Large compact clusters and fast dynamics in coupled nonequilibrium systems
Shauri Chakraborty, Sukla Pal, Sakuntala Chatterjee, Mustansir Barma
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y Large compact clusters and fast dynamics in coupled nonequilibrium systems
Shauri Chakraborty , Sukla Pal , Sakuntala Chatterjee and Mustansir Barma Department of Theoretical Sciences, S.N. Bose National Centre for Basic Sciences,Block - JD, Sector - III, Salt Lake, Kolkata 700098, India TIFR Centre for Interdisciplinary Sciences, 21 Brundavan Colony,Osman Sagar Road, Narsingi, Hyderabad 500075, India.
We demonstrate particle clustering on macroscopic scales in a coupled nonequilibrium systemwhere two species of particles are advected by a fluctuating landscape and modify the landscape inthe process. The phase diagram generated by varying the particle-landscape coupling, valid for allparticle density and in both one and two dimensions, shows novel nonequilibrium phases. Whileparticle species are completely phase separated, the landscape develops macroscopically orderedregions coexisting with a disordered region, resulting in coarsening and steady state dynamics ontime scales which grow algebraically with size, not seen earlier in systems with pure domains.
PACS numbers: 05.40.-a, 64.75.Gh, 68.43.Jk
Particle clustering is important in many natural physical and biological phenomena, for instance, the formation ofsediments [1] and protein clustering on a biological membrane [2]. Evidently, it is important to understand processesthat cause clustering in different physical contexts, and how these processes influence the properties of the cluster andthe time taken to form it. Often, large-scale clustering is driven by interactions with an external medium which itselfhas correlations in space and time [3–5]. An important physical effect in such systems is the back-influence of theparticles on the medium. This interaction can aid clustering, or destroy it. If a cluster does form, it may be compactand robust, or a dynamic object that undergoes constant reorganization. The formation time may grow exponentiallywith the size, or as a power law. Given this wealth of possibilities, it is important to look for an understanding, withinsimple models, of the circumstances under which different sorts of macroscopically clustered states occur.In this letter, we derive the phase diagram of a simple model system as we vary the interaction between theenvironment and particles. In the process, we unmask a novel non-equilibrium phase of particles with compactclustering and rich and rapid dynamics coexisting with a macroscopically organized landscape. The model has partialoverlap with the lattice gas model of Lahiri and Ramaswamy (LR) for sedimenting colloidal crystals [6, 7], but thenew phases manifest themselves outside the LR regime. Our results hold in both one and two dimensions.The model consists of two sets of particles moving stochastically in a fluctuating potential energy landscape.Particles try to minimize their energy by (a) moving along the local potential gradient of the landscape and (b)modifying the landscape around them in such a way as to lower the energy further. The model is generic but wediscuss it in the language of particles confined to move on a fluctuating surface in the presence of gravity, where theparticles can locally distort the surface shape to further lower the energy (see Fig. 1). One of the particle species isconsidered lighter and the other is heavier; we use the name LH (Light-heavy) model to describe the system. Process(b) affects the landscape dynamics quite differently in parts which are rich or poor in one species of particle, ultimatelyresulting in the formation of distinct macroscopic regions, each corresponding to a phase. Our study reveals a rich setof phenomena: strong phase separation with fluctuationless phases for particles, but a different sort of organizationfor the landscape; a rapid approach to the steady state; and intricate steady state dynamics of the interfaces betweenphases, with three distinct temporal regimes.There has been a recent surge of activity in the field of coupled driven diffusive systems [8–10], and it is usefulto view our work in this context. This activity has resulted in a catalogue of universality classes which describehow propagating modes in these systems decay in time. The modes themselves are defined by diagonalizing coupledhydrodynamic equations to linear order, with eigenvalues giving their speeds of propagation. The disordered phase ofour system is indeed described by this theory. But the ordered phases of primary interest to us correspond to complex eigenvalues at the linear level [11]; the imaginary parts signal instabilities, heralding macroscopic phase separation.However, such linear analysis cannot reveal the characteristics of the final phases, which can and do differ fromeach other in fundamental respects. Our results provide the necessary characterization and thus contribute to theimportant goal of classifying ordered phases in coupled driven diffusive systems.In a nutshell, the quintessentially nonequilibrium states found here exhibit phase separation, with qualitativelydifferent types of ordering for the particles and landscape, quite unlike systems known earlier. In particular, particlesdisplay strong phase separation [7] characterized by pure, fluctuationless phases, which cohabit with three macroscopicregions of the surface, two of which possess long range order, while the third does not. These findings differ markedlyfrom the strongly phase separated states found earlier in the LR and ABC models [7, 12] and imply strong changesfor both static and dynamical properties. Notably, the approach to our steady state is rapid, with a coarsening timethat grows as a power law of size, as opposed to the much slower exponentially growing time scales found earlier. (a) T i m e (b) (c) FIG. 1: Schematic presentation of different phases in one and two dimensions. (a) shows the coarsening mechanism in onedimension for b ′ ≤ H ( L ) particles are shown by solid (hollow) circles. (b) and (c) show typical configurations in twodimensions. In (b) H ( L ) cluster shown by red (blue) color and (c) shows the equal height contours. The LH model consists of two coupled driven diffusive systems, with conserved quantities. This is a lattice modelof H (heavier) and L (lighter) particles with damped motion under gravity and residing on a fluctuating surface. Thelocal dynamics of the particles and the surface are coupled: H and L particles at neighboring sites may interchangelocations, and do so preferentially if the local tilt of the surface favors a downward move for H . Particles reside onlattice sites and interact via hard-core exclusion: a site holds at most one particle ( H or L ). If the symbols / and \ indicate upward and downward tilts of the surface, respectively, then the particles follow the dynamics: W ( H \ L → L \ H ) = D + aW ( L \ H → H \ L ) = D − aW ( H/L → L/H ) = D − aW ( L/H → H/L ) = D + a (1)where W denotes the probability per unit time for a particular process to occur. This dynamics conserves the totalnumber of H (and L ) particles. Under the weight of the particles, a local hill on the surface gets pushed downward,at a higher rate by H than by L . In one dimension, the surface consists of a chain with N sites. The lattice bondsrepresenting discrete surface elements, can have two possible orientations with slopes τ i +1 / = ±
1, which are calledupslope and downslope bonds, respectively. In one dimension surface dynamics can be represented as W ( /H \ → \ H/ ) = E + bW ( \ H/ → /H \ ) = E − bW ( /L \ → \ L/ ) = E − b ′ W ( \ L/ → /L \ ) = E + b ′ (2)This dynamics conserves the overall slope. In two dimensions the surface consists of a square lattice and height of asite can change, provided all four neighboring sites are at the same height [13, 14]. We consider periodic boundaryconditions with no overall tilt of the surface.Figure 2 shows the phase diagram of the system in bb ′ plane, with a taken to be positive. It follows from Eq. 2that interchanging b and b ′ is tantamount to interchanging the H and L species. Hence we consider positive b , while b ′ can be positive or negative or zero. FIG. 2: Phase diagram in the b − b ′ plane. For b > b ′ >
0, the system shows SPS. The dotted horizontal and vertical linesare related to each other via interchange of the two particle species. On these lines the system is in IPS phase. The stripedregion ( − b < b ′ <
0) represent FPS phase. b = − b ′ line corresponds to FDPO phase. The dotted region ( b ′ < − b ) correspondto disordered phase. Strong Phase Separation (SPS) : The right half of the phase diagram, b ′ >
0, corresponds to the LR model [6, 7], asappropriate to sedimenting colloidal crystals [6]. In this regime, the light particles tend to move the surface upward.In steady state, the upslope and downslope surface bonds phase separate to form a single macroscopic valley andhill, which hold all the H and L particles, respectively, in separated clusters. Both particles and tilts show SPS;the approach to steady state is extremely slow (involving times T relax ∼ exp( αN )) owing to the formation of largemetastable barriers.In the left half of the phase diagram in Fig. 2 we have b ′ <
0, which means that the part of the surface occupiedby L particles shows a downward drift. As discussed below, different phases are obtained depending on whether thisvelocity is larger than, smaller than, or the same as the velocity imparted by the H particles. Disordered Phase : When the L particles push the surface faster (dotted region in Fig. 2), neither of the particlespecies nor the tilts shows long-range order. Interesting dynamical aspects of this wave-carrying disordered phase willbe presented elsewhere [15]. Fluctuation-dominated Phase Ordering (FDPO) : On the line b = − b ′ , both L and H particles push the landscapedown at the same rate. This implies Kardar-Parisi-Zhang dynamics for the surface while L - H exchange rules (Eq.1) imply that H -particles tend to collect in local valleys. This reduces to the passive scalar problem studied earlier,in which particles exhibit FDPO, characterized by singularities of the 2-point correlations and giant fluctuations ofthe density [16, 17]. Interestingly, this phase boundary can be identified exactly by looking for the onset of complexeigenvalues in a linear stability analysis of the coupled hydrodynamic equations for the landscape and the particles[14]. Fast-fall with Phase Separation (FPS) : When L particles push the surface down at a slower rate than H ’s, a newphase ensues (shown striped in Fig. 2). In steady state, L and H particles separate completely as in SPS. The surfaceunderlying the H -cluster forms a macroscopic valley but unlike SPS, the phases are not pure, e.g. the macroscopicmajority-upslope region accommodates a finite fraction (1 − m ) of downslope bonds. The majority-upslope region inturn acts as a ‘tilt reservoir’ which drives a finite tilt current through the part of the surface which holds L particles.By identifying an upslope (downslope) bond with a particle (hole) we identify the phase as the maximal current phasein an open-chain asymmetric exclusion process [14, 18]. Consequently, near the edges of the L -domain the tilt density ρ shows an algebraic 1 / √ r variation, while in the bulk ρ ≈ / J = 2 bm (1 − m ) inthe two arms holding the H particles with that in the maximal current phase J ′ = 2 b ′ / L -rich portion, wededuce 4 m (1 − m ) = b ′ /b , a relation we have verified numerically. The presence of a finite tilt current through thesystem results in a finite downward velocity of the surface and in steady state, the full surface moves downward atfinite speed, preserving the macroscopic valley and disordered tilt region, along with the pure domains of H and L particles (Fig. 1a). Infinitesimal-fall with Phase Separation (IPS) : For b ′ = 0 (the vertical dashed line in Fig. 2) the local fluctuationsin the surface occupied by L particles are of the symmetric Edwards-Wilkinson type. In this phase, the H and L particles again form pure domains. The surface beneath the H -cluster has the shape of a deep valley consisting of t σ D
28 32 36 40 N D D ~ / N D ~ e - α N N = N = N = FIG. 3: Occurrence of three regimes in the steady state dynamics of the deepest point of the valley. Main plot: Mean squareddisplacement of the deepest point of the valley as a function of time. The displacement shows an initial diffusive growth,followed by a plateau, and finally another diffusive regime at large time. Bottom Inset: Small time diffusivity D ∼ /N . TopInset: For large time, the diffusivity D ∼ e − αN , with α ≃ .
26. These data correspond to b ′ = 0, b = E , a = D and have beenaveraged over 5000 steady state configurations. pure domains of upslope and downslope bonds. By contrast, the surface occupied by the L -cluster in this case behaveslike an open-chain symmetric exclusion process connected to the two reservoirs of upslope and downslope tilts at thetwo ends. Thus the tilt density varies linearly in this region with a gradient ∼ /N [14, 19], leading to a tilt currentand an infinitesimal downward velocity ∼ /N of the entire landscape. We schematically show a typical configurationin Fig. 1a.The general properties of the phases discussed above remain valid even in two dimensions, where H and L particlesform compact clusters. The shape of these two dimensional clusters depends on the topography of the surface heights.As in one dimension, we find a deep valley that holds the H -cluster. Measured from the bottom-most point, theheight increases linearly in both x and y directions and it is easy to see that in this case the equal height contourshave the shape of a diamond. In Fig. 1b and 1c, we show some representative configurations.The way in which the landscape is organized in the IPS and FPS phases has a profound influence on dynamicalproperties. For instance, although the centre of mass of the H cluster remains stationary for a long time, the landscapeimmediately below it undulates in time, leading to three distinct temporal regimes in steady state. These are capturedby monitoring the mean-squared displacement σ of the deepest point of the valley. Figure 3 presents the data for b ′ = 0. At small time t ≪ N , we find σ grows diffusively with a diffusion constant D ∼ /N . But after timesof order N , a plateau for σ is reached at a value ∼ N . From a simple consideration of the total (gravitational)energy of the H particles, it is easy to show that when the deepest point coincides with the centre of mass of the H cluster, the energy is minimum. Any displacement from this position gives rise to a restoring force that scales linearlywith the displacement. The motion of the deepest point is thus described by an Ornstein-Uhlenbeck process [20];consequently, the deepest point diffuses within a region of size √ N around the H cluster centre of mass [14]. Finally,at very large t , the H cluster itself moves diffusively around the system and the valley moves along with it (see [14]).The mean-squared displacement of the deepest point in this regime has a diffusion coefficient D ∼ e − αN .Another important aspect of the dynamics concerns the relaxation to steady state starting from an initially disor-dered state. Interestingly, IPS and FPS phases show an enormous reduction in this relaxation time, as compared to r/t C ( r , t) t=5X10 t=1X10 t=2X10 t=4X10 r/t C ( r , t) t=2X10 t=4X10 t=6X10 t=8X10 r/t C ( r , t) t=5X10 t=7X10 t=1X10 t=1.4X10 r/t C ( r , t) t=5X10 t=7X10 t=1X10 t=1.4X10 (a) (b)(c) (d) FIG. 4: Scaling of particle density correlation in the coarsening phase. in both one and two dimensions. The equal timedensity correlation for the particles C ( r, t ) shows a collapse when r is scaled by L ( t ) ∼ t /z . (a) and (c) show scaled data for b = 0 . , b ′ = 0 in one and two dimensions, respectively. Here we find z ≃
2. (b) and (d) show similar plots for b = 0 . , b ′ = − . z ≃ z ≃ .
6. We have used N = 1024(a), N = 16384(b) and N = 256 × earlier known examples of SPS as in the LR and ABC models [7, 12]. For b ′ > L -cluster tends to move upwards and forms a hill, while an H -cluster pushes the landscape down and forms avalley. For two adjacent valleys with H -clusters to merge, the time to dissolve the intermediate hill containing the L -cluster grows exponentially with the size of the L -cluster, and hence the final SPS state is reached over a time-scale e γN . By contrast, in the FPS and IPS ( b ′ ≤
0) phases, the landscape is organized differently, and this leads to fastrelaxation, with times growing as N z . This is because the part of the landscape beneath the intervening L clustereither shows symmetric fluctuations (for b ′ = 0) or gets pushed downward (for b ′ < H particles when separations are scaled by the coarseninglength scale L ( t ), which is found to grow as t /z . In IPS phase ( b ′ = 0) we find z = 2 in both one and two dimensions.The FPS phase ( − b < b ′ <
0) shows z ≃ N and t while for smaller values of thesevariables, our data show finite size effects (see [14] for details). In two dimensions for FPS, we measured z ≃ . N and t we could access. Our plots in Fig. 4 constitute the first observation of algebraiccoarsening for completely phase separated systems, and stand in strong contrast to the ultraslow logarithmic coars-ening observed in the LR and ABC models [7, 12]. Underlying the speed-up of the coarsening process is a simplemechanism, namely the formation of disordered segments of the landscape between ordered clusters. These segmentsgenerate fluctuations which allow mergers of ordered regions to occur on a rapid time scale.To summarize, we have considered coupled driven systems consisting of two species of particles being advected byan energy landscape whose dynamics is in turn influenced by the particles. The differential interaction between thelandscape and the two particle species gives rise to different phases in the system as the interaction parameters arevaried. We have demonstrated the occurrence of new phases with fast dynamics, where both the particles and thelandscape show long range order and the composite system is in a nonequilibrium current-carrying state.We conclude with a discussion of the implications of our work for modeling in biophysical contexts. It is knownthat on the cell membrane, various proteins and lipids like integrins, T-cell receptors are present in the form of nano-clusters [21, 22]. This clustering is shown to be induced by the cortical actin cytoskeleton [2, 23], and within a recenttheoretical model of the process, an FDPO state has been observed [24]. There is now experimental evidence that theactin cytoskeleton also gets reorganized by these membrane components [22, 25]. This raises the interesting possibilityof new phases arising if the treatment of [24] is extended to account for the back action of membrane components onthe cytoskeleton.We acknowledge useful discussions with T. Sadhu, M. Rao, and A. Das. The computational facility used in thiswork was provided through Thematic Unit of Excellence on Computational Materials Science, funded by Nanomission,Department of Science and Technology, India. [1] S. Ramaswamy, Advances in Physics , 297 (2001)[2] K. Gowrishankar, S. Ghosh, S. Saha, C. Rumamol, S. Mayor, and M. Rao, Cell , 1353 (2012).[3] J. M. Deutsch,
J. Phys. A , 1449 (1985)[4] M. Wilkinson and B. Mehlig, Phys. Rev. E , 040101 (2003)[5] B. Drossel and M. Kardar, Phys. Rev. Lett. , 614 (2000); A. Nagar, M. Barma and S. N. Majumdar, Phys. Rev. Lett. , 240601 (2005)[6] R. Lahiri, S. Ramaswamy, Phys. Rev. Lett. , 1150 (1997).[7] R. Lahiri, M. Barma and S. Ramaswamy, Phys. Rev. E , 1648 (2000).[8] H. van Beijeren, Phys. Rev. Lett. , 180601 (2012).[9] P. Ferrari, S. Sasamoto and H. Spohn,
J. Stat. Phys.
377 (2013).[10] V. Popkov, A. Schadschneider, J. Schmidt and G. Sch¨utz,
Proc. Natl. Acad. Sci. USA , , 12645 (2015)[11] S. Ramaswamy, M. Barma, D. Das and A. Basu, Phase Transit. , 363 (2002)[12] M.R. Evans, Y. Kafri, H.M. Koduvely, and D. Mukamel, Phys. Rev. Lett. , 425 (1998).[13] G. Manoj and M. Barma, J. Stat. Phys. in preparation [16] D. Das and M. Barma,
Phys. Rev. Lett. , 1602 (2000); D Das, M Barma and S. N. Majumdar, Phys. Rev. E , 046126(2001); S. Chatterjee and M. Barma, Phys. Rev. E , 011107 (2006).[17] R. Kapri, M. Bandopadhyay and M. Barma, Phys. Rev. E , 012117 (2016).[18] B. Derrida, E. Domany and D. Mukamel, J. Stat. Phys ,
667 (1992).[19] B. Derrida, J.L. Lebowitz, and E.R. Speer,
J. Stat. Phys. , 599 (2002).[20] G.E. Uhlenbeck and L.S. Ornstein
Phys. Rev. , 823 (1930).[21] G. J. Bakker, C. Eich, J. A. Torreno-Pina, R. Diez-Ahedo, G. Perez-Samper, T. S. van Zanten, C. G. Figdor, A. Cambiand M. F. Garcia-Parajo, Proc. Natl. Acad. Sci. USA , , 4869 (2012).[22] C. Yu, J. B. K. Law, M. Suryana, H. Y. Low and M. P. Sheetz, Proc. Natl. Acad. Sci. USA , , 20585 (2011).[23] D. Goswami, K. Gowrishankar, S. Bilgrami, S. Ghosh, R. Raghupathy, R. Chadda, R. Vishwakarma, M. Rao, and S.Mayor, Cell , 1085 (2008).[24] A. Das, A. Polley and Madan Rao,
Phys. Rev. Lett. , 068306 (2016).[25] R. Varma, G. Campi, T. Yokosuka, T. Saito and M. L. Dustin,
Immunity , 117 (2006). Appendix A: Simulation details in two dimensions and finite size effects
The surface in two dimensions is simulated through a discrete solid-on-solid algorithm, where the height differencebetween the nearest neigbours on a square lattice is maintained at ±
1. Let h ( i, j ) denote the height of the site ( i, j )on the lattice. This site is said to be on a local hill, if all its four neigbours with coordinates ( i ± , j ) and ( i, j ± h ( i, j ) −
1. Similarly, the site ( i, j ) is said to be in a valley when the neighbors have height h ( i, j ) + 1.The two dimensional surface evolves in time by switching between the hills and the valleys. In our model, a site isselected at random. If it is on a hill, then it can flip to a valley when its height gets reduced by two units. Thisflipping rate is ( E + b ) when the site is occupied by an H particle and ( E − b ′ ) when it is occupied by an L particle.Similarly, if the chosen site happens to be in a valley, then its height can increase by two units and it becomes a hill.The switching rate in this case is ( E − b ) when an H particle is present, and ( E + b ′ ) when an L particle is present.For updating the particles, we select a bond (horizontal or vertical) of the square lattice. If the two sites adjacent tothe bond are occupied by two different species of particles, then their positions are exchanged with rate ( D + a ) ifafter the exchange, height of the H particle decreases. The reverse exchange occurs with rate ( D − a ).In our simulation, we start with a flat initial configuration of the surface and random configuration for the particlesand evolve the system in time, following the algorithm described above. In steady state the surface develops a singlevalley where the equal height contours resemble diamonds, shown in Fig. 1c in the main paper. However, in certaincases, we also observe another type of state, where a completely different height topography emerges. Instead of asingle minimum seen for diamond shaped contours, we find a line of minima and height increases along the directionperpendicular to this line. In other words, the surface has the shape of a trench. The equal height contours andthe perimeter of the H -cluster for such a configuration is shown in Fig. A-1. We have verified that such wedge-likeconfigurations result from finite size effects in our system with periodic boundary conditions and for larger systemsonly diamond-shaped contours are found. (A) (B) FIG. A-1: Representative plots for wedge type configurations in the IPS phase in two dimensions on a 64 ×
64 square lattice.In (A) H ( L ) clusters are shown by red(blue) color, while (B) shows equal height contour plots. Appendix B: Linear stability analysis of the coupled hydrodynamic equations
From the dynamical rules shown in Eqs. 1 and 2 in main text, the particle current and the tilt current can bewritten as J ρ = 2 aρ ( x, t )(1 − ρ ( x, t ))(1 − m ( x, t )) J m = m ( x, t )(1 − m ( x, t ))[2 ρ ( x, t )( b + b ′ ) − b ′ ] (B-1) ρ ( x, t ) and m ( x, t ) denote the coarse-grained densities of the H particles and the upslope bonds respectively. We writeeach density as sum of the average value and a small perturbation: ρ ( x, t ) = ρ + δρ ( x, t ) and m ( x, t ) = m + δm ( x, t )and expand currents in Eq. B-1 upto linear order in δρ and δm . This gives us the continuity equation ∂~u∂t + A ∂~u∂x = 0 (B-2)where we have used the compact notation ~u , denoting a column vector with components ρ and m . The matrix A isdefined as A = " ∂J ρ ∂ρ ∂J ρ ∂m∂J m ∂ρ ∂J m ∂m = (cid:20) a (1 − ρ )(1 − m ) − aρ (1 − ρ )2 m (1 − m )( b + b ′ ) 2 ρ (1 − m )( b + b ′ ) − b ′ (1 − m ) . (cid:21) For an untilted surface, m = 1 /
2, and the eigenvalues are λ = ± p − aρ (1 − ρ )( b + b ′ ) (B-3)For b > − b ′ , one has imaginary eigenvalues, indicating linear instability. In this case, the system shows macroscopicallyordered phases. For b < − b ′ , the eigenvalues are real and the system is disordered. The b = − b ′ straight line separatesthe ordered and the disordered phase and on this line, the system exhibit FDPO. Appendix C: Static Correlations in Steady State
In Figs C-1 and C-2, we present our results on the static correlation functions for the particles and for the surfacebonds in one dimension. Figure C-1 corresponds to IPS phase and Fig. C-2 correspond to FPS phase. Here we haveconsidered equal number of H and L particles and all distances have been measured from the centre of mass of the H -cluster. ρ ( r, N ) and S + ( r, N ) denote, respectively, the density of H particles and the probability to find an upslopebond at a distance r from the centre of mass. -100 -50 0 50 100 (r-N/4) ρ ( r , N ) L=512N=1024N=2048 r ρ ( r , N ) N=3000N=3250N=3500N=4000
500 750 1000 1250 r S + ( r , N ) N=3000N=4000N=5000N=6000 -1 -0.5 0 0.5 1 r/N S + ( r , N ) -80 -40 0 40 80 (r-N/4) ( - S + ( r , N )) N (a) (b)(c) (d) FIG. C-1: Static correlation functions in IPS phase. (a): Density of H particles ρ ( r, N ) at q = 0. The density changes sharplyfrom 1 to 0 indicating pure phases of H and L particles. (b): For q = ( D − a ) / ( D + a ) = 0 .
99 ( see Eq. 1 in the main text), H particle density approaches 1 as N is increased. This confirms the existence of a pure phase in the thermodynamic limit forall q <
1. (c) Main plot: For r ≫ √ N a scaling collapse is obtained for different system sizes when [1 − S + ( r, N )] N is plottedagainst ( r − N/ r changes sign, S + ( r, N ) shows a transition from 0 to 1 across the domain boundary,which is at the deepest point of the valley. Due to diffusive motion of the domain boundary, S + ( r, N ) is a scaling functionof r/ √ N in this region. In main plot and inset we show data for N = 512 (black circles), 1024 (red squares) and 2048 (bluediamonds). (d): For q = 0 .
99 and 0 ≪ r ≪ N/ S + ( r, N ) → N is increased. These data have been averaged over at least3 × steady state configurations. -50 -25 0 25 50 (r-N/4) ρ ( r , N ) N=4096N=8192 r ρ ( r , N ) N=384N=512N=768
40 60 80 100 (r-N/4) S + ( r , N )- . r/N S + ( r , N ) N=1024N=2048 (a) (b)(c) (d) ~ ( r - N / ) - / FIG. C-2: Static correlation functions in FPS phase with b = 0 . b ′ = − .
2. (a): Density of H particles ρ ( r, N ). The densitychanges sharply from 1 to 0 indicating pure phases of H and L particles. (b): For q = ( D − a ) / ( D + a ) = 0 .
92 (see Eq. 1 in maintext), ρ ( r, N ) approaches 1 as N is increased. This demonstrates existence of pure L and H domains in the thermodynamiclimit, as long as q < i.e. the H particles preferentially slide downward. (c): S + ( r, N ) for small r remains constant at a valuewhich is less than unity. This indicates that the upslope and downslope bonds underlying H -cluster undergo phase separationbut no pure phases are formed. The variation of S + ( r, N ) in the region occupied by L cluster shows an algebraic decay to adisordered phase, as expected for maximal current phase of an open-chain ASEP. (d): The algebraic decay of S + ( r, N ) occurswith an exponent 1 /
2, consistent with earlier known results for maximal current phase. These data have been averaged overat least 10 steady state configurations. Appendix D: Mechanisms responsible for two distinct time scales in valley dynamics
In this section we explain the mechanisms for the short and large time valley dynamics. The deepest point of thevalley lies at the boundary between two pure domains of upslope and downslope surface bonds and these pure domainsextend upto the edges of the H particle cluster. Beyond those edges, S + ( r, N ) has a linear gradient of order 1 /N and it is easy to show by a simple mean-field calculation that the average distance of the nearest downslope (upslope)bond from the right (left) edge of the particle cluster ∼ √ N . Therefore this downslope (upslope) bond can diffusethrough this distance over a time-scale ∼ N , and form a local hill, as shown in Fig. D-1A and then the downslope(upslope) bond moves ballistically down the valley, over a time-scale ∼ N , and reaches the deepest point causinga unit displacement of the deepest point towards right (left). Thus the process illustrated in Fig. D-1 takes placeover a time-scale ∼ N which explains the D ∼ /N behavior for short times. The mechanism also shows that theparticles do not undergo any displacement in this case. The deepest point of the valley slides back and forth beneaththe H particle cluster. This motion, however has an energy cost, and as discussed in the main text, for a distance ∆between the deepest point and the centre of mass of the H cluster, the energy cost ∼ ∆ . Thus the motion of thevalley can be described by an Ornstein-Uhlenbeck process. This prevents the valley from taking a large excursion ineither direction and as a result the mean-squared displacement reaches a plateau.In the limit of very long time, the H particle cluster will start moving around the system diffusively and the valleywill naturally move along with it. This mechanism has been illustrated in Fig. D-2 which shows movement of an L particle through the H particle cluster. When the L particle reaches the bottom of the valley, the deepest pointundergoes a displacement, along with the center of mass of the H particle cluster. Note that the time-scale for this1 (A) (B)(C) (D) FIG. D-1: Motion of the deepest point over ∼ N time-scale. (A) An occupied local hill forms at the right edge of the particlecluster. (B) The occupied local hill, being unstable, zips through the pure domain of upslope bonds. (C) The local hill reachesthe bottom of the valley. (D) The deepest point shifts one site to the right. process is rather large because the probability that the L particle, starting from the HL domain boundary, reachesthe bottom of the valley, is very low and decays exponentially with the domain size (see our data in Fig. D-3). Asa result, the diffusivity of the valley in this regime ∼ e − αN . Existence of an exponentially large time-scale breaksthe translational invariance of the system but the relaxation time-scale is still algebraic, as demonstrated in our maintext. Appendix E: Dynamical exponent for the FPS phase
Two point density correlation function for the H particles is defined as C ( r, t ) = < ρ i ( t ) ρ i + r ( t ) > − ρ (E-1)where, ρ i ( t ) is 1(0) if the i th site is occupied by an H ( L ) particle at time t during the coarsening phase. The angularbrackets denote average over initial conditions. ρ denotes the global density of H particles in the system. Define r as C ( r , t ) = 0 and the dynamical exponent z can be estimated from the variation of r with t , using the scalingrelation r ∼ t /z . The plots of r vs t for FPS phase in one and two dimensions are presented in Figs E-1 and E-2,respectively.2 (A) (B)(C) (D)(E) (F)(G) (H) FIG. D-2: Motion of the deepest point over ∼ e N time-scale. (A) Formation of a local hill at the right edge of the H particlecluster. (B) The rightmost H particle slides down the hill and gets detached from the cluster. (C) The resulting hill with L particle flips and another local hill with H particle is formed at the adjacent site. (D) Another H particle slides down the hilland detaches from the H cluster. (E) The L particle propagates down the valley. (F) At the bottom of the valley a local hillwith L particle is formed. (G) This local hill flips and the deepest point of the valley shifts one site to the right. (H) The H particles to the left of the deepest point slide down, one after another, moving the center of mass of the particle cluster to onesite to the right.
15 20 25 30 35 N P r o b a b ili t y t o f i nd a h o l e a t t h e b o tt o m o f t h e va ll e y ~ e x p (- α N ) FIG. D-3: The probability to find an L particle at the deepest point of the surface falls off as e − αN with system size where α = 0 . histories. t r N=2048N=4096N=8192 ~ t . ~ t . FIG. E-1: Plot of r vs time during the coarsening phase for b = 0 . b ′ = − . r ∼ t . . For t < ∼ , we observe a crossover region where r ∼ t . . We have used ρ = 1 / t r L=128X128L=256X256 ~ t . FIG. E-2: Plot of r vs t during the coarsening phase for b = 0 . b ′ = − . r ∼ t . . We haveused ρ = 1 //