Orbital Magnetism of Active Viscoelastic Suspension Is Exclusive for the Active System with Signiﬁcant Inertia

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case of inertia number In of mechanics. dynamics effect the everyday body the of that rigid top evident spinning in is fluid, a it a to important life, or bodies celestial equally body From rigid be both. a forming can particles inertia of system a be ∗ † orepnigato [email protected] : author Corrresponding [email protected] : author Corresponding huh ihrsett h nrildnmc men- dynamics inertial the to respect with Though, nri a aepoon ffc ndnmc.I can It dynamics. on effect profound have can Inertia eateto hsc,Uiest fClut,9 cay P Acharya 92 Calcutta, of University Physics, of Department ihrsett nraigpritneo self-propulsion of v which persistence moment, increasing magnetic to su the respect processes, that with physical find underlying also of we diamagne Interestingly, scales from time transition novel the a tuning undergoes system is (par the medium diamagnetic field, the is system of moment the scale particles, time magnetic self-propelling dissipation orbital o elastic The classical the when finite that have inertia. can finite system contains the system that shown have Lor driv we down, via GFDR, breaks motion (GFDR) their relation harm to fluctuation-dissipation dimensional perpendicular two a applied in field move magnetic to constrained and charged are Abstract: eateto hsc,Uiest fKrl,Kariavattom, Kerala, of University Physics, of Department active .INTRODUCTION I. ria ants fAtv iceatcSuspension Viscoelastic Active of Magnetism Orbital ytm ofri oei hslow this in done is far so systems ecnie iuesseso fatv (self-propelling active of suspension dilute a consider We − 7 eod 8.Therefore, [8]. seconds usnadMSahoo M and Muhsin M Dtd ue2,2021) 29, June (Dated: ra Saha Arnab − 4 hs eaain(IS.I ssrnl nune and influenced strongly is It (MIPS). induced fundamen- motility separation the is systems of phase Brownian One active of [26]. feature the system tal on the in depends present crucially states inertia discontinuous, dynamical these or between continuous dynamical transition is The different confined trap. particles inertia, a self-propelling of in by presence developed the are dynam- states In long-time their [29]. in ics particles influence active profound of has velocity which and as dynamics between experimentally delay orientation induce shown the can been inertia modi- that has theoretically qualitatively as it well are example, For systems active fundamen- fied. the of of some ob- properties inertia, is tal tuning It the fine by [34–36]. describe that well Ac- served can self-propellers effects. inertial inertia inertial of including dynamics the macroscopic model explore Self the Langevin to tive in 33] [32, [26–31]. fabricated scales be particles length can propelled robots self propelling of on focusing in the- effects perspectives, from as experimental inertial studies well as are well as as there level oretical Recently particle level. single collective their the the in in role significant both a dynamics, play apt can are inertia [25] where insects examples flying [10–20]), [21–24], swimmers etc. macroscopic mini-robots self- vi- plate, internal vibrating in-built motor, (e.g. with bration mechanisms material self-vibrating or granular propelling are of Macro- onwards), particles torques. and scopic (and forces inertial millimeter-sized by me- influenced strongly viscosity from prominent. low starting a become in dia will moving with particles effect medium active Typically, inertial a in viscosity, moving lower objects self-propelling larger including motion [5]. torques Brownian and forces over-damped self-propelling flow and/or hydrodynamics Stokes active [9]) (namely, consider stresses one active where including limit number Reynolds lal,i eps h neo ute considering further envelop the push we if Clearly, mgei) hrfr,fragvnsrnt ftemagnetic the of strength given a for Therefore, amagnetic). htdie h ytmoto equilibrium of out system the drives that , † aul hnr od okt-009 India Kolkata-700009, Road, Chandra rafulla n h ytmaa rmeulbim hl breaking While equilibrium. from away system the ing ha,atv utain n ic-lsi dissipation. visco-elastic and fluctuations active as, ch agr(mle)ta h essec iesaeo the of scale time persistence the than (smaller) larger nzfre u otefiieatvt,tegeneralised the activity, finite the to Due force. entz i oprmgei tt advc-es)sml by simply vice-versa) (and state paramagnetic to tic nse teulbim eae non-monotonically behaves equilibrium, at anishes Thiruvananthapuram- a encluae xcl.Rmral,w find we Remarkably, exactly. calculated been has ncta.Terdnmc sculdt constant a to coupled is dynamics Their trap. onic ∗ btlmgeimol hntednmc fthe of dynamics the when only magnetism rbital atce navsoeatcfli.Particles fluid. visco-elastic a in particles ) 695581 India , 2 suppressed by the presence of inertia. It has also been ered to be negligible in comparison to other forces present shown that due to inertia, the coexisting phases of high in the system. We will focus on how such system respond and low particle density, obtained by MIPS, have widely to the presence of external magnetic field. We quan- different kinetic temperatures, which is in contrast with tify the response by evaluating the magnetic moment of equilibrium phase separation [37]. the system and its characteristic features with respect Being motivated by the above recent findings on in- to the time scales signifying various physical processes ertial active systems, here we will report the magnetic occurring within the system, such as : (1) time scale properties of a dilute suspension of charged active par- related to the correlation of active fluctuations (thanks ticles under external magnetic field. In particular we to self propulsion) and (2) time scale related to dissipa- will show that if the particles posses finite inertia, only tion. The time scale associated with active fluctuations then such system can exhibit orbital magnetism and goes originate from the persistence of the self-propellers to through a transition from the paramagnetic to the the move along a certain direction despite random collision diamagnetic state in different regime of parameter space from surrounding fluid particles. The time scale associ- of the model. The transition between paramagnetic and ated with dissipation signifies the characteristic time of diamagnetic states depends crucially on the interplay of the surrounding viscoelastic fluid within which the elas- different time scales related to the physical processes (e.g. tic dissipation dominates and beyond which the viscous active fluctuations and dissipation) involved in the dy- dissipation dominates. Note that here we consider the namics of the system. elastic dissipation being transient, decays within a finite Before going into the details of our findings, we will time whereas the viscous dissipation prevails for large introduce here the medium in which the active particles time. When these two time scales are equal, the gener- are suspended. We consider the medium to be viscoelas- alised fluctuation-dissipation relation (GFDR) [53] holds tic with transient elasticity. The elastic forces exerted and the system remains non-magnetic with zero magnetic by the medium to the self-propelling particles dissipate moment. Conversely, when the fluctuation and dissipa- within a finite time beyond which, particles are dragged tion time scale are unequal, GFDR breaks down and the only by the viscous forces. Theoretically one can also system shows non-zero magnetic moment under the in- consider the viscous limit of the problem where elastic fluence of external magnetic field. Depending on these forces dissipate very fast and only viscous forces are left two competing time scales, our analysis reveals that the to drag the suspended particles. In experiments, usu- suspension can exhibit either paramagnetic or diamag- ally polymers are added to the viscous fluids to make netic behaviour. In particular, we show that when the it transiently viscoelastic[38]. Viscoelasticity can Trig- persistence in active fluctuation dominates over the dis- ger fast transitions of a Brownian particle in a double sipation, the system manifests paramagnetism and when well optical potential[39]. It can add remarkable features the dissipation dominates over the self-propulsion, the to the dynamics of the active system suspended in it. system exhibits diamagnetism. Therefore, fixing the ex- For example, it enhances rotational diffusion of the ac- ternal magnetic field to a non-zero constant value, when tive particles[40, 41]. In a viscoelastic environment, the the time scales of these two physical processes, namely, self-propelling colloids exhibit transition from enhanced active fluctuations and dissipations (thanks to elastic- angular diffusion to persistent rotational motion beyond ity and viscosity of the medium) are tuned, the system a critical propulsion speed [42]. Viscoelasticity can en- undergoes a transition from diamagnetic states to para- hance or retard the swimming speed of a helical swim- magnetic states and vice-versa. Importantly, it has also mer, depending on the geometrical details of the swim- been shown that all these magnetic characteristics of the mer and the fluid properties [43]. Natural active systems system crucially depend on the presence of inertia. If are often found in viscoelastic environment. It is im- inertia of the system is negligibly small in compared to perative to study viscoelastic effects on their dynamics the dissipative as well as active forces, all the existing [44, 45]. Theoretically it has been shown that, elastic- forces in the system cancel each other, providing a zero ity in a non-Newtonian fluid can suppress cell division net-force, and then the magnetic moment vanishes. As a and cell motility [46]. In viscoelastic environment active result the system looses its magnetic characteristics. pulses can reverse the flow [47]. In case of chemically In the next section, we address the model by which the powered self-propelling dimers, fluid elasticity enhances problem is described in detail. The results are systemat- translational and rotational motion at the single parti- ically detailed in the subsequent sections and finally we cle level whereas in the multi-particle level, it enhances conclude. alignment and clustering[48]. In this article we consider a dilute viscoelastic suspen- II. MODEL AND METHOD sion of inertial, self-propelling (thereby, active) particles in two dimensions (2D). The suspension is confined in a 2D harmonic potential. The particles are charged and Model: We consider N non-interacting active (self- subject to a constant magnetic field perpendicular to the propelling) particles suspended in the viscoelastic plane in which the particles are moving [49–52]. Since the medium at temperature T . The particles are at positions system is dilute, the inter-particle interactions are consid- ri(t)= xi(t)ˆx + yi(t)ˆy and at velocities r˙i = vi at time t, 3 where i =1, 2, 3, ....N is the particle index, (ˆx, yˆ) are the where (α, β) (X, Y ). Here tc is the noise correlation unit vectors along X and Y . Each particle has charge q time. The effective∈ noise in the dynamics has finite cor- and constrained to move on X-Y plane. Particles are con-| | relation and it decays exponentially with time constant 1 2 2 fined within a 2D harmonic trap, U(xi,yi)= 2 k(xi +yi ) tc, representing the persistence of the self-propellers. Up where k is the spring-constant. The particles are also to t = tc, the self-propellers remain quite persistent to subjected to an external constant magnetic field B = Bzˆ move along a direction same as its previous steps, de- wherez ˆ is the unit vector along Z. The equation of mo- spite making random collisions with surrounding fluid tion of i-th particle is given by, particles. When t>tc, change of the direction with respect to the previous step becomes more probable. In the t limit of tc 0, with D =2γKBT , the active fluctuations r ′ v ′ ′ q v B r → mï = γ g(t t ) i(t )dt +| |( i ) k i+√Dξi(t). become thermal and the system becomes passive. There- − 0 − c × − fore, in the current model for active (self-propelling) par- Z (1) ticles, finite and non-zero t quantifies the activity of the Here m is the mass of the particle and r˙ = v is its c i i system. Thus we model the dynamics of the particle as an velocity. To take inertia into account, in the equation of Active Ornstein-Uhlenbeck Process (AOUP)[56, 57]. But motion of the particle ( Eq.(1)), we consider ¨r = v˙ as i i with inertia and elastic forces from the medium, the dy- the acceleration of the particle. namics can be represented by the generalized Langevin’s As the particles are self-propelling, despite the random ′ equation [54]. One may note that when tc = tc, we get collision of the particles of the surrounding viscoelastic ′ ′ ξα(t)ξβ(t ) = δαβg(t t ), which is the generalized fluc- fluid, they remain persistent to move along a certain di- h i − rection up to a finite time scale. Moreover the dynam- tuation dissipation relation (GFDR). Here we will explore both the situations where GFDR holds and where it does ics of the particles contains finite memory due to the ′ ′ not hold ( i.e. both for tc = t and tc = t ). A special elasticity of the fluid and therefore the dynamics is non- c 6 c Markovian [54]. The first term in RHS of Eq. (1) repre- case where the elasticity of the fluid dissipates very fast compared to the active fluctuations (i.e. for t′ 0 and sents the drag force on the particle because of the fric- c → tion with the surrounding medium. Due to the elasticity tc > 0), will be discussed in greater detail. present in the medium, the drag at time t not only de- Method: Now we solve the model Eq. (1) to evaluate |q| pends on the velocity of the particle at that particular the magnetic moment of i-th particle, Mi = ri vi 2c × time t, rather it depends on the weighted sum of all the in steady states. Introducing the complex variable zi = past velocities within the time-interval between 0 and t. xi + jyi (j = √ 1), we rewrite Eq. (1) as, As we consider the time-evolution of the particles to be − stationary, the weight function g should be a function of (t t′), with t t′. In particular we choose the weight- t − ≥ function (in other words, friction kernel) as ′ ′ ′ 2 zï(t)+ Γg(t t )z ˙i(t ) dt j ωcz˙i(t)+ ω0zi(t)= εi(t) t t′ − − 1 − ( − ) Z0 g(t t′)= e tc′ t t′ (2) ′ (4) − 2tc ≥ where the parameters are: Γ = γ ,ω = |q|B , ω = The above kernel gives the maximum weight to the cur- m c mc 0 v k 2ΓKB T x y rent velocity i(t), whereas the weight to the past ve- m and εi(t)= m (ξi (t)+ j ξi (t)). ′ locities decays exponentially with the rate 1/tc. The q q ′ By performing the Laplace transform of the complex time t tc is the time required for elastic dissipation ∞ ∼ ′ variable z (t) andz ˙ (t) we get: z (s)= e−stz (t) dt to decay substantially. Therefore, for time t>tc, the i i i i ′ L{ } 0 viscous dissipation dominates. In the limit tc 0, → and z˙i (s)= s zi (s), where R g(t t′) = δ(t t′) and consequently the system is left L{ } L{ } with− only viscous− dissipation. The friction kernel g(t t′) captures the Maxwellian viscoelasticity formalism, where− 1 at large enough time the fluid becomes viscous through ′ + s εi (s) tc L{ } a transient viscoelasticity, allowing the elastic force to zi (s)= 2 L{ } 3 1 2 Γ i ωc 2 ω0 relax down to zero [54]. s + ′ iωc s + ′ ′ + ω0 s + ′ tc − tc − tc tc The second term in the RHS of Eq. (1) represents (5) Lorentz force [55] caused by the magnetic field which The denominator in the Eq. (5) can be factorized as couples X and Y . The third term in the RHS of Eq. (1) follows: represents the harmonic confinement. The term with ξ appeared in Eq. (1) represents active, coloured (and D = (s s1)(s s2)(s s3) (6) thereby athermal) noise. The moments of ξα(t) are given − − − by, The roots of Eq (6) can be found using Cardano’s

|t−t′| method. Substituting Eq. (6) in Eq. (5) and using ′ δαβ − t ξα(t) =0, ξα(t)ξβ (t ) = e c (3) method of partial fraction, h i h i 2tc 4

3 ak zi (s)= εi (s) (7a) L{ } s sk L{ } k X=1 − 3 bl z˙i (s)= εi (s) (7b) L{ } s sl L{ } l X=1 − where (ai,bi)’s are found solving, 1 a1 + a2 + a3 = 0; a1(s2 + s3)+ a2(s3 + s1)+ a3(s2 + s1)= 1, a1s2s3 + a2s3s1 + a3s2s1 = ′ − tc 1 b1 + b2 + b3 = 1; b1(s2 + s3)+ b2(s3 + s1)+ b3(s2 + s1)= ′ , b1s2s3 + b2s3s1 + b3s2s1 =0 −tc

By taking inverse Laplace transform of Eq. (7a) and The average orbital magnetic moment of the particle is Eq (7b), we get zi(t) andz ˙i(t) as rewritten as

3 t ′ sk (t−t ) ′ ′ zi(t)= ak e εi(t ) dt (8a) M t q r v q ∗ k=1 Z ( ) = | | i i zˆ = Mzˆ = | |Im( zi(t)z ˙i (t) )ˆz X 0 h i 2c h| × |i 2c h i (9) 3 t ′′ sl(t−t ) ′′ ′′ Here, Im(... ) denotes the imaginary part, and ‘ ’ denotes z˙i(t)= bl e εi(t ) dt (8b) the complex conjugation. Using Eq. (8), the average∗ or- l X=1 Z0 bital magnetic moment is calculated to be

t t 3 ′ ′′ q ΓKBT ′ ∗ ′′ |t −t | ∗ sk(t−t ) sl (t−t ) − tc ′ ′′ M(t)= | | Im ak bl e e e dt dt (10) mctc   k,l X=1 Z0 Z0   Since we are interested in the time asymptotic limit or steady state, by straightforward integration in the long time (t ) limit, we obtain the average orbital magnetic moment as: → ∞

3 2 q ΓKBT ∗ tc 2tc M( )= | | Im ak bl ∗ + ∗ 2 2 (11) ∞ mctc  (1 + sitc)( 1+ s tc) (si + s )( 1+ s t )  k,l " j j i c # X=1 − −  

III. DISCUSSION ′ ′ (b)On tc tc plane where tc >tc, the system becomes diamagnetic− as M < 0. In this regime (elastic) dissipative forces dominate over the self-propulsion. Moreover, In Fig.1 we have plotted M as a function of tc and ′ the externally applied magnetic field induces a magnetic tc with different values of (ω0,ωc, Γ). From this exact field within the system of charged particles which opposes result, the following magnetic features of the system be- the external magnetic field itself. Therefore the system come apparent: ′ is diamagnetic when tc >tc. ′ ′ ′ (a) First of all, along the diagonal tc = tc, where GFDR However on tc tc plane where tc > tc, the self- holds, the average orbital magnetic moment M = 0. propulsion persists− for longer time as compared to the Therefore, along the diagonal, the system becomes non- elastic dissipation of the particles. As a result the in- magnetic even if the dynamics contains memory. It is duced magnetic field within the system follows the exter- the reminiscent of the exact result stating that the ther- nal magnetic field. Hence the system becomes paramag- mal average of magnetisation of any classical system in netic. equilibrium is zero [58]. Therefore, for a given strength of the external mag- 5

(a) (b)

′ FIG. 1. hM(∞)i [Eq. (11)] as a function of tc and tc; (a) with ωc = 1, ω0 = 1, and Γ = 1, (b) with ωc = 1.5, ω0 =1, Γ=1 (c) |q| ω0 . ω ω0 ω K T with = 1 5, c = 1, Γ = 1, (d) with Γ = 3, = 1, c = 1. Both B and mc are assumed to be unity here. netic field, the system exhibits a transition from the dia- to a maximum value beyond which M decreases with magnetic state to the paramagnetic state and vice-versa further increase in ωc. Therefore M has a non-monotonic simply by tuning the rate of the dissipation and the per- dependence on ωc. The other two parameters, namely Γ sistence time scale of self-propulsion. Moreover, if the and ω0 can only reduce M monotonically. direction of the magnetic field is reversed, the system For further insight, we consider the following special shows opposite behaviour regarding the phases. case where the friction kernel is a delta function (note that in the limit t′ 0, the exponential friction kernel (c) While passing from diamagnetic states to the para- c → magnetic states or vice-versa, the system undergoes a Eq (2) becomes delta function) and the noise correlation considerably large regime of non-magnetic states with is still exponential with correlation time, tc. In this limit, ′ only viscous dissipation takes place and the equation of M = 0 on the tc-tc plane. This regime includes the di- ′ motion [Eq (1)] reduces to agonal tc = tc. However, it also includes a regime across ′ the diagonal where tc = t (but they are comparable). It 6 c is to be noted that across this regime, the magnetization q B mzï = γz˙i + j | | z˙i kzi + εi(t), (12) of the system still remains zero. This regime grows as one − c − proceeds along the diagonal. It confirms that if GFDR where j = √ 1. This dynamics can be considered as in- holds good, it implies that the average orbital magnetic − moment is always zero, however the converse is not true. ertial active Ornstein Uhlenbeck process (IAOUP). The Average magnetic moment can still be zero even if the over-damped version of which, namely active Ornstein system is driven out of equilibrium where GFDR does Uhlenbeck process (AOUP) is now commonly used to not hold good. represent over-damped motion of active Brownian parti- ′ cles and successfully used to describe important features Similarly, we also note that when tc >> tc, GFDR like MIPS [56] and dynamical heterogeneities of active is obviously broken and the system is far from equilib- systems at high densities [60]. We have exactly solved rium conditions. Remarkably, in this regime, the para- the dynamics both analytically as well as using computer magnetic moment get reduced further and approaches to simulation. Following the similar procedure as before, zero. Hence the system becomes non-magnetic. This is the solution of the dynamics Eq (12), zi(t) andz ˙i(t) can because for large enough tc, due to high persistence in the be obtained as dynamics, the self propulsion of the particles overcomes t the influence of the external magnetic field and induces a ′ sk(t−t ) ′ ′ persistent rectilinear motion into them. This hinders the zi(t)= ak e εi(t ) dt (13a) k , particles to form closed current loops which are essential X=1 2 Z0 to exhibit finite magnetic moment [59]. t ′′ (d) Apart from the fluctuation and dissipation time sl(t−t ) ′′ ′′ z˙i(t)= bl e εi(t ) dt . (13b) scales, the parameters Γ,ω0 and ωc can also alter the ′ l=1,2 Z profile of M(tc,tc). With increasing ωc, M increases up X 0 6

Where, sk’s are given by trap). The value of the magnetic moment at τ0 is given |q|KB T ωc by Mr(τ ) = 2 2 . Clearly Mr(τ ) can 0 mc ωc +(Γ+2ω ) 0 1 2 2 0 s = (Γ i ωc) (Γ i ωc) 4ω (14a) (1,2) 2 − − ± − − 0 only decrease with ωh0 whereas it isi non-monotonic with q ωc (For small ωc, Mr(τ0) increases with ωc but for large enough ωc it decreases and finally limωc→∞ Mr(τ0) = 0). In figure [2] we show the behaviour of M with t and ak’s and bl’s are given by r c for different values of ω0 and ωc both from theoretical calculation as well as from simulation. Numerical 1 simulation of the dynamics has been carried out using a1 = , a2 = a1 (15a) s1 s2 − Heun’s method algorithm. We perform the simulation − s1 s2 with a time step of 0.001 second and run the simulation b = , b = b (15b) 4 1 s s 2 −s 1 upto 10 seconds. For each realization the results are 1 − 2 1 taken after ignoring the initial 103 transients in order Using the solutions in Eq (13) and Eq[9], the average for the system to approach steady state. The averages magnetic moment in the long time limit is given by are taken over 104 realizations.

Mr = M( )t′ ∞ c→0 2 q KBT t ωc = | | c (16) (a) (b)(b) mc  2 2 2 2  (1 + tc (Γ + tcω0)) + tc ωc   It is evident from the aforementioned exact result in Eq.16 that for tc = 0, Mr = 0 which is consistent with equilibrium result, namely Bohr-van Leeuwen theorem (BvL [58]). When tc > 0, the system goes away from equilibrium and therefore, it is intuitive that Mr also |q|KB Tωc 2 shoot up. When tc > 0 but small, Mr = t , mc c M t t′ 2 FIG. 2. r [Eq. (16)] is plotted with c (at c → 0 limit) implying that in this limit Mr increases with tc. from analytics as well as from numerical simulation (green dotted lines), for (a) various values of ω0, keeping ωc and Γ On the other hand, It is also evident from the afore- to be fixed as unity (b) for various values of ωc , keeping ω0 |q| mentioned exact result in Eq.16 that for tc , Mr = 0. and Γ as fixed to unity. Both KB T and are also assumed → ∞ mc From the point of view of equilibrium physics, it is to be unity here. counter intuitive. This is because in this limit the system is very far from equilibrium and hence the equilibrium result (Mr = 0) should not hold good. However, on a closer inspection one may note that in this limit, From Eq.16, one can also analyse the behaviour of Mr with respect to Γ, ω and ωc. AsΓ , it is evi- the persistence of the self-propulsion is so high that it 0 → ∞ is not allowing the particles to form a close current loop dent from the expression that Mr = 0. It implies that under the influence of the magnetic field, which is essen- in high viscous limit where inertia is negligibly small, the magnetic moment reduces to zero and it vanishes as tial to build up finite magnetic moment within the realm −2 of classical equilibrium physics. Therefore the magnetic Mr Γ . Therefore the active particles can exhibit non-zero∼ magnetic moment only when the dynamics of moment vanishes in tc limit. But for finite and |q|KB Tω→c ∞−2 the system contains considerable inertia. Similarly from large tc, Mr = 4 tc signifying that for large mcω0 Eq.16, it is evident that for large ω0, Mr approaches to- −2 −4 tc, Mr decreases with t . wards zero as Mr ω . Physically this occurs because, c ∼ 0 Taking together, we find Mr has a non-monotonic de- as ω0 increases the particles are constrained to move in pendence on tc. We exactly determine the optimum tc smaller area and therefore the area of the current loop ( τ ) at which Mr is maximum by solving formed by the particles decreases. This eventually leads ≡ 0 to zero magnetic moment. The dependence of Mr on ωc is non-monotonic. When ωc = 0, Mr = 0 and for dMr 4 4 2 3 = ω0τ0 +Γω0τ0 Γτ0 1=0. (17) small but finite ωc, Mr increases with ωc (for small ωc, B dtc tc=τ − − |q|K T 2 0 Mr t ωc). On the other hand, for large ωc, ≃ mc c 1 |q|KB T 1 Here τ0 = solves the equation, which implies the Mr mc ωc . Therefore, Mr decreases with large ω0 ≃ optimum tc or τ0 is independent of ωc (i.e. independent ωc and eventually it vanishes as ωc . This non- → ∞ of magnetic field), but it is inversely proportional to ω0 monotonic dependence of Mr on ωc for different Γ and tc (i.e. varies inversely with the strength of the harmonic is depicted in Fig.[3], both from analytics and simulation. 7

(a) (b)

FIG. 3. Mr [Eq. (16)] is plotted with ωc for different (a) Γ and (b) for different tc, keeping other parameters fixed as unity.

IV. CONCLUSION tence time scale), still it can have zero magnetic moment, as in case of equilibrium. In this work, we consider a system of non-interacting It is important to explore how the results qualifies for charged particles, self-propelling in 2D, being suspended relatively denser system with significant inter-particle in- in a Maxwellian viscoelastic medium, with considerable teractions. Work in this direction is in progress. We inertia. They are confined in a harmonic trap and sub- believe all the aforementioned theoretical results are ject to Lorentz force due to externally applied constant amenable to suitable experiments and they are impor- magnetic field, perpendicular to the plane of their mo- tant to implement magnetic control on the the dynamics tion. Due to the imbalance between elastic as well as of an active suspension by fine tuning external magnetic viscous dissipation (thanks to the medium) and active field. fluctuations (thanks to self-propulsion), the system goes V. ACKNOWLEDGEMENT out of equilibrium. The magnetic moment of the system can become non-zero and the system undergoes an M.M. and M.S. acknowledge the INSPIRE Faculty re- interesting transition from diamagnetic phase to para- search grant (IFA 13 PH-66) by the Department of Sci- magnetic phase and vice-versa. The transition depends ence and Technology, Govt. of India. M.S. also ac- on the interplay between the time scales involved in the knowledge the UGC start-up grant and UGC Faculty dissipative processes and active fluctuations. recharge program (FRP-56055), UGC. A.S. thanks the We have also determined how the magnetic moment of start-up grant from UGC, UGCFRP and the Core Re- the system depends on the parameters like cyclotron fre- search Grant ( CRG/2019/001492) from DST, Govt. of quency ω related to the strength of the magnetic field, c India. We thank Sourabh Lahiri for critical reading of natural frequency ω related to the strength of the har- 0 the manuscript. monic trap and friction coefficient Γ. As Γ , the magnetic moment vanishes, suggesting that the→ ∞ orbital magnetism of active viscoelastic suspension is exclusive for the active system with significant inertia. VI. AUTHOR CONTRIBUTION Interestingly, we find that even if the system remains under deep non-equilibrium conditions (in particular, when the particles are self-propelling with large persis- Both MS and AS have contributed equally to this work.

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