Jerky active matter: a phase field crystal model with translational and orientational memory
JJerky active matter: a phase field crystal model with translational and orientationalmemory
Michael te Vrugt, Julian Jeggle, and Raphael Wittkowski ∗ Institut f¨ur Theoretische Physik, Center for Soft Nanoscience,Westf¨alische Wilhelms-Universit¨at M¨unster, D-48149 M¨unster, Germany
Most field theories for active matter neglect effects of memory and inertia. However, recentexperiments have found inertial delay to be important for the motion of self-propelled particles.A major challenge in the theoretical description of these effects, which makes the application ofstandard methods very difficult, is the fact that orientable particles have both translational andorientational degrees of freedom which do not necessarily relax on the same time scale. In thiswork, we derive the general mathematical form of a field theory for soft matter systems with twodifferent time scales. This allows to obtain a phase field crystal model for polar (i.e., nonsphericalor active) particles with translational and orientational memory. Notably, this theory is of thirdorder in temporal derivatives and can thus be seen as a spatiotemporal jerky dynamics. We obtainthe phase diagram of this model, which shows that, unlike in the passive case, the linear stabilityof the liquid state depends on the damping coefficients. Moreover, we investigate sound wavesin active matter. It is found that, in active fluids, there are two different mechanisms for soundpropagation. For certain parameter values and sufficiently high frequencies, sound mediated bypolarization waves experiences less damping than usual passive sound mediated by pressure wavesof the same frequency. By combining the different modes, acoustic frequency filters based on activefluids could be realized.
I. INTRODUCTION
Almost all field theories for active matter systems arederived in the overdamped limit and neglect the inertiaof the active particles. However, there is a current in-crease of interest in the role of memory and (inertial)delay in active matter [1–7]. Experiments have showninertial [1–3] and sensorial [4, 5] delay to be importantfor the dynamics of self-propelled particles. Inertia canlead to interesting effects that are not present in over-damped active systems [1–3, 8–11], such that inertial ac-tive matter models, as presented in Refs. [10, 12], forman important extension of usual models. A particularlyinteresting aspect that remains to be explored is the factthat active particles have translational and rotational de-grees of freedom which both are associated with inertia,but might relax on different time scales [3, 13]. Memoryand inertia effects are intimately connected and some-times equivalent, as can be seen from an analysis in theMori-Zwanzig framework (see Section II).The problem of memory effects has been discussed inother contexts, very notably in particle physics [14–16].Most dissipative transport equations (such as the diffu-sion equation) are acausal, since they assume that signalspropagate with an infinite velocity. This is typically justi-fied by arguing that microscopic relaxation processes oc-cur infinitely rapidly compared to the relevant dynamics.However, this assumption is not consistent with specialrelativity where the maximum propagation speed of sig-nals is the speed of light. Relativistic descriptions are re-quired for phase transitions in heavy ion collisions, where ∗ Corresponding author: [email protected] the relevant dynamics itself is very fast, such that ne-glecting memory effects and causality constraints is anapproximation that cannot be justified [15]. For this rea-son, more general theories have been derived that takethese effects into account [14–17].This problem can also arise in soft matter systemssuch as polymer solutions [17]. Although relativistic ef-fects are typically irrelevant here, it is still an approx-imation to assume that the macroscopic order parame-ters change much slower than the microscopic degrees offreedom. The time scale will then be set by the damp-ing coefficients rather than by the speed of light. Asnoted by Archer [18, 19], who suggested this as a topic offurther investigation within classical dynamical densityfunctional theory (DDFT) [20], the mathematical struc-ture of transport equations for relativistic heavy ion col-lisions and underdamped soft matter is identical. Thereis, however, an additional difficulty that can occur in thelatter case, namely that two different relaxational timescales are relevant. A typical example would be soft mat-ter systems consisting of particles with orientational de-grees of freedom – in particular active matter – whereposition and orientation relax on different time scales.However, different time scales can also be relevant forother systems. Here, we discuss this problem in a verygeneral way and then specialize our results to the case ofactive field theories.Phase field crystal (PFC) models [21], which provide acoarse-grained description of crystallization and patternformation in materials, are a particularly suitable frame-work for such investigations. They were first proposedphenomenologically [22–24] and then derived from (dy-namical) density functional theory [25, 26]. The earlyforms have been extended into a variety of directions in-cluding orientational degrees of freedom [27–30] and ac- a r X i v : . [ c ond - m a t . s o f t ] F e b tive matter [31, 32]. Moreover, second-order models in-volving inertia have been derived [33–37], these are some-times called “modified PFC” (MPFC) models [33, 38].Consequently, PFC models are a suitable framework forincorporating delay into active field theories, as recentlyshown by Arold and Schmiedeberg [12]. PFC models arereviewed in Ref. [21], their relation to DDFT is also dis-cussed in Refs. [20, 39].While these models are of second order in tempo-ral derivatives, memory effects have also been linked tothird-order models. A typical example is jerky dynam-ics, which provides differential equations for the rate ofchange of the acceleration. These can be relevant inNewtonian mechanics if the force is memory-dependent.The resulting third-order differential equations are use-ful for obtaining simple models of chaos [40–43]. More-over, third-order dispersions have been studied in optics[44, 45], where they naturally appear in delay systems[46].In this work, we discuss how to incorporate finite re-laxation times in systems where two different time scalesare present. This is done in a very general way, sincethis problem can occur in a variety of contexts. We thenspecialize to the case of active PFC models and obtaina generalization of the traditional active PFC model [31]that incorporates translational and rotational memory.Notably, the resulting theory is of third order in tempo-ral derivatives and can thus be viewed as a spatiotem-poral jerky dynamics. Field theories of this form arealmost completely unexplored, and can – like their afore-mentioned counterparts from other areas of nonlinear dy-namics – be expected to show very interesting behavior.As shown in the present work, this is relevant for exper-iments on active matter. In contrast to the passive case,it is found that the damping can affect the linear stabilityof the fluid state in the PFC model.An interesting application of the new PFC model isthe investigation of sound waves in active matter. In pas-sive systems, sound propagates due to translational iner-tia. However, it has been found that propagating densitywaves are also possible in overdamped active systems asa consequence of the coupling between density and po-larization [47, 48]. Our model incorporates both transla-tional inertia and activity. It therefore allows to describetwo different mechanisms of sound propagation that arerelevant for active matter. In particular, we show thatat higher frequencies, the usual “passive” sound can ex-perience a stronger damping than “active” sound waves,such that waves of the latter type can propagate furtherinto the medium.This article is structured as follows: In Section II, weexplain how finite relaxation times can be incorporatedinto transport equations. We provide a brief introductionto jerky dynamics in Section III. A general model for By “ n -th-order model” we always denote a model that is of n -thorder in temporal derivatives if not stated otherwise. systems with two time scales is developed in Section IV.In Section V, we use these results to obtain an activePFC model with memory. We perform a linear stabilityanalysis of this model in Section VI. Sound propagationis studied in Section VII. We conclude in Section VIII. II. CAUSAL FIELD THEORIES
For a conserved order-parameter field φ ( (cid:126)r, t ), where (cid:126)r and t denote position and time, respectively, the timeevolution is given by the continuity equation ∂ t φ ( (cid:126)r, t ) = − (cid:126) ∇ · (cid:126)J ( (cid:126)r, t ) (1)with the current (cid:126)J ( (cid:126)r, t ). A typical example is traditionalDDFT [20, 49–52], where the current is (cid:126)J ( (cid:126)r, t ) = − ˜ M φ ( (cid:126)r, t ) (cid:126) ∇ δF [ φ ] δφ ( (cid:126)r, t ) (2)with the mobility ˜ M (which for traditional DDFT isgiven by ˜ M = βD with the thermodynamic beta β andthe diffusion constant D ), the density φ (which is theorder-parameter field), and the free-energy functional F .Another example is the Cahn-Hilliard equation [53, 54],where the current is (cid:126)J ( (cid:126)r, t ) = − ˜ M (cid:126) ∇ δF [ φ ] δφ ( (cid:126)r, t ) . (3)A third example is Active Model B [55], where the currentis (cid:126)J ( (cid:126)r, t ) = − ˜ M (cid:126) ∇ (cid:18) δF [ φ ] δφ ( (cid:126)r, t ) + λ ( (cid:126) ∇ φ ( (cid:126)r, t )) (cid:19) + (cid:126) Λ( (cid:126)r, t ) (4)with the activity parameter λ and the noise (cid:126) Λ( (cid:126)r, t ).All these currents share a common property: They are Markovian , i.e., they only depend on the value of φ attime t , but not on values of φ at times s < t . Moreover,they are local , i.e., the time derivative of φ at position (cid:126)r does not depend on the value of φ at a different position (cid:126)r (cid:48) .This is not the most general case. A good startingpoint for a systematic analysis is the Mori-Zwanzig for-malism [56–61]. It allows to derive macroscopic trans-port equations for an arbitrary set of relevant variablesfrom the microscopic dynamics and thereby, as discussedin Ref. [61], gives very general insights into the generalstructure of these equations. As exploited in recent workby Meyer et al. [62–65], the formalism is also useful foranalyzing memory effects in a systematic way. In gen-eral, ignoring some degrees of freedom of a system leadsto a transport equation in which the dynamics dependson the state of the system at previous times. Approxi-mating this equation by a Markovian equation is possibleif the set of relevant variables one has chosen captures thecomplete macroscopic dynamics [52, 61].Let us assume that we know the microscopic equationsgoverning a passive many-particle system, but are onlyinterested in a conserved scalar order parameter φ ( (cid:126)r, t )that is a function of the microscopic degrees of freedom.Then, we write the general time evolution as ∂ t φ ( (cid:126)r, t ) = (cid:126) ∇ · (cid:90) t d s (cid:90) d r (cid:48) M ( t, s, (cid:126)r, (cid:126)r (cid:48) ) (cid:126) ∇ (cid:48) δF [ φ ] δφ ( (cid:126)r (cid:48) , s ) , (5)where M is the memory kernel. We have ignored here anorganized drift term, which often vanishes for reasons ofsymmetry if there is only one relevant variable, and wehave dropped a noise term. Apart from this, the form(5) is completely general.The rather complicated general form (5) can be sim-plified significantly if the relevant variable φ is slow. Inthis case, it can be assumed that φ is constant on thetime scales on which the microscopic degrees of freedomrelax. For this to be possible, the relaxation of these mi-croscopic degrees of freedom has to occur very rapidly.If we also assume that the nonlocality can be neglected(e.g., because the system is dilute [52]), the exact trans-port equation (5) can be approximately written as ∂ t φ ( (cid:126)r, t ) = (cid:126) ∇ · (cid:18) D ( (cid:126)r, t ) (cid:126) ∇ δF [ φ ] δφ ( (cid:126)r, t ) (cid:19) (6)with the diffusion tensor D . This is the so-called Marko-vian approximation .Although it is made in almost all practical cases, theMarkovian approximation is not innocent. From a foun-dational perspective, it introduces the thermodynamicirreversibility not present in the time-reversal-invariantmicroscopic laws of physics [66]. In the Markovian limit,an H-theorem corresponding to an increase of entropycan be proven [67, 68]. From a more practical point ofview, the Markovian approximation corresponds to theassumption that the set of relevant variables we havechosen gives a complete description of the macroscopicstate. For example, if φ ( (cid:126)r, t ) is the number density, mak-ing the Markovian approximation implies that the mo-mentum density (cid:126)g ( (cid:126)r, t ) relaxes very rapidly, i.e., that weare working in the overdamped limit [52].What is also relevant here is that Eq. (6) is a diffusiveequation which is of first order in temporal and secondorder in spatial derivatives. This leads, in general, to aninfinite propagation speed of signals, i.e., to an acausal equation. Early treatments of this problem include in-ertial extensions of the heat equation [69] (see Ref. [17]for a review). This issue is particularly relevant in thecase of relativistic systems. If we wish to describe phase In our notation, the operators (cid:126) ∇ and ∂ t are always understoodas acting on the whole term to the right of them. For example,in the expression ∂ t f ( t ) g ( t )+ h ( t ) with time-dependent functions f , g , and h , the operator ∂ t acts on the product f ( t ) g ( t ) and notjust on f ( t ). Obviously, ∂ t does not act on h ( t ). transitions in the early universe or in heavy-ion collisions,we generally have to take into account that signals canonly propagate with a finite velocity (namely the speed oflight) [15]. Thus, when applying the Mori-Zwanzig for-malism to relativistic systems, we generally have to bevery careful in handling memory effects [14]. However,as we shall see below, the same problem can arise in softmatter physics, even though the velocities and time scalesare very different there.We start discussing this issue by presenting the the-ory derived by Koide et al. [15] for describing phase-separation processes in relativistic high energy physics.For a conserved order parameter, the Cahn-Hilliard equa-tion is a very successful theory for phase separation. Inorder to take into account the finite propagation speedin relativistic systems, the Cahn-Hilliard current (3) ismodified as (cid:126)J ( (cid:126)r, t ) = − M (cid:90) t d s e − γ ( t − s ) (cid:126) ∇ δF [ φ ] δφ ( (cid:126)r, s ) (7)with the damping coefficient (inverse relaxation time) γ and the modified mobility M = γ ˜ M . We havehere inserted as the memory kernel a memory function γ exp( − γ ( t − s )), which can be motivated by certain as-sumptions about the noise. In the Mori-Zwanzig formal-ism, the memory kernel is related to the correlation ofthe noise [60]. The assumption of white noise leads toa Markovian dynamics, whereas colored noise gives theform (7) [15].The time derivative of Eq. (7) is given by ∂ t (cid:126)J ( (cid:126)r, t ) = − M (cid:126) ∇ δF [ φ ] δφ ( (cid:126)r, t ) + γM (cid:90) t d s e − γ ( t − s ) (cid:126) ∇ δF [ φ ] δφ ( (cid:126)r, s )= − M (cid:126) ∇ δF [ φ ] δφ ( (cid:126)r, t ) − γ (cid:126)J ( (cid:126)r, t ) . (8)Differentiating Eq. (1) with respect to time and insertingEq. (8) then gives ∂ t φ ( (cid:126)r, t ) + γ∂ t φ ( (cid:126)r, t ) = M (cid:126) ∇ δF [ φ ] δφ ( (cid:126)r, t ) . (9)This is a second-order causal Cahn-Hilliard equation thatis similar in form to the telegrapher’s equation. Theoriesof this form can be used to study hyperbolic spinodaldecomposition [70–73]. There are two important limit-ing cases of Eq. (9) one can consider. The first one isthe overdamped limit γ → ∞ with M/γ = const. (cor-responding to fixed ˜ M ), in which case we recover thestandard Cahn-Hilliard equation ∂ t φ ( (cid:126)r, t ) = ˜ M (cid:126) ∇ δF [ φ ] δφ ( (cid:126)r, t ) . (10)The second one is the underdamped limit γ → M , which gives the inertial Cahn-Hilliard equation ∂ t φ ( (cid:126)r, t ) = M (cid:126) ∇ δF [ φ ] δφ ( (cid:126)r, t ) . (11)For the DDFT current (2), Archer [18, 19] has notedthat this procedure leads to a causal DDFT that has thesame structure as the DDFT ∂ t φ ( (cid:126)r, t ) + γ∂ t φ ( (cid:126)r, t ) = 1 m (cid:126) ∇ · (cid:18) φ ( (cid:126)r, t ) (cid:126) ∇ δF [ φ ] δφ ( (cid:126)r, t ) (cid:19) (12)with particle mass m for particles with inertia. A sim-ilar result was obtained by Chavanis [74]. Within ourabove considerations about the Mori-Zwanzig formalism,we can give a physical explanation: For systems whereinertia is relevant, such as atomic fluids, it is no longerpossible to assume that the number density is the onlydegree of freedom that is relevant, such that we also re-quire the momentum density. Hence, if we want to derivea transport equation for a system with inertia in whichthe number density is the only relevant variable (i.e., aDDFT for an atomic fluid), we can no longer make theapproximation of infinitely fast relaxations. Instead, weneed to take into account that in an underdamped systemthe velocities need a finite time to relax. From Eq. (12),we can also see why the over- and underdamped lim-its have to be taken in the form (10) and (11), respec-tively: The overdamped limit of Eq. (12) then leads totraditional DDFT [19], whereas the underdamped limitleads to a generalized Euler equation (without convectiveterm). From Eq. (12), we find that M = 1 /m such that˜ M = 1 / ( γm ). Since we know from DDFT that ˜ M = βD ,we can infer γ = 1 / ( βmD ). Therefore, varying γ at fixed M corresponds to changing D , whereas varying γ atfixed ˜ M corresponds to changing m .Another way to look at this issue is to take into ac-count that equations of motion containing memory ef-fects only arise if we do not consider all degrees of free-dom of a system, but only a reduced set [61] (Hamilton’sequations or the Heisenberg equation of motion have nomemory). The memory terms then incorporate (alongwith the noise) the dynamics of those parts of the sys-tem that we do not wish to model explicitly. This alsoimplies that we can, instead of considering the memory,obtain a Markovian dynamics if we enlarge our set of rel-evant variables [20, 59]. In this case, we can, rather thanusing a transport equation for the density that containsmemory, also write down coupled memoryless equationsfor mass and momentum density. Mathematically, thisis reflected by the fact that the equations with memoryare of second order. We could alternatively obtain twoequations of motion with the usual first-order structureby using ∂ t φ as an additional relevant variable. The rea-son why ∂ t φ appears is that it is no longer negligible ifmemory terms are relevant. In the example presentedhere, we would have ∂ t φ = − (cid:126) ∇ · (cid:126)g , such that using ∂ t φ as One can also vary γ at fixed M by changing β . However, achange of the temperature will typically also affect the free en-ergy, whereas the diffusion coefficient only appears in the mobil-ity. a relevant variable is equivalent to adding the momentumdensity (cid:126)g to the set of relevant variables. III. JERKY DYNAMICS
For analyzing the problem at hand, the theory of jerkydynamics [41–43, 75–77] will prove to be very useful. Ajerky dynamics is an ordinary differential equation of theform [43] ... x = J ( x, ˙ x, ¨ x ) (13)with a time-dependent variable x ( t ) and a function J .Jerky dynamics is very important for chaos theory [41,75]. Mechanically, the “jerk” is the rate of change ofacceleration [78]. Intuitively, one might expect jerks to beof no importance in classical mechanics, since Newton’sequation of motion ¨ x = 1 m F ( x, ˙ x ) (14)with the force F is of second order. (Although a jerkydynamics can obviously be obtained by taking the timederivative of Eq. (14), it would not provide any physicalinsights.) However, if the force has memory, i.e., if itdepends on values of x or ˙ x at previous times, takingthe time derivative of Eq. (14) can lead to an interestingjerky dynamics that contains additive terms dependingsolely on x . Such models can allow for chaos [43].This relation to memory is what makes jerky dynamicsrelevant for the present investigation. What will also beuseful is that, in some cases, a three-dimensional dynam-ical system can be written in the form (13). An exampleis Sprott’s model R [79]˙ x = a − y, (15)˙ y = b + z, (16)˙ z = xy − z (17)with the dynamical variables x , y , and z and the con-stants a and b , which is a simple model for chaos. Theproblem is addressed in detail in Ref. [40], where theconditions under which such a transformation is possibleare discussed. A useful strategy [40] is to calculate thefirst time derivative of Eqs. (16) and (17) and the firstand second time derivatives of Eq. (15). This gives sevencoupled equations for x, ˙ x, ¨ x, ... x , y, ˙ y, ¨ y, z, ˙ z , and ¨ z . Thesecan be used to eliminate y, ˙ y, ¨ y, z, ˙ z , and ¨ z , which givesa closed equation for ... x that solely depends on x , ˙ x , and¨ x . For Eqs. (15)–(17), one obtains [42]... x = − ¨ x − x ( a − ˙ x ) − b. (18) IV. SOFT MATTER WITH TWO TIME SCALES
In systems of active (or nonspherical) particles, oneneeds to take into account both translational and orien-tational degrees of freedom. As a toy model for describ-ing the physics of such systems, we use the dynamicalequation ∂ t φ ( (cid:126)r, t ) = T ( (cid:126)r, t ) + R ( (cid:126)r, t ) . (19)Later, we interpret T and R as the contributions fromtranslational and orientational degrees of freedom, re-spectively. However, the considerations in this sectionapply to any physical system whose dynamics can bewritten in the form (19). We are assuming nothing aboutthe form of T and R here – it can be conserved or non-conserved, active or passive, and it can depend on φ aswell as on spatial derivatives of φ . The field φ can be ascalar, a vector, or a tensor. As in Section II, we general-ize Eq. (19) towards a causal dynamics with time delay : ∂ t φ ( (cid:126)r, t ) = (cid:90) t d s (cid:0) e − γ T ( t − s ) T ( (cid:126)r, s ) + e − γ R ( t − s ) R ( (cid:126)r, s ) (cid:1) . (20)There is a very important difference to the models knownfrom the literature which we have discussed in Section II:In general, we cannot assume that translational and ori-entational degrees of freedom relax on the same timescale. Therefore, we have introduced two different re-laxation parameters γ T and γ R for translation and ori-entation, respectively. This has important consequencesfor the resulting dynamics. To see this, we calculate thetime derivative of Eq. (20), which gives ∂ t φ ( (cid:126)r, t ) = − (cid:90) t d s (cid:0) γ T e − γ T ( t − s ) T ( (cid:126)r, s )+ γ R e − γ R ( t − s ) R ( (cid:126)r, s ) (cid:1) + T ( (cid:126)r, t ) + R ( (cid:126)r, t ) . (21)For γ T = γ R = γ , we could write Eq. (21) as ∂ t φ ( (cid:126)r, t ) = − γ∂ t φ ( (cid:126)r, t ) + T ( (cid:126)r, t ) + R ( (cid:126)r, t ) . (22)In general, however, the sum of the first two terms inEq. (21) is not proportional to ∂ t φ ( (cid:126)r, t ), since the transla-tional and rotational contributions appear with differentprefactors due to the different time scales on which thecontributions change. Therefore, Eq. (21) cannot be writ-ten as a second-order partial differential equation withouttime convolution.Physically, this is due to the fact that we require threerather than two variables for a complete description ofthe system – in addition to mass and momentum densitythe angular momentum density is needed. Therefore, werequire a third-order partial differential equation to de-scribe the dynamics of φ ( (cid:126)r, t ) without time convolution. It is also possible that, in this step, the form of T and R changescompared to Eq. (19), e.g., due to a rescaling of the mobility fordimensional reasons as discussed in Section II. This can be obtained using the procedure introduced inSection III.First, we make the definitions x ( (cid:126)r, t ) = φ ( (cid:126)r, t ) , (23) y ( (cid:126)r, t ) = (cid:90) t d s e − γ T ( t − s ) T ( (cid:126)r, s ) , (24) z ( (cid:126)r, t ) = (cid:90) t d s e − γ R ( t − s ) R ( (cid:126)r, s ) . (25)For the remainder of this section, we drop the depen-dence on (cid:126)r since it is not important for the further cal-culations. With the definitions (23)-(25), we can obtainfrom Eq. (20) the dynamical system˙ x = y + z, (26)˙ y = − γ T y + T, (27)˙ z = − γ R z + R. (28)The problem has thus been reduced to the derivationof the jerky dynamics corresponding to the dynamicalsystem given by Eqs. (26)–(28). For this purpose, wecompute the derivatives¨ x = ˙ y + ˙ z, (29)¨ y = − γ T ˙ y + ˙ T , (30)¨ z = − γ R ˙ z + ˙ R, (31)... x = ¨ y + ¨ z. (32)One can now use Eqs. (26)–(31) to express the unknownvariables ¨ y and ¨ z in Eq. (32) in terms of x , ˙ x and ¨ x .From Eqs. (27) and (30), we get¨ y = γ T y − γ T T + ˙ T . (33)Similarly, Eqs. (28) and (31) give¨ z = γ R z − γ R R + ˙ R. (34)From Eqs. (27)–(29), we get¨ x = − γ T y + T − γ R z + R. (35)Eqs. (32)–(34) lead to... x = γ T y + γ R z − γ T T − γ R R + ˙ T + ˙ R. (36)From Eq. (26), we obtain y = ˙ x − z. (37)Moreover, Eq. (35) gives y = − γ T ¨ x + 1 γ T T − γ R γ T z + 1 γ T R. (38)Equating Eqs. (37) and (38) allows to find z = − γ T γ R − γ T ˙ x − γ R − γ T (¨ x − T − R ) . (39)At this point, we have made the assumption γ R (cid:54) = γ T .From Eqs. (37) and (39), we get y = ˙ x + γ T γ R − γ T ˙ x + 1 γ R − γ T (¨ x − T − R )= γ R γ R − γ T ˙ x + 1 γ R − γ T (¨ x − T − R ) . (40)Finally, combining Eqs. (36), (39), and (40) leads to... x = γ T γ R γ R − γ T ˙ x + γ T γ R − γ T (¨ x − T − R ) − γ T T + ˙ T − γ R γ T γ R − γ T ˙ x − γ R γ R − γ T (¨ x − T − R ) − γ R R + ˙ R, (41)which can be simplified to the final result... x = − ( γ T + γ R )¨ x − γ T γ R ˙ x + γ R T + γ T R + ˙ T + ˙ R. (42)It is easily verified that Eq. (42) also holds for γ T = γ R . V. JERKY ACTIVE MATTER
Phase field crystal (PFC) models are a useful frame-work for the description of soft and active matter [21,31, 80–83]. They can be obtained as a limiting caseof the more complex and more general case of DDFT[26, 32, 39]. Reviews are given by Refs. [21] (PFC mod-els) and [20] (DDFT), both reviews discuss the derivationof PFC models from DDFT. The order parameters arethe rescaled density ψ ( (cid:126)r, t ) and the polarization (cid:126)P ( (cid:126)r, t ),which arise through an orientational expansion [84, 85]of the one-body density ρ ( (cid:126)r, ˆ u, t ) depending on position (cid:126)r and orientation ˆ u .The active PFC model reads [31, 32] ∂ t ψ ( (cid:126)r, t ) = ˜ M (cid:126) ∇ δFδψ ( (cid:126)r, t ) − ˜ v (cid:126) ∇ · (cid:126)P ( (cid:126)r, t ) , (43) ∂ t (cid:126)P ( (cid:126)r, t ) = ( ˜ M (cid:126) ∇ − ˜ D r ) δFδ (cid:126)P ( (cid:126)r, t ) − ˜ v (cid:126) ∇ ψ ( (cid:126)r, t ) (44)with the free-energy functional F , activity parameter (arescaled self-propulsion velocity) ˜ v , and rotational diffu-sion constant ˜ D r . For generality and later convenience,we have introduced a constant mobility ˜ M >
0, which isset to one in most treatments of the active PFC model,but which here will be useful for taking under- and over-damped limits.A generalization towards time-delay dynamics reads ∂ t ψ ( (cid:126)r, t ) = (cid:90) t d s e − γ T ( t − s ) (cid:18) M (cid:126) ∇ δFδψ ( (cid:126)r, s ) − v (cid:126) ∇ · (cid:126)P ( (cid:126)r, s ) (cid:19) , (45) ∂ t (cid:126)P ( (cid:126)r, t ) = (cid:90) t d s e − γ T ( t − s ) (cid:18) M (cid:126) ∇ δFδ (cid:126)P ( (cid:126)r, s ) − v (cid:126) ∇ ψ ( (cid:126)r, s ) (cid:19) − (cid:90) t d s e − γ R ( t − s ) D r δFδ (cid:126)P ( (cid:126)r, s ) , (46) where we have introduced the rescaled coefficients M = γ T ˜ M , v = γ T ˜ v , and D r = γ R ˜ D r . This ansatz assumesthat the terms arising from convection and translationaldiffusion relax on a time scale γ − T , whereas the termarising from rotational diffusion relaxes on a time scale γ − R . Note that Eq. (45) still has the form of a continuityequation for ψ ( (cid:126)r, t ). Since all further steps are formallyexact, the transformation to the jerky form therefore doesnot affect the fact that ψ ( (cid:126)r, t ) is conserved.While Eq. (45) can be treated as a second-order dy-namics in the standard way since it only depends on onetime scale (see Section II), Eq. (46) gives rise to a third-order dynamics. To connect to the general structure pre-sented in Section IV, we identify (cid:126)x ( (cid:126)r, t ) = (cid:126)P ( (cid:126)r, t ) , (47) (cid:126)T ( (cid:126)r, t ) = M (cid:126) ∇ δFδ (cid:126)P ( (cid:126)r, t ) − v (cid:126) ∇ ψ ( (cid:126)r, t ) , (48) (cid:126)R ( (cid:126)r, t ) = − D r δFδ (cid:126)P ( (cid:126)r, t ) . (49)We combine Eqs. (47)–(49) with Eq. (42) to the jerkyactive matter model ∂ t ψ ( (cid:126)r, t ) = − γ T ∂ t ψ ( (cid:126)r, t ) + M (cid:126) ∇ δFδψ ( (cid:126)r, t ) − v (cid:126) ∇ · (cid:126)P ( (cid:126)r, t ) , (50) ∂ t (cid:126)P ( (cid:126)r, t ) = − ( γ T + γ R ) ∂ t (cid:126)P ( (cid:126)r, t ) − γ T γ R ∂ t (cid:126)P ( (cid:126)r, t ) − ( γ T + ∂ t ) D r δFδ (cid:126)P ( (cid:126)r, t )+ ( γ R + ∂ t ) (cid:18) M (cid:126) ∇ δFδ (cid:126)P ( (cid:126)r, t ) − v (cid:126) ∇ ψ ( (cid:126)r, t ) (cid:19) . (51)As a consistency check, we confirm that the dynamics of (cid:126)P is of second order for γ T = γ R = γ . In this case, by theline of argument presented in Section II, it should read ∂ t (cid:126)P ( (cid:126)r, t ) = − γ∂ t (cid:126)P ( (cid:126)r, t ) + ( M (cid:126) ∇ − D r ) δFδ (cid:126)P ( (cid:126)r, t ) − v (cid:126) ∇ ψ ( (cid:126)r, t ) . (52)Equation (51) can, for γ T = γ R = γ , be written as (cid:126) ∂ t + γ ) (cid:18) − ∂ t (cid:126)P ( (cid:126)r, t ) − γ∂ t (cid:126)P ( (cid:126)r, t )+ ( M (cid:126) ∇ − D r ) δFδ (cid:126)P ( (cid:126)r, t ) − v (cid:126) ∇ ψ ( (cid:126)r, t ) (cid:19) . (53)If we impose Eq. (52) as an initial condition at t = 0,Eq. (53) ensures that Eq. (52) is satisfied at all times,such that we have a second-order dynamics. (The ne-cessity of imposing an additional initial condition arisesbecause a differential equation of third order in tempo-ral derivatives requires an initial condition for the secondtemporal derivative.)We can also consider the over- and underdamped lim-its. The overdamped limit corresponds to γ T , γ R → ∞ with fixed ˜ M , ˜ v , and ˜ D r . Thereby, the standard ac-tive PFC model given by Eqs. (43) and (44) is recovered.The underdamped limit, on the other hand, correspondsto γ T , γ R → M , v , and D r . In this case, afterintegrating Eq. (51) over t , we find ∂ t ψ ( (cid:126)r, t ) = M (cid:126) ∇ δFδψ ( (cid:126)r, t ) − v (cid:126) ∇ · (cid:126)P ( (cid:126)r, t ) , (54) ∂ t (cid:126)P ( (cid:126)r, t ) = ( M (cid:126) ∇ − D r ) δFδ (cid:126)P ( (cid:126)r, t ) − v (cid:126) ∇ ψ ( (cid:126)r, t ) . (55)Other limits are also possible. For example, we can con-sider the case in which translational degrees of freedomare underdamped, whereas rotational degrees of free-dom are overdamped. This corresponds to γ T → γ R → ∞ with keeping M , v , and ˜ D r fixed. We thenobtain ∂ t ψ ( (cid:126)r, t ) = M (cid:126) ∇ δFδψ ( (cid:126)r, t ) − v (cid:126) ∇ · (cid:126)P ( (cid:126)r, t ) , (56) ∂ t (cid:126)P ( (cid:126)r, t ) = ( M (cid:126) ∇ − ˜ D r ∂ t ) δFδ (cid:126)P ( (cid:126)r, t ) − v (cid:126) ∇ ψ ( (cid:126)r, t ) . (57)Note that the dynamics of (cid:126)P , given by Eq. (57), is stillof second order in the case of completely overdampedorientational dynamics as long as the translational dy-namics is underdamped. This, however, is plausible sincethe local polarization changes not only due to rotationsof the individual particles, but also due to translationalmotion.It is interesting to compare Eq. (57) to the polarizationdynamics given by Eq. (5) in Ref. [12], which reads (inour notation) ∂ t (cid:126)P ( (cid:126)r, t ) = ( C (cid:126) ∇ − ˜ D r ) (cid:126)P − ˜ v (cid:126) ∇ ψ ( (cid:126)r, t ) (58) with a constant C . This equation, which is of first orderin temporal derivatives, also arises in the context of anunderdamped PFC model. However, Eq. (58) is derivedby neglecting orientational convection and by introducinga gradient term proportional to ( (cid:126) ∇ (cid:126)P ) in the free energy.Consequently, the gradient term in Eq. (58) has a differ-ent physical origin than the gradient terms in Eqs. (44)and (57), despite the fact that it looks very similar. VI. LINEAR STABILITY
In the following, we drop the dependence on space andtime and restrict ourselves to one spatial dimension. Fol-lowing Ref. [80], we use the Swift-Hohenberg [86] free-energy functional F = (cid:90) d x (cid:16)
12 ( ψ ( (cid:15) +(1+ ∂ x ) ) ψ )+ 14 ( ψ + ¯ ψ ) + b P (cid:17) (59)with the constant coefficients (cid:15) (shifted rescaled temper-ature [82]), ¯ ψ (mean density), and b . This leads to δFδψ = ( (cid:15) + (1 + ∂ x ) ) ψ + ( ψ + ¯ ψ ) , (60) δFδP = bP. (61)Next, we consider small deviations from a referencestate ( ψ, P ) = (0 ,
0) in the form ψ = ψ exp( λt − i kx ) , (62) P = P exp( λt − i kx ) (63)with amplitudes ψ and P , growth rate λ , imaginaryunit i, and wave number k . We insert Eqs. (60) and (61)together with the ansatz given by Eqs. (62) and (63) intoEqs. (50) and (51), linearize, and obtain λ ψ = − γ T λψ − M k ( (cid:15) + 3 ¯ ψ + (1 − k ) ) ψ + i v kP , (64) λ P = − γ T γ R λP − ( γ T + γ R ) λ P − ( γ T + λ ) D r bP − ( γ R + λ ) M k bP + i v ( γ R + λ ) kψ . (65)When writing Eqs. (64) and (65) as an eigenvalue problem in the form λ (cid:18) ψ P (cid:19) = M (cid:18) ψ P (cid:19) (66)with the matrix M = (cid:18) − γ T λ − M k ( (cid:15) + 3 ¯ ψ + (1 − k ) ) λ i v λk i v ( γ R + λ ) k − γ T γ R λ − ( γ T + γ R ) λ − ( γ T + λ ) D r b − ( γ R + λ ) M k b (cid:19) , (67)we get the characteristic polynomial determining the dispersion :0 = ( − γ T λ − M k ( (cid:15) + 3 ¯ ψ + (1 − k ) ) − λ )( − γ T γ R λ − ( γ T + γ R ) λ − ( γ T + λ ) D r b − ( γ R + λ ) M k b − λ )+ v ( γ R + λ ) k . (68)To investigate the properties of the jerky active mat-ter model, we started by performing a linear stabilityanalysis of the homogeneous state ( ψ, P ) = (0 , λ for different values of the real wavenumber k . IfRe( λ ) > k , the system is unstable.Since Eq. (68) is a fifth-order polynomial, we have solvedit numerically.The results can be found in Fig. 1, which shows thedispersion relation for various values of the damping pa-rameters γ T and γ R . The other parameters have beenchosen as M = 1, v = 0 . D r = 0 . (cid:15) = − .
5, ¯ ψ = − b = 0 .
1, similar to the values used in Ref. [80]. There,it was argued that only ¯ ψ < (cid:15) is negative. In this case,the liquid state is stable for larger | ¯ ψ | .In our extended model, we find that for γ T = γ R = 0 . γ T = γ R = 0 . λ = − γ , since for γ T = γ R = γ , one can (due toEq. (53)) factor out a contribution ( γ + λ ) in Eq. (68).Since γ >
0, this does not affect the stability.A particularly interesting aspect of our model (inwhich the jerky form is relevant) is the case γ T (cid:54) = γ R .For γ T = 0 . γ R = 0 . γ T = γ R = 0 . γ T ) while fixing the other one can make thesystem unstable. Finally, for γ T = 0 . γ R = 0 . λ ) also increasesslightly above the stability boundary compared to thesituation for γ T = γ R = 0 . γ R has a much smaller effect on the stability ofthe system than changing γ T .The phase diagram, which is shown in Fig. 2, givesa broader picture. It shows the maximum of Re( λ ( k )), To be able to write Eqs. (64) and (65) as the eigenvalue problem(66), we have multiplied Eq. (64) by λ . In Eq. (68), we have thendivided by λ again. Although this trick can only be applied for λ (cid:54) = 0, we can easily show that Eq. (68) is also correct for λ = 0:In this case, we have λ = λ such that Eqs. (64) and (65) formalready an eigenvalue problem whose characteristic polynomialis easily found to be given by Eq. (68). i.e., the maximal growth rate, as a function of γ T and γ R for (a) v = 0 .
5, (b) v = 0 .
75, and (c) v = 1.As can be seen, stronger translational damping (larger γ T ) stabilizes the homogeneous state, whereas rotationaldamping (measured by γ R ) has only weak effects on thelinear stability. Moreover, comparing the phase diagramsfor different values of v shows that activity tends todestabilize the system. It is a very remarkable observation that the values ofthe damping parameters affect the linear stability of thesystem. In the passive case, Eq. (64) leads to the disper-sion relation λ ( k ) = − γ T ± (cid:114) γ T − M k f ( k ) (69)with the function f ( k ) = (cid:15) + (1 − k ) + 3 ¯ ψ that is inde-pendent of γ T and γ R . The largest eigenvalue λ , obtainedby choosing the “+” sign in Eq. (69), has a positive realpart if and only if f ( k ) < k (cid:54) = 0). This reflectsthe fact that the passive system with damping always ap-proaches the minimum of the free-energy functional F ,implying that whether the homogeneous state is stablesolely depends on whether it is a minimum of F . Thedamping coefficient in the passive case can only deter-mine how and how rapidly equilibrium is approached,but not the equilibrium state itself. The active system,however, does not obey such a minimization principle. VII. SOUND WAVES
Next, we discuss sound propagation, which is a fur-ther possible application of underdamped PFC models[33, 34]. Sound waves (propagating density perturba-tions) in active matter are known to differ from those inpassive systems in interesting ways [87]. In the passivecase, sound waves propagate due to the presence of iner-tia, and studying them requires hydrodynamic equationsin which the momentum density is a relevant variable[18, 68]. However, in active systems, sound propagation Note that we fix M , v , and D r rather than ˜ M , ˜ v , and ˜ D r . Arold and Schmiedeberg [12], who also studied underdampeddynamics, have reported that the damping parameter does notaffect the state diagram. In our notation, the calculations inRef. [12] fix ˜ M , ˜ v , and ˜ D r rather than M , v , and D r . Asdiscussed in Section II for the case of DDFT, fixing M corre-sponds, roughly speaking, to changing the damping coefficient γ = 1 / ( βmD ) by changing the diffusion coefficient D , whereasfixing ˜ M corresponds to changing γ by changing the mass m . -0.8-0.6-0.4-0.20 R e ( λ ) γ T = 0 . , γ R = 0 . γ T = 0 . , γ R = 0 .
40 0.2 0.4 0.6 0.8 1 k -0.8-0.6-0.4-0.20 R e ( λ ) γ T = 0 . , γ R = 0 . k γ T = 0 . , γ R = 0 . ac bd M = 1 , v = 0 . , D r = 0 . , (cid:15) = − . , ψ = − , b = 0 . γ T and γ R and with fixed values for the other parameters. A dashed lineindicates Re( λ ) = 0. The homogeneous state is unstable at low and stable at high damping. is also possible in an overdamped system, as can be shownusing the Toner-Tu model [88]. Here, the density is cou-pled to a dynamic equation for the polarization, whichallows to derive a second-order equation for linear densityvariations and thus for travelling sound waves [47, 48].This effect can also be described in our model, since italso couples the density to the polarization. However,since our model can also incorporate effects of inertia,it still contains sound modes in the passive limit, whichthen correspond to usual sound waves. Hence, under-damped active systems, as described by the jerky activematter model, should allow for different types of soundwaves.Sound can also be studied by considering a linear Strictly speaking, since sound waves are an adiabatic process, oneshould also include temperature fluctuations [18]. Interestinginsights can, however, already be gained from the isothermalmodel considered here. For an accurate microscopic descriptionof sound in a nonisothermal system, see Ref. [68]. perturbation, i.e., by using the characteristic polyno-mial (68), obtained from the ansatz given by Eqs. (62)and (63). However, the physical interpretation and thusthe procedure is slightly different: In Section VI, we havesolved Eq. (68) for the complex growth rate λ , given afixed real value of k . Physically, this means that the sys-tem is subject to a perturbation ∝ exp(i kx ) at t = 0,and we then analyze whether this perturbation grows ordecays in time. By considering all values of k (disper-sion relation), we then obtain the linear stability, sinceall linear perturbations can be decomposed in Fouriermodes exp(i kx ). In the study of sound waves, on theother hand, we are interested in the way in which an os-cillation exp(i ωt ) with angular frequency ω propagatesthrough the system. This implies that we solve Eq. (68)for the complex wavenumber k = q + i κ (with q, κ ∈ R ),given a fixed λ = i ω that is purely imaginary . The imag-inary part κ of k then gives the inverse of the length scaleon which the sound wave decays (attenuation length).Equation (68), which contains only even powers of k , then produces four types of solutions, namely (1) ω ,0 γ T γ R v = 0 . un s t a b l e stablehomogeneousstate γ T v = 0 . unstablehomogeneousstate stable γ T v = 1 unstablehomogeneousstate m a x ( R e ( λ )) a b c M = 1 , D r = 0 . , (cid:15) = − . , ψ = − , b = 0 . γ T and γ R for different values of v and fixed values of the other parameters. The color indicates the maximum ofthe real part of λ ( k ). Black regions correspond to a stable, colored regions to an unstable fluid reference state. The phaseboundary is indicated as a dashed line. An increased activity v destabilizes the system, whereas a larger value of γ T stabilizesit. q >
0, (2) ω , q <
0, (3) ω > q <
0, and (4) ω < q >
0. Here, solutions of types (1) and (2) describewaves travelling to the right (to + ∞ ), whereas solutionsof types (3) and (4) describe waves travelling to the left(to −∞ ). For reasons of symmetry, we can restrict our-selves to solutions of type (1). In this case, solutions with κ < x = 0 and travelto the right with an amplitude that decreases in the di-rection of propagation. On the other hand, a solutionwith κ > κ < v = 0) with veryweak damping ( γ T = 0 . γ R = 0 . v = 0, Eqs. (50) and (51) describ-ing ψ and P , respectively, decouple. Both equations havetravelling waves as a solution. The ψ -waves correspondto usual passive sound, i.e., to density perturbations thatpropagate due to the coupling of density and momentum.On the other hand, the P -waves are propagating devia-tions from the equilibrium polarization. We here assumethat the polarization can still be defined in a meaningfulway for v = 0. For an active Brownian sphere, the orientation might only arisethrough the activity [84], such that there are no orientationaldegrees of freedom left for v = 0 and (cid:126)P cannot be defined.In practice, however, an asymmetry can still be present even ifit does not result in self-propulsion. As an example, consider anonspherical particle that is propelled by ultrasound [89] – it stillhas a (geometrical) orientation if the ultrasound is switched off.If the polarization cannot be defined for v = 0, modes linked tothe polarization have a physical meaning only for v (cid:54) = 0. The results ( ω ( q ) and κ ( ω )) for parameters (a) (cid:15) = − . b = 0 .
1, (b) (cid:15) = − . b = 0 .
1, and (c) (cid:15) = − . b = 0 . M = 1, D r = 0 .
5, and¯ ψ = −
1. In the first row, ω ( q ) is shown (Figs. 3a(i)-3c(i)), the second row shows the corresponding curvesfor κ ( ω ) (Figs. 3a(ii)-3c(ii)). Four modes can be found:First, there is a red mode that propagates almost withoutdamping ( κ ≈ q ( ω ) is small at small ω , followed by a region withstronger growth at larger ω . This mode is damped atsmall ω and propagates further into the medium at larger ω . Third, there are a blue mode and an orange modewhere ω ( q ) has (for these parameters) a strong negativeslope and reaches ω = 0 for q (cid:54) = 0. The damping of theorange mode has no strong dependence on the frequency,whereas the blue mode has positive κ and is thereforenot a physically reasonable sound wave (we plot it as adashed line to indicate this).When comparing the various plots, the physical originof the different modes becomes transparent: The greenmode is identical in Fig. 3a and Fig. 3b, whereas theother modes show a significant change: In Fig. 3b(i), theyintersect at ω ≈ .
6, which is not the case in Fig. 3a(i). At ω ≈ .
6, one also observes a maximum for the orange anda minimum for the blue mode in Fig. 3b(ii), which is notthe case in Fig. 3a(ii). On the other hand, the red, blue,and orange modes do not change between Fig. 3a andFig. 3c, whereas the growth of q with ω in the green modenow starts at ω ≈ . ω ≈ . (cid:15) , a parameter affecting ψ , hasbeen changed between Figs. 3a and 3b, it follows thatthe red, blue, and orange modes correspond to densitywaves. The red mode is usual passive sound, whereas theorange and blue modes arise from the higher-order spatialgradients in the free energy (59). (We have verified thatthe orange and blue modes disappear if these terms are1 q ω i (cid:15) = − . , b = 0 . q i (cid:15) = − . , b = 0 . q i (cid:15) = − . , b = 0 .
50 0.2 0.4 0.6 0.8 1 ω -1.5-1-0.500.51 κ ii ω ii ω ii γ T = 0 . , γ R = 0 . , M = 1 , v = 0 , D r = 0 . , ψ = − a b c FIG. 3. Dispersion relations showing ω ( q ) and κ ( ω ) for sound propagation in a passive system with weak damping. Only modeswith positive q and ω are shown. Different colors are used to distinguish the different modes. The red, blue, and orange modesare unaffected by a change of b , the green mode is unaffected by a change of (cid:15) . For the blue and orange modes, the curves ω ( q )coincide. Since it has positive κ , the blue mode is shown as a dashed curve. not present in Eq. (59).) As far as ω ( q ) is concerned,the orange and blue modes coincide, which is not thecase for κ ( ω ). On the other hand, only the parameter b , which affects P , has been changed between Fig. 3aand Fig. 3c, such that the green mode corresponds topolarization waves.The situation is quite different in the active case ( v =0 . γ T = 0 . γ R = 1. Apart from the change indamping and activity, we use the same parameters as inFig. 3.When comparing Figs. 4a-4c, it is found that a changeof both (cid:15) and b affects all four modes. The reason is that,in the active case, oscillations of ψ and P are coupled.Moreover, the shapes of the modes change. For the redmode, the most significant change in ω ( q ) is a sharp bend observed at ω ≈ . ω , whereas it can propagate al-most without damping at small ω (Fig. 4a(ii)-Fig. 4c(ii)).When considering the green mode, strong variations canbe found for both ω ( q ) and κ ( ω ). For (cid:15) = − . (cid:15) = − . ω ( q ) is not even a function due toa local maximum in q ( ω ) (Fig. 4b(i)) and κ ( ω ) reachesa local maximum at ω ≈ . ω ( q ) has a “hockey-stick shape”,where for larger frequencies the mode grows to the right(and not to the left as in the passive case). Moreover, ω ( q ) does not coincide for the blue and orange mode inthe active case.Consequently, if we define “sound” as “propagatingdensity oscillation”, we find two different mechanisms forsound propagation in active matter: First, there are os-cillations due to translational inertia as in the passivecase. These are still present in active matter, although2 q ω i (cid:15) = − . , b = 0 . q i (cid:15) = − . , b = 0 . q i (cid:15) = − . , b = 0 .
50 0.2 0.4 0.6 0.8 1 ω -1.5-1-0.500.51 κ ii ω ii ω ii γ T = 0 . , γ R = 1 , M = 1 , v = 0 . , D r = 0 . , ψ = − a b c FIG. 4. Analogous to Fig. 3, but now for sound propagation in an active system with strong damping. All modes are nowaffected by changes of (cid:15) and b . their properties are modified by the coupling to polar-ization degrees of freedom. Second, there are oscillationsthat arise from the coupling of the density to the po-larization. This mechanism is not present in fluids con-sisting of passive spherical particles. As can be seen inFigs. 4a(ii) and 4c(ii), the damping of the green mode de-creases for larger frequencies if we set (cid:15) = − .
5, whereasthe “passive” sound modes are more strongly dampedin this case. Consequently, sound can propagate furtherinto the medium via the second, “active” mechanism athigher frequencies at these parameter values.We summarize our observations for sound propagationin active matter: • For v = 0, there is one mechanism for sound prop-agation (coupling of density and momentum), as iswell known from passive fluids. • For active systems, stronger damping is required toensure that the system is still stable against pertur-bations. • A second type of sound waves exists in the activecase, arising from the coupling of density and po-larization. (For v = 0, these waves only affect thepolarization, not the density.) • For certain parameter values, waves of the first typepropagate further into the medium at small fre-quencies, whereas waves of the second type propa-gate further at higher frequencies.The last effect is of particular interest for possible ap-plications, since it allows for the development of acousticfrequency filters: If one excites oscillations in the redmode, smaller frequencies will propagate further into themedium, i.e., the system constitutes a low-pass filter. Onthe other hand, if the green mode is excited, higher fre-quencies will experience less damping, such that we havea high-pass filter. The properties of the filters can betuned by changing the system parameters as shown inFig. 4.3
VIII. CONCLUSIONS
We have shown how the presence of two relaxationaltime scales changes a general first-order soft mattermodel into a third-order spatiotemporal jerky dynam-ics. This general structure has then been used to derivea general active PFC model in which, in contrast to thestandard case, relaxational and orientational time scalesare both finite and different.The resulting model as well as the method by whichit was obtained allow to describe active matter systemsin which inertia is important with greater accuracy thanprevious theories. Since our model is of third order intemporal derivatives (unlike usual theories which are offirst or second order), it can be expected to show a num-ber of interesting effects that are not present in othermodels. This is also plausible by comparison to the caseof ordinary dynamical systems, where jerky dynamics al-lows for chaotic behavior that is not possible in second-order models.From the dispersion relation, we were able to investi-gate the linear stability and the resulting phase diagram,showing that in active systems, where the phase bound-aries are not determined by the free energy, changes of the damping coefficients can affect the linear stability of thefluid phase. The fluid state is destabilized by increasedactivity. Moreover, it is found that active fluids can ex-hibit two different mechanisms for sound propagation:waves arising from density-momentum coupling as inthe passive case, which can propagate with small damp-ing at low frequencies, and waves arising from density-polarization coupling, which can propagate further athigh frequencies. This effect has potential applicationsin the development of acoustic frequency filters based onactive fluids.Possible extensions of this work include the considera-tion of translational-rotational coupling (as discussed forparticles with general shape in Ref. [90]). Moreover, onecould consider spherical domains (as done previously forsimpler PFC models [91, 92]) or more general memorykernels.
ACKNOWLEDGMENTS
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