ∀ τ < t}.
(6)
Agreement with the FT, indicated by R2 ≈ 1, holds over the largest range of τ at fd ≈ 0.34, in the plastic flow region coinciding with peaks in both Dx and Dy. Here, the colloids flow in plastic fluctuating channels, as shown in Fig. 2(b), which are rapidly changing over time. Previous studies of vortex plastic flow have shown that motion just above threshold occurs in the form of a small number of flowing channels that are generally static in time and that produce a periodic time signal as the vortices repeat the same path over and over again [15]. We observe such a static channel plastic flow regime for fd < 0.335. In this case, the chaotic hypothesis does not apply even though the system is undergoing plastic flow, and we find that the fits to the FT are poor. At higher drives where more flow channels open, it was shown previously that the velocity fluctuations become much more random, the velocity power spectrum is no longer periodic but has a 1/f noise characteristic, and that the flow becomes chaotic [8,12,15,25]. In the regime 0.335 < fd < 0.355, R2 comes the closest to reaching R2=1 for all the ranges of τ we analyze (with the exception of very short τ as noted above), indicating that this regime gives the best agreement with the FT. For higher drives fd >~0.4, Dx and Dy drop and we find an onset of dynamical ordering where the particles form an anisotropic crystal structure that is similar to the dynamical reordering observed in vortex systems at sufficiently high drive [3,5,7,8]. As fd increases, the FT continues to hold for small τ, with the maximum value of τ for which R2 > 0.5 decreasing with increasing fd. This is a result of the fact that on short time scales, the particles experience a "shaking temperature" Ts as they move over the pinning sites which decreases as Ts ∝ 1/fd [3,16]. At longer times, however, the formation of a partially ordered crystal structure limits the size of the fluctuations the particles can experience and eliminates retrograde trajectories, causing the agreement with the FT to fail at long times.
Fig4.png
Figure 4: Limits of regime in which FT is verified in the system from Fig. 3. (a) Solid curve: mean dissipation η〈v〉/fd vs fd, relating colloid displacements to applied drive. Upper crosses: longitudinal diffusivity Dx vs fd. Lower crosses: transverse diffusivity Dy vs fd. (b) Coefficient of determination R2 of the fit log(p(Jτ)/p(−Jτ))=mJτ as a function of τ and fd. Values closer to 1 indicate better agreement with the FT. The FT holds over the largest range of τ in the fluctuating plastic flow regime near fd ≈ 0.34 illustrated in Fig. 2(b); however, curves for τ <~10 are poor fits in this region given the discontinuous p.d.f. (see text).
Fig5.png
Figure 5: (a) Nonequilibrium temperature βτ in a nonthermal system with quenched disorder at fd=0.335, 0.340, 0.345, and 0.350 (from top to bottom). In all cases βτ relaxes to the intrinsic noise level within τ <~10. (b) β20=〈β15 < τ < 25〉 vs fd for ρ = 0.1 (x), 0.2 (+), 0.3 (\Diamond), 0.5 ([¯]), and 0.6 (\bigcirc).

5  Nonequilibrium Temperature

As described previously, the FT allows the definition of a "nonequilibrium temperature" βτ→∞ when sufficient retrograde trajectories of duration exceeding the microscopic time scales of the system can be sampled. This necessarily involves a balance of time scales since the second law of thermodynamics guarantees that p(Jτ < 0)=0 as τ→∞. In Fig. 5(a), we plot βτ versus τ showing the existence of an asymptotic nonequilibrium temperature β∞ in a nonthermal system with quenched disorder. We find that the time scale of relaxation to the asymptotic value is unchanged when varying both the applied drive and particle density (data not shown). We note that other approaches have been used to establish effective temperatures for systems of vortices driven over random disorder using generalized fluctuation-dissipation relationships [16].
Having established that we are able to measure a nonequilibrium temperature β∞ within the time scale explored by the simulation, we observe how this quantity changes as we vary fd and colloid density ρ by calculating β20=〈β15 < τ < 25〉. As predicted, we see in Fig. 5(b) that β20 decreases with increasing fd and with decreasing ρ; however, the variance with ρ appears secondary since the value of fd required for depinning is anticorrelated with ρ. We cannot measure a broader range of fd as we can only define a nonequilibrium temperature for those values of fd where the FT holds over a wide range of τ, limiting us to drives near the depinning threshold where plastic flow occurs.
Fig6.png
Figure 6: Nonequilibrium temperature βτ/β100 for a system with fd=0.345 and varied pinning radius rp=0.45, 0.55, 0.65, 0.75, 0.85, and 0.95.

6  Large Pinning Trap Regimes and Negative Events

We next examine βτ for a system with the same parameters but for varying trap radii up to rp = 0.95 as shown in Fig. 6. The time required to reach an asymptotic value of βτ increases with increasing rp. This occurs when a portion of the colloids are able to undergo significant rotational motion within the traps while still remaining confined, producing an excess of retrograde trajectories. We also observe an interesting overshoot effect at intermediate times for the intermediate trap sizes. The overshoot arises because each colloid spends only a finite amount of time trapped in a well. For the smaller trap radii, the colloids are trapped only for a very short time which is less than τ, while for larger trap radii, some colloids remain within a particular trap for the entire duration of the simulation. At the intermediate trap radii of 0.45 < rp < 0.85 for 10 − 20τ, the colloids each spend a portion of the time in the trap and than hop. For rp > 0.75, the colloids located within the traps almost never leave during the duration of the simulation and the dynamics are dominated by the flow of unpinned colloids. This result shows that the time evolution of βτ can be an effective way to probe the dynamics of the system.

7  Relation to Experiments

We have demonstrated a connection between the range of τ over which fits to the FT can be performed and the nature of the disordered flow in our interacting particle system driven over quenched disorder. This suggests that the FT can provide an additional tool for quantifying differences between different types of disordered flow. The flow ranges from individual filaments of motion near depinning, where the motion is too cyclic to permit good fits to the FT; highly braiding chaotic flow slightly higher above depinning, where the FT can be fit over a wide range of τ but not at extremely short τ due to a discontinuity in the injected power p.d.f.; and increasingly ordered flow for drives well above depinning, where the FT can be fit only at short τ for times less than the caging time of individual particles. Experimental tests of the FT for systems of collectively interacting particles in the presence of quenched disorder could be performed in several ways. The trajectories of superconducting vortices could be directly imaged using various techniques [51], permitting extraction of the quantity v(t) for individual vortices over a time span that does not exceed the time required for the vortex to cross the field of view. The recent demonstration of dragging a single vortex through a sample [52] makes it possible to perform a vortex experiment analogous to the colloid experiment of Wang et al. [38]; this would permit access to a much larger range of τ than can be obtained from the imaging experiments. For example, a single vortex could be held in place while the remaining vortices flow plastically around it, and the fluctuations in force required to hold the vortex in one spot could be measured and analyzed as described in our work. The most straightforward measurement would be to measure local voltage fluctuations with a Hall probe [14] at a constant drive to determine the power dissipation. Local probes could also be used to study fluctuations in Wigner crystal [18] and magnetic domain wall [22] systems.

8  Summary

We have shown that the fluctuation theorem can be used to characterize the class of far from equilibrium nonthermal systems of interacting particles driven over quenched disorder where there is a transition from a stationary to a moving state. The FT holds for the regime near the onset of motion where the transport is nonlinear, the particle trajectories are disordered with 1/f noise characteristics, and the particle displacements exhibit diffusive behavior, in agreement with the expectation that regimes that agree best with the FT should be chaotic. As a function of the trajectory length τ and the magnitude of the external drive, we use the goodness of fit to the FT to construct a dynamic phase diagram showing that in the plastic flow regime the FT fits well out to long trajectory lengths. In the dynamical reordering regime, we find that the FT fits well only for short trajectory lengths due to the short time effective temperature experienced by the particles as a result of being driven over the quenched disorder, but that for longer trajectory lengths, the partial ordering of the particle lattice cuts off the fluctuations and gives poor fits to the FT when the system drops out of the chaotic regime. For drives right at the onset of motion, where the flow occurs in a small number of static channels through the system, the FT does not hold since the particle orbits are periodic and the chaotic behavior is lost. The FT cannot be applied to the pinned phase since there are no fluctuations. We demonstrate that it is possible to extract an effective temperature based on the FT in the plastic flow regime, and that this temperature complements the effective shaking temperature defined in earlier work which cannot be applied to the plastic flow regime. Systems in which these results could be tested with local transport measurements include magnetic domain walls, vortices in type-II superconductors, sliding Wigner crystals, driven stripe systems, and earthquake models. It would also be interesting to apply this approach to analyze other non-thermal systems that exhibit similar crackling noise, such as dislocation dynamics.
This work was carried out under the auspices of the NNSA of the U.S. DOE at LANL under Contract No. DE-AC52-06NA25396. J.A.D. was supported by the Krell Institute Computational Science Graduate Fellowship, U.S. DOE grant DE-FG02-97ER25308.

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