Focusing by blocking: repeatedly generating central density peaks in self-propelled particle systems by exploiting diffusive processes
FFocusing by blocking: repeatedly generating central density peaks in self-propelledparticle systems by exploiting diffusive processes
Andreas M. Menzel ∗ Institut f¨ur Theoretische Physik II: Weiche Materie,Heinrich-Heine-Universit¨at D¨usseldorf, D-40225 D¨usseldorf, Germany
Over the past few years the displacement statistics of self-propelled particles has been intenselystudied, revealing their long-time diffusive behavior. Here, we demonstrate that a concerted com-bination of boundary conditions and switching on and off the self-propelling drive can generateand afterwards arbitrarily often restore a non-stationary centered peak in their spatial distribution.This corresponds to a partial reversibility of their statistical behavior, in opposition to the above-mentioned long-time diffusive nature. Interestingly, it is a diffusive process that mediates and makespossible this procedure. It should be straightforward to verify our predictions in a real experimentalsystem.
PACS numbers: 82.70.Dd,47.63.Gd,87.16.Uv
I. INTRODUCTION
Diffusion [1] is one of the most studied and most fa-miliar phenomena in physics. Regular diffusion is wellaccessible on the different levels of description in statis-tical mechanics, making it a prominent textbook exam-ple [2–5]. On the single-particle level, the situation istypically modeled by corresponding Langevin equations[3–5]. These contain a stochastic force term to reproducethe experimentally observed Brownian motion. Likewise,the scenario can be characterized by a dynamic equationfor the probability density to find a particle at a certainposition at a certain time. This type of dynamic equa-tion is usually termed Smoluchowski equation or, moregenerally, Fokker-Planck equation [3–6]. The connectionsbetween the particle picture and these continuum equa-tions for the probability density are well established. Fi-nally, on a more macroscopic level, the phenomenon isaddressed by monitoring the time evolution of concen-tration profiles using diffusion equations [2, 3].One particular feature of diffusive phenomena is theirirreversibility. In general, it is not observed that an ini-tial density peak that has flattened over time will sud-denly rearise. This is reflected by the nomenclature inthe theoretical approaches. One refers to the irreversiblepart of the Fokker-Planck operator [5, 6] in the probabil-ity density picture and to the irreversible currents [3] inmacroscopic descriptions. A deeper theoretical founda-tion is provided by the H-Theorem [3–5], which allows aconnection to the increase of entropy or decrease in freeenergy.Here, we report on a globally diffusive system that isdemonstrated to at least partially break this paradigm ofirreversibility. Diffusive systems of initially peaked den-sity profiles generally do not reestablish such a densitypeak at a later time. Nevertheless, the example systemanalyzed below shows exactly such behavior, at least to ∗ [email protected] some degree. There, a non-stationary centered densitypeak can be generated independently of the initial distri-bution. Afterwards it can be restored arbitrarily often.It is the non-equilibrium nature of the investigated self-propelled particle system that allows for these effects,together with the possibility to switch on and off theself-propulsion. II. MODEL SYSTEM
Self-propelled particles feature a mechanism of acti-vated motion [7–11]. On the one hand, this can for ex-ample be the crawling or swimming machinery of motilebiological microorganisms such as bacteria and algal cells[12–18]. On the other hand, self-propelled particles wereartificially realized for instance in the form of granularhoppers [19–22] or self-propelled droplets [23–25]. A par-ticularly interesting example are Janus particles that al-low localized heating by light illumination of only onehalf of their body [26–30]; or they catalyze chemical re-actions on only part of their surface [31–33]. The re-sulting build-up of temperature or concentration gradi-ents along their surfaces induces stress differences thatcan effectively drive the particles forward in a type ofphoretic motion [34]. As a central common property, themigration direction of self-propelled particles is not im-posed or fixed from outside but results from symmetrybreaking [10]. Light-activated self-propulsion offers theadvantage of being easily switchable by turning on andoff the external illumination [26–30, 35].We consider for our purpose mostly the statistics of themotion of non-interacting self-propelled particles. Thisimplies that either dilute systems are investigated, or theaveraged statistical behavior of single objects is analyzed.At the end we will briefly address the effect of basic stericinteractions. As is natural e.g. for microswimmers, we ad-dress their overdamped motion. For simplicity and sincemany experiments are performed in this way [15–24, 26–32], we confine ourselves to two spatial dimensions. The a r X i v : . [ c ond - m a t . s o f t ] M a y corresponding Langevin equations for a single particlethen read (compare e.g. Refs. [32, 36]) d r dt = βD t [ F ˆu − ∇ V ( r )] + (cid:112) D t ξ r , (1) dφdt = (cid:112) D r ξ φ . (2)Here, r denotes the particle position and ˆu the orienta-tion of its self-propulsion direction. F sets the strengthof self-propulsion that drives the particle forward along ˆu . The potential V ( r ) leads to a force acting only onthe particle position. In our case, it will be nonzero onlyat the system boundaries to include the steric effect ofa confining cavity. t denotes time and β = ( k B T ) − with k B the Boltzmann constant and T the tempera-ture. In the two-dimensional plane, the self-propulsiondirection can be parameterized by a single angle φ via ˆu = (cos φ, sin φ ). D t sets the translational and D r the rotational diffusion constant. Finally, ξ r and ξ φ are Gaussian white noise terms with averages (cid:104) ξ r ( t ) (cid:105) = , (cid:104) ξ φ ( t ) (cid:105) = 0, (cid:104) ξ r ,i ( t ) ξ φ ( t (cid:48) ) (cid:105) = 0, (cid:104) ξ r ,i ( t ) ξ r ,j ( t (cid:48) ) (cid:105) = δ ij δ ( t − t (cid:48) ), and (cid:104) ξ φ ( t ) ξ φ ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ), where δ ij is theKronecker delta and δ ( t − t (cid:48) ) the Dirac delta function.We now switch to the probability density descriptionby deriving from Eqs. (1) and (2) the correspondingSmoluchowski or Fokker-Planck equation [4, 6, 37]. Indimensionless units, this dynamic equation for the prob-ability density ψ ( r , φ, t ) to find a particle at time t andposition r with orientation φ of the self-propulsion direc-tion becomes ∂ t ψ = − v (cos φ ∂ x ψ + sin φ ∂ y ψ ) + ∇ ψ + ∂ φ ψ + ∇ · [( ∇ V ) ψ ] . (3)Here, we rescaled time, space, and the potential by t (cid:48) = tD r , r (cid:48) = r (cid:112) D r /D t , and V (cid:48) = βV , respec-tively, with the primes omitted in the equation, and weconfine ourselves to the x - y plane. The only remain-ing parameter determining the system behavior is now v = βF (cid:112) D t /D r , which measures the relative strengthof the self-propelling drive. From the data listed inRef. [27] we infer that v = 30 constitutes a reasonableorder of magnitude.In our case, the potential V ( r ) only contains the effectof the cavity boundaries. Within the bulk of the cav-ity, the potential vanishes and the particle is completelyfree. Furthermore, V does not act on the orientation ofthe self-propulsion direction ˆu . This direction is solelydetermined by the independent rotational diffusion pro-cess Eq. (2). It has been demonstrated in various worksthat, as a consequence, the global and long-time transla-tional behavior of such particles is itself diffusive, with aneffective diffusion constant increased by self-propulsion[26, 27, 33, 36]. Despite these facts, we identify in thefollowing a procedure that breaks the irreversibility im-plied by this long-time diffusive behavior, at least par-tially. Remarkably, our procedure is mediated by andonly becomes possible due to rotational diffusion as de-scribed by Eq. (2). III. CHANNEL
We start by considering an infinitely extended chan-nel. This implies two parallel and infinitely extendedconfining walls, in our case along the ˆy direction. Forsimplicity, we choose the confining potential as V ( r ) = V ( x ) = (cid:40) k (cid:0) | x | − L (cid:1) if | x | > L , | x | < L , (4)where the width of the channel between the wall regionsis given by L >
0. The precise magnitude of the exponentand of the potential strength k > k = 50.As an initial condition, we consider self-propelledparticles located in the center of the channel withequally distributed orientations ˆu . The correspond-ing probability density is independent of the y coordi-nate, ψ ( x, φ, t = 0) = δ ( x ) / π . Consequently, the prob-lem becomes translationally invariant in the ˆy direction.The spatial probability profile ψ x ( x, t ) across the chan-nel is obtained by integrating out the φ -dependency of ψ ( x, φ, t ). We choose a channel width of L = 20, whichin the framework of Ref. [27] corresponds to about 11particle diameters.Our protocol is the following. We numerically integratethe resulting dynamic equation for ψ ( x, φ, t ) forward intime. Fig. 1 (a) shows the sharply-peaked initial proba-bility density profile ψ x ( x, t ≈
0) at a very early stage,together with the confining potential V ( x ). Due to self-convection and since all orientations of the self-propulsiondirection are equally probable at t = 0, this peak splitsand two fronts propagate outward towards the channelboundaries, see Fig. 1 (b). At the boundaries, the prob-ability density piles up and after a while reaches a sta-tionary profile as displayed in Fig. 1 (c). Such a blockingeffect due to a confining potential has been observed invarious forms before [27, 38–40]: the self-propelling drivepushes the particles towards the walls, and the only wayof escape is a reorientation of the self-propulsion direc-tion. In our case, this reorientation is solely due to rota-tional diffusion.We now switch off self-propulsion, setting v = 0. Asmentioned above, for light-activated mechanisms this cansimply be achieved by turning off the external illumina-tion [26–30, 35]. This has two effects as displayed inFig. 1 (d). First, the potential pushes the particles backtowards the edges x = ± L/
2. And second, translationaldiffusion leads to a broadening of the density peaks. Nev-ertheless, rotational diffusion is still active and generateswithin this localized particle cloud equally strong sub-populations of inward and outward oriented particles.When then self-propulsion is switched on again, v (cid:54) = 0,the peaks at the boundaries split. Outward oriented par-ticles return towards the channel boundaries. However,inward oriented particles form peaks that start to propelback towards the channel center, see Fig. 1 (e). Whenthey meet in the channel center, they overlay and again x/L . . . . . . ψ x , V / k (a) ψ x ( x,t ≈ / V ( x ) /k x/L (b) ψ x ( x,t =0 . V ( x ) /k x/L . . . . . . ψ x , V / k (c) ψ x ( x,t =2 . V ( x ) /k x/L (d) ψ x ( x,t =7 . V ( x ) /k − / − / / / x/L . . . . . . ψ x , V / k (e) ψ x ( x,t =7 . V ( x ) /k − / − / / / x/L (f) ψ x ( x,t =7 . V ( x ) /k FIG. 1. Time evolution of the spatial probability densitydistribution ψ x ( x, t ) across the channel of width L , boundedby a confining potential V ( x ) of strength k . From the initialdensity peak (a) fronts migrate outward (b) and get blocked atthe confining boundaries (c), where stationary density peaksbuild up. Switching off the self-propelling drive, these peaksbroaden due to spatial diffusion (d); reorientation is possibledue to rotational diffusion. Turning on the drive again (e), thepeaks at the boundaries rebuild, but also two further peaksstart to head back towards the channel center. When theyoverlay in the channel center (f), the initial spatial densitydistribution is partially restored. From here, the cycle can berepeated arbitrarily often without further losses. Dominatingself-propulsion directions are marked by arrows. Parametervalues are L = 20, v = 30, and k = 50 in rescaled units. form a (smaller) central peak, as shown in Fig. 1 (f).Consequently, the initial distribution has been restoredpartially, with significantly lower magnitude of the cen-tral peak (we show below that the restored peak heightcan be notably increased by steric interactions). Af-terwards, following the same protocol, this central peakcan be restored arbitrarily often without further losses.The restoration is possible despite the irreversible diffu-sive processes involved. In fact, we have explicitly ex-ploited a diffusive process to achieve our goal, namelyrotational diffusion during the period of switched-off self-propulsion.Fig. 2 shows the whole process in the x - φ plane. There,we explicitly observe the initial splitting into outwardpropagating peaks centered around φ = 0 , π and φ = π ,see Fig. 2 (a). The outward oriented particles get trappedat the boundaries, leading to the peaks localized both in x and φ direction in Fig. 2 (b). After switching off thedrive, rotational diffusion leads to equally φ -distributedlocalized states at the boundaries. In other words, thepeaks broaden to walls extended in the φ direction inFig. 2 (c). Finally, when the drive is turned on again,the inward oriented parts of these walls start to propagateback to the channel center, while the outward orientedones return towards the wall regions, see Fig. 2 (d).During this cycle, it is necessary to switch off the -1/2 -1/4 0 1/4 1/2 0 π π π π π π π π φx/L ψ ( x , φ , t = . ) (a) φx/L ψ ( x , φ , t = . ) (b) φx/L ψ ( x , φ , t = . ) (c) φx/L ψ ( x , φ , t = . ) (d) FIG. 2. Detailed probability density distribution ψ ( x, φ, t )corresponding to Fig. 1 (b)–(e) in the x - φ plane. At earlytimes (a), the initial front at x = 0 has split, leading to apeak at φ = 0 , π that travels to the right, while the oneat φ = π migrates to the left. The peaks get blocked andbuild up at the confining boundaries (b), centered aroundthe self-propulsion directions that point outwards. When theself-propelling drive is switched off (c), these peaks broaden,leading to fronts of equally distributed orientations φ . Fi-nally, when self-propulsion is turned on again (d), those partsof the fronts corresponding to outward orientations reformthe blocked peaks of (b); those parts of the fronts containinginward orientations start to propagate towards the channelcenter. Arrows mark the inward and outward self-propulsiondirections. drive to generate with sufficient probability particle ori-entations towards the channel center. Turning on thedrive, such particles immediately head towards the cen-ter, forming the propagating fronts that rebuild the cen-tral peak. If the drive were not switched off, a parti-cle at a wall would always leave the boundary as soonas it slightly turns away. At this moment, due to thetypically gradual rotational diffusion, the main com-ponent of the drive points along the bounding surfaceand not towards the channel center; pronounced inward-propagating fronts could not form.The time scales for the individual processes are de-termined by the intrinsic system parameters. It takesthe ballistic time L/ v ≈ .
33 for the edge of the out-ward propagating fronts in Fig. 1 (b) to reach the chan-nel boundaries. One should wait an order of magnitudelonger for the stationary peaks in Fig. 1 (c) to build up.Then, after switching off the drive, rotational diffusionover a time span ∆ t leads to a mean squared angular dis-placement of (cid:104) (∆ φ ) (cid:105) = 2∆ t in rescaled units. To achieve (cid:112) (cid:104) (∆ φ ) (cid:105) (cid:38) π to smoothen out the orientational distri-bution, a period of ∆ t (cid:38) π / ≈ (cid:104) (∆ x ) (cid:105) ∼ ∆ t in rescaled unitsand requiring (cid:112) (cid:104) (∆ x ) (cid:105) (cid:46) L/
8, we obtain an upper con- x/L . . ψ x (a) ψ x ( x,t ≈ / x/L (b) ψ x ( x,t =1) x/L . . ψ x (c) ψ x ( x,t =3) − − x/L (d) ψ x ( x,t =3 . − − x/L . . ψ x (e) ψ x ( x,t =3 . − − x/L (f) ψ x ( x,t =4 . FIG. 3. Same process as in Fig. 1, but now without theconfining channel walls, i.e. V ( x ) ≡
0. Spatial distances arerescaled by the same length L for comparison. From the ini-tial density peak (a) fronts migrate outward (b). Switchingoff the self-propelling drive, these free-standing peaks broadendue to spatial diffusion (c); reorientation is possible due to ro-tational diffusion. Turning on the drive again (d), the peakssplit into two counterpropagating fronts. When the inward-traveling peaks overlay in the center (e), the initial spatialdensity distribution is partially restored. Over time, the den-sity profile continuously flattens (f) due to the absence ofconfining boundaries. straint for the waiting time ∆ t (cid:46) L / ≈
6. After turn-ing on the drive again, the ballistic time L/ v ≈ . L/ (cid:46) v .We should mention that a similar process could alsobe achieved without boundaries. In Fig. 3 we switch offthe drive after the initial separation, let the rotationaldiffusion work, and then switch on the drive again. Thislikewise leads to an overlay and partially restored peakat the initial starting point. However, over time, thepeaks will more and more broaden due to diffusive pro-cesses, and density is lost towards the open boundaries.In contrast to that, a stationary state is involved in theform of the built-up peaks at the boundaries in Fig. 1 (c).From here, the central peak in Fig. 1 (f) can in principlebe generated in an infinite number of repeatable cycleswithout further loss of restored peak intensity. The pro-cess becomes independent of initial conditions. Intervalsof self-propulsion v (cid:54) = 0 are necessary for this process tobe observed. IV. CIRCULAR GEOMETRY
In an experiment using a channel structure, the chan-nel ends may influence the statistics. Therefore, we addi- r/L . . . . ψ r , V / k (a) ψ r ( r,t ≈ / V ( r ) /k r/L (b) ψ r ( r,t =0 . ψ r ( r,t =0 . ψ r ( r,t =0 . r/L . . . . ψ r , V / k (c) ψ r ( r,t =2 . V ( r ) /k / / r/L (d) ψ r ( r,t =7 . V ( r ) /k / / r/L . . . . ψ r , V / k (e) ψ r ( r,t =7 . V ( r ) /k / / r/L (f) ψ r ( r,t =7 . V ( r ) /k FIG. 4. Time evolution of the spatial probability density dis-tribution ψ r ( r, t ) in the circular cavity of diameter L , boundedby a confining potential V ( r ) of strength k . From the initialcentral density peak at r = 0 (a) a front migrates outward (b).It gets blocked at the cavity boundary (c), where a station-ary density peak builds up. When the self-propelling drive isswitched off, this peak broadens due to spatial diffusion (d);simultaneously, the orientations become equally distributedwithin the peak due to rotational diffusion. Turning on thedrive again (e), the peak at the boundary rerises; an addi-tional second peak emerges and heads back towards the cavitycenter. When it arrives there (f), the initial spatial densitydistribution is partially restored. From here, the cycle can berepeated arbitrarily often without further losses. Dominatingself-propulsion directions are marked by arrows. Parametervalues are L = 20, v = 30, and k = 50 in rescaled units. tionally turn to a finite confinement. We consider a two-dimensional circular cavity of the same diameter L asthe channel width above. The confining potential V ( r ),with r = (cid:107) r (cid:107) , has the same form as V ( x ) in Eq. (4).We assume an initial density distribution peaked in thecenter of the cavity at r = 0, again with equally dis-tributed orientations of the self-propulsion direction ˆu .Thus the whole process becomes radially symmetric. Wecan reduce the number of variables measuring the self-propulsion direction by an angle ϑ relatively to the radialdirection. For the resulting probability density ψ ( r, ϑ, t )we obtain ∂ t ψ = − v (cid:104) cos ϑ ∂ r ψ − r sin ϑ ∂ ϑ ψ (cid:105) + 1 r ∂ r [ r∂ r ψ ] + (cid:20) r (cid:21) ∂ ϑ ψ + 1 r ∂ r [ r ( ∂ r V ) ψ ] . (5)We now follow the same protocol as before. The pro-cess is displayed in Fig. 4 for the radial probability den-sity ψ r ( r, t ) obtained by integrating out the angle ϑ in ψ ( r, ϑ, t ). Starting from a density peak around r = 0in Fig. 4 (a), particles migrating outward from the cen-ter of the cavity as in Fig. 4 (b) get trapped at theboundary. There, a localized peak of elevated proba-bility density forms, see Fig. 4 (c). Switching off the / / x/L . . . . . . ψ x ( x , t = . ) / (a) † =0 † =0 . † =1 † =5 † =30 − / − / / / x/L . . . . . . ψ x ( x , t = . ) † (b) FIG. 5. Same system as in Fig. 1 for the channel geometry,but now with an additional steric pair interaction v ( r , r (cid:48) ) = (cid:15)δ ( r − r (cid:48) ). For increasing interaction strength (cid:15) , starting from (cid:15) = 0, (a) the height of the stationary boundary peaks for v (cid:54) = 0 decreases, whereas (b) the height of the restored centralpeak increases. The peaks are smeared out at elevated valuesof (cid:15) . self-propelling drive, rotational and spatial diffusion re-main active and lead to equally distributed orientationswithin the broadening density peak, see Fig. 4 (d). Whenthe self-propelling drive is turned on again in Fig. 4 (e)inward-pointing particles start to head towards the cav-ity center. After a while, a density peak is observed inthe center of the confinement in Fig. 4 (f). As before, theinitial distribution has been partially restored. Further-more, the situation in Fig. 4 (f) can afterwards be com-pletely restored arbitrarily often and does not depend onthe initial condition any more. This is because an inter-mediate stationary state was involved in Fig. 4 (c). V. STERIC INTERACTIONS
In Refs. [26–28], statistical properties of isolated light-switchable self-propelled particles were experimentallydetermined for dilute systems. We now briefly discusswhat happens to the above results when particle interac-tions start to become important.The nature of steric [41, 42] as well as of possiblehydrodynamic interactions [43, 44] depends on detailsof the particular system under consideration. To re-main at this stage as general as possible, we only con-sider the most central feature of hard steric pair interac-tions: a diverging interaction energy when two particlesare found at the same position. The simplest way toinclude this general property is by a steric pair inter-action of the shape v ( r , r (cid:48) ) = (cid:15)δ ( r − r (cid:48) ), where r and r (cid:48) mark the positions of the two particles and (cid:15) is therescaled interaction strength. In the derivation of thestatistical equations, we apply the mean-field approx-imation for the two-particle density ψ (2) by inserting ψ (2) ( r , φ, r (cid:48) , φ (cid:48) , t ) = ψ ( r , φ, t ) ψ ( r (cid:48) , φ (cid:48) , t ) [45]. This leadsto an additional contribution (cid:15) ∇· (cid:2) ψ ( r , φ ) ∇ (cid:82) ψ ( r , φ (cid:48) ) dφ (cid:48) (cid:3) in Eq. (3).There is now a certain pressure on the particles to leavehighly populated areas. It is reflected by a lower mag-nitude of the stationary boundary peaks for v (cid:54) = 0, seeFigs. 5 (a) and 6 (a) for the two considered geometries,respectively. Remarkably, close to the dilute limit, where / / r/L . . ψ r ( r , t = . ) (a) † =0 † =10 † =50 † =100 † =250 / / r/L . . ψ r ( r , t = . ) † (b) FIG. 6. Same system as in Fig. 4 for the circular geometry,but now with a steric pair interaction v ( r , r (cid:48) ) = (cid:15)δ ( r − r (cid:48) )included. Starting from (cid:15) = 0, (a) the height of the stationaryboundary peaks for v (cid:54) = 0 decreases, while (b) the height ofthe restored central peak increases for increasing interactionstrength (cid:15) . At pronounced magnitudes of (cid:15) the peaks aresmeared out. the above approximations are restricted to, the steric in-teractions as a result enhance the build-up of the restoredcentral peak, see Figs. 5 (b) and 6 (b). At higher inter-action strengths, the density distributions are smearedout. VI. CONCLUSIONS
In this work, we have demonstrated how in a confinedself-propelled particle system a central density peak canbe repeatedly generated. Our procedure implies switch-ing on and off the self-propulsion mechanism in combina-tion with appropriate confining boundary interactions. Ifwe start from an accordingly generated (or otherwise pre-pared) initial central density peak, this initial peak can(partially) be restored. After the first cycle of restora-tion, the central density peak can be repeatedly restoredarbitrarily often without further losses in peak magni-tude. In that sense, we break the irreversibility of thislong-time diffusive system.Naturally, there are other examples where diffusiveprocesses can be reversed. For instance, the diffusionof dipolar electric or magnetic particles can be counter-acted by a guiding drift force resulting from an electricor magnetic field gradient. Yet, in such situations, theexternal forces directly pull the particles, imposing andpre-setting the drift directions. The present case is dif-ferent. Here, each particle itself individually selects itsmigration direction in a stochastic process of rotationaldiffusion.Many experiments on self-propelling particles are per-formed in two-dimensional set-ups [15–24, 26–32]. Ac-cordingly, we here concentrated on two spatial dimen-sions. In principle, the analogous effect could also beanalyzed in three dimensions. However, this renders theanalysis considerably more tedious and requires a well-resolved three-dimensional detection technique on the ex-perimental side.It should be straightforward to verify our results incorresponding experiments. Switching on and off theself-propulsion can easily be achieved for light-activatedmechanisms [26–30, 35]. As briefly demonstrated, stericinteractions among the particles can promote the ef-fect. Details of interactions between particles [10, 11, 30],steric alignment interactions between non-spherical par-ticles and cavity walls [10, 11, 38, 46–48], or hydrody-namic interactions among the particles and with cavitywalls [10, 11, 40, 44, 46, 49, 50] may influence the statis-tics in different ways. The above theoretical descriptionshould then be adjusted to the specific situation under investigation.h
ACKNOWLEDGMENTS
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