Large deviation statistics of non-equilibrium fluctuations in a sheared model-fluid
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Large deviation statistics of non-equilibriumfluctuations in a sheared model-fluid
Pritha Dolai and Aditi Simha
Department of Physics, Indian Institute of Technology Madras, Chennai 600036,IndiaE-mail: [email protected] and [email protected]
Abstract.
We analyse the statistics of the shear stress in a one dimensional modelfluid , that exhibits a rich phase behaviour akin to real complex fluids under shear.We show that the energy flux satisfies the Gallavotti-Cohen FT across all phases inthe system. The theorem allows us to define an effective temperature which deviatesconsiderably from the equilibrium temperature as the noise in the system increases.This deviation is negligible when the system size is small. The dependence of theeffective temperature on the strain rate is phase-dependent. It doesn’t vary much atthe phase boundaries. The effective temperature can also be determined from the largedeviation function of the energy flux. The local strain rate statistics obeys the largedeviation principle and satisfies a fluctuation relation. It does not exhibit a distinctkink near zero strain rate because of inertia of the rotors in our system.
Keywords : Large deviations in non-equilibrium systems, fluctuation phenomena,stochastic particle dynamics, numerical simulations.
Contents1 Introduction 22 Model and phase behaviour 33 Stress fluctuations and the fluctuation theorem 5 arge deviation statistics of non-equilibrium fluctuations in a sheared model-fluid
1. Introduction
The first quantitative statement of heat production in finite systems was provided bythe Fluctuation theorem (FT) of Evans, Cohen and Morris in 1993 [1]. The theoremrelates the probabilities of observing a positive value of the time averaged dissipativeflux and a negative one of the same magnitude in a thermostatted dissipative system. Itwas first demonstrated in a molecular dynamics simulation of a sheared two-dimensionalfluid of hard disks [1].Several classifications of the FT exist. The steady state and transient
FTs differin the ensemble of trajectory segments considered. In the stationary state Fluctuationtheorem (SSFT), such as in the original work of Evans et al [1], the trajectories of fixedduration τ belong to the driven steady state and the FT becomes valid in the limit τ → ∞ . In the transient FT (TFT) [2], the segments belong to a system initially in anequilibrium state evolving into a non-equilibrium steady state [3].Another classification is based on whether the fluctuation statistics are for a global or local quantity. The FT of Evans et al was for fluctuations in the total entropyproduction which is a global quantity. Gallavotti and Cohen [4, 5] provided a rigorousmathematical derivation of this theorem and in addition rederived a FT for the average oflocal observables. This local FT is easier to test experimentally as negative fluctuations,too rare to be observed experimentally in global quantities, are more prevalent in localquantities. Moreover, it does not require steady state conditions to hold globally andcan be applied when it holds only locally.To put our work in context, it is essential to distinguish between the FT of Evansand Searles [3] and that of Gallavotti and Cohen [4, 5]. The former is a statement offluctuations in the dissipative flux of thermostatted non-equilibrium states. It providesan expression for the probability of a dissipative flux in the direction opposite to thatrequired by the second law of thermodynamics. It requires that the initial state ordistribution satisfies the condition of ergodic consistency. The Gallavotti-Cohen FT[4, 5] is more general and applies to non-equilibrium systems driven far from equilibriuminto nonlinear chaotic regimes. It depends crucially on the Chaotic Hypothesis whichimposes certain conditions on systems to which it can be applied. Our system which isa dissipative stochastic system driven far from equilibrium falls under this category. Weshow that the energy flux averaged over a duration τ , W τ , satisfies the Gallavotti-CohenFT lim τ →∞ τ ln P (+ W τ ) P ( − W τ ) = βW τ (1)in the steady state. β = ( k B T eff ) − defines an effective temperature.Both FTs have been validated in several systems. The Evans-Searles FT has beenverified in a number of molecular dynamics simulations, electrical resistor circuits [8],and for a colloidal particle in an optical trap [6, 7]. The Gallavotti-Cohen FT has beensatisfied in many real systems such as in turbulent flows [9–11], turbulent Rayleigh-B´enard convection [12], vertically agitated granular gas [13] and in a sheared micellar2 arge deviation statistics of non-equilibrium fluctuations in a sheared model-fluid gel [14].Our system mimics a fluid under shear at constant strain rate and exhibits a richphase behaviour similar to that seen in real complex fluids under shear [15]. Unlike in theisoenergetic simulations of a sheared two dimensional fluid [1], the heat and energy fluxhave different statistics in our system and the dynamics is not time reversal invariant.We show that eqn.(1) is satisfied in all phases of the system and study the variationof the effective temperature, T eff , with the strain rate, strength of the stochastic forceand relation to changes in phase. We show that T eff can also be obtained from thelarge deviation function (LDF) of the energy flux. Finally, we analyse the statistics offluctuations in the local strain rate.
2. Model and phase behaviour
Our model fluid is the classical one-dimensional XY model of a lattice of spins or rotors(Figure 1), each of which can rotate perpendicular to the lattice. The angle θ i of thespin s i = (cos θ i , sin θ i ) is the only degree of freedom of each rotor or spin. Each spininteracts only with its nearest neighbours via torsional forces. In the equilibrium modelthese are forces arising from the interaction potential U = − P N − s i · s i +1 . υ θ Figure 1.
The 1D driven XY model. The angle θ and velocity ˙ θ of each rotorcharacterizes the state of the system at each instant. To generate the non-equilibrium steady states of this model, the rotors are subjectedto Langevin dynamics. In addition to the conservative force given by U , frictional andstochastic forces are included that conserve total angular momentum. The net torsionalforce on each rotor determines its angular acceleration according to Newton’s law ofmotion, i.e. , I ¨ θ i = τ i = τ i,i +1 + τ i,i − , (2)where I is the moment of inertia of each rotor which is unity in the chosen units and τ i the total torque acting on rotor i . This has three contributions: conservative, dissipativeand random, each of which is pairwise additive and acts along ˆ θ ij where θ ij = θ i − θ j isthe relative angular separation between rotors i and j . τ i = X
The conservative torque, τ Ci = − ∂U ( θ ij ) ∂θ i . The frictional torque depends only on therelative velocity between rotors and has the simple form τ Dij = − Γ( ˙ θ i − ˙ θ j ) where Γ isthe frictional co-efficient. The random torque τ Rij = σζ ij , where σ is the amplitude ofthe random torque and ζ ij is a Gaussian random variable of unit variance. The systemis driven by rotating one of its boundaries relative to the other. This is done usingLees-Edwards boundary condition for the angle variable which allows us to impose arelative velocity υ between the boundaries of the system along with periodic boundaryconditions so that edge effects are eliminated. The imposed strain rate is defined as˙ γ = υ /N , where N is the number of rotors. The model has been used earlier tomeasure transition rates in non-equilibrium steady states and its rich phase behaviourlikened to the phenomenology of real complex fluids under shear [15–17]. A model thatbears some similarity to ours is the sheared solid model [18].The equations of motion (2) are integrated forward in time using a self-consistentDissipative Particle Dynamics (DPD) algorithm [19, 20]. The time step of integration dt is chosen such that the relative motion between rotors per time step is not greaterthan 0.24.The system exhibits four qualitatively distinct phases. Figure 2(a) depicts thevarious phases observed as a function of the imposed strain rate, ˙ γ and noise amplitude(expressed in terms of T = σ / .
04 . We distinguish between thesephases based on the distribution of the time-averaged local strain rate ( h dθ i,i +1 /dt i )in the system. The following four phases are observed : (I) Uniform shear phase - theaverage local shear rate is the same through out the system. This phase exists at high σ and Γ ˙ γ where thermal and frictional forces are much larger than the conservative force ∂U ( θ ij ) /∂θ i . Uniform shear implies a linear average velocity profile. All Newtonianfluids shear uniformly. In complex fluids, a uniform shear flow regime always exists(Figure 2(b) ). (II) Slip plane phase - the shear in this phase is localized to a fewneighbouring rotors (or planes) while the majority move without any relative motionbetween their neighbours as in a elastic solid (Figure 2(d) ). This phase is observedwhen the thermal energy is small and the average torque is less than the maximumpotential gradient. Relative motion at the slip plane produces an oscillatory torquethat propagates in the solid region as damped torsional waves. Slip planes have beenobserved in surfactant cubic phases [21]. (III) Solid-fluid coexistence - This phase iscreated when the yield event at a slip plane triggers more yeild events locally giving riseto a finite region thats fluid (where the rotors overcome the potential barrier), co-existingwith solid regions (Figure 2(e) ). This co-existence is possible only when the local timeaveraged stress is just below the yield point. Such phases have been observed in foams[22, 23]. (IV) Shear banding - the local shear rate assumes two or more values, formingregions of different effective viscosities (Figure 2(c) ). Each of these regions is called aband. Shear banding has been observed in polymers [24, 25]. A precise characterizationof these phases along with a mean field analysis of the model’s phase behaviour can befound in [15].For the range of ˙ γ and σ studied (at fixed value of Γ = 0 .
04 ), we do not see a4 arge deviation statistics of non-equilibrium fluctuations in a sheared model-fluid turbulent phase in this system. However, we expect to see turbulence for large ˙ γ , largesystem size, and by reducing the value of the frictional co-efficient Γ .We restrict our study here to properties of the driven steady state which is attainedafter an initial transient behaviour. In the steady state, the rate of change of the totalenergy ˙ E = W − ˙ Q , where the energy flux or power W is the rate at which work isdone on the system by the driving force and the heat flux ˙ Q is the heat dissipated perunit time. Since the system is sheared at constant strain rate, W ( t ) = f ( t ) υ where f is the net torque between neighbouring rotors on either side of the sheared boundary.The distribution of f is therefore the distribution of the shear stress in the system. Thefrictional coefficient and the interaction potential are kept fixed in our simulations whilethe strain rate ( ˙ γ ) and noise amplitude ( σ ) are varied. T I II IIIIV γ (s -1 ) . (a) -0.2 0 0.2050100 R o t o r -1 0 1 Angular velocity (s -1 ) R o t o r T = 0.208 γ = 0.004 s -1 . γ = 0.02 s -1 T = 0.075 . (b)(c) -10 0 10 R o t o r -100 0 100 Angular velocity (s -1 ) R o t o r T = 0.075 γ = 0.204 s -1 . T = 0.208 γ = 2.04 s -1 . (d)(e) Figure 2. (a)Phase diagram of the model fluid for Γ = 0 . s − and I = 1indicating four distinct phases. (I) uniform shear phase, (II) slip-planes, (III) solid-fluid coexistence, (IV) shear banding. Average velocity profile in - (b) Phase I, (c) PhaseIV , (d) Phase II and (e) Phase III.
3. Stress fluctuations and the fluctuation theorem
The shear stress of the system fluctuates about a positive mean value and assumesboth positive and negative values. Shown in Figure 3(a) is a typical time evolution5 arge deviation statistics of non-equilibrium fluctuations in a sheared model-fluid of the shear stress f τ averaged over duration τ for a system of 100 rotors in the shearbanding phase. To find the probability distribution P ( f τ ), f ( t ) is recorded as the systemevolves in the steady state over a long duration of time (corresponding to a few hundredrevolutions of each rotor) and over many realizations of the random torque. 2 π/ ˙ γ ,which is the time it takes for a rotor experiencing a local shear rate ˙ γ to completeone revolution, defines a timescale. The data is then averaged over different durations τ larger than any correlation time ( t c ) in the system. In some cases the window ofaveraging is shifted from a previous one by a time larger than the correlation timeto improve sampling. The distribution of the resulting averaged data is P ( f τ ). Theaverage energy flux into the system in a duration τ is W τ = τ R t + τt W ( t ′ ) d t ′ = f τ υ ;and P ( W τ /υ ) = P ( f τ ). We define the dimensionless quantity X τ = W τ / h W τ i . Thedistribution P ( X τ ) for different values of τ corresponding to the fluctuations in Fig. 3(a)is shown in Fig. 3(b) . Its deviation from a Gaussian distribution is shown in the inset. f Time (s) (a) -2 -1 0 1 2 3 X τ -5 -4 -3 -2 -1 P ( X τ ) c c c c c -1 0 100.030.06 τ =3 t c (b) X τ l n ( R ) c c c (c) X τ ( / τ ) l n ( R ) c c c (d) Figure 3. (a) Typical shear-stress fluctuations for ˙ γ =0.327 s − , σ = 0 . X τ for different τ ’s. (c) Plotof ln[ P (+ X τ ) /P ( − X τ )] vs X τ for different durations τ expressed in terms of thecorrelation time t c . Solid lines are the straight line fits to data. (d) Plot of τ ln[ P (+ X τ ) /P ( − X τ )] vs X τ . All collapse into a straight line passing through theorigin as shown by the fitted solid line. arge deviation statistics of non-equilibrium fluctuations in a sheared model-fluid In terms of X τ , eq.(1) for finite τ isln( R ) ≡ ln P (+ X τ ) P ( − X τ ) = β h W τ i X τ τ. (4)We set k B = 1 . The straight lines in Fig. 3(c) validate this and their collapse onto oneline on scaling with 1 /τ (Fig. 3(d) ) validates eqn. (1). T eff can be calculated from theslope of the collapsed line. It quantifies the probability of observing negative shear inthe system; a large T eff corresponds to a higher probability of finding the system withnegative shear stress. -2 -1 0 1 2 3 W τ -6 -5 -4 -3 -2 -1 P ( W τ ) Uniform shearShear bandingSlip-planeSolid-fluid (a) -2 -1 A v er ag e s t re ss T =0.3T =0.408T =0.533T =1.302T =5.208 γ (s -1 ) . (b) Figure 4. (a) Probability distributions in different phases. The distributionsin uniform shear flow, shear banding, slip-plane and solid-fluid co-existence phasecorrespond to following sets of parameters: (i) T = 0 . γ = 0 . s − , t c = 4 . s .(ii) T = 0 . γ = 0 . s − , t c = 4 . s . (iii) T = 0 . γ = 0 . s − , t c = 3 . s .(iv) T = 0 . γ = 1 . s − , t c = 1 . s . (b) Average stress as a function of ˙ γ fordifferent T . When the fluctuations are Gaussian distributed, the probability distribution P ( W τ ) ∝ exp − ( W τ − h W τ i ) / (2 σ W τ ). This would then imply P (+ W τ ) P ( − W τ ) = exp (2 h W τ i W τ /σ W τ ) . (5)Comparing this with the ratio of the probabilities according to the Gallavotti-CohenFT (1), we get2 h W τ i /σ W τ = β τ ⇒ σ W τ / h W τ i = q T / h W τ i τ (6)The left hand side (LHS) is a quantity that involves the standard deviation and meanof the Gaussian distribution. The temperature T is the thermodynamic temperaturewhen the system is in the linear response regime. It is the effective temperature, T eff ,when the system is not in the linear response regime and the fluctuations are Gaussiandistributed. The quantity on the LHS of (6) is also related to the curvature of the LDFat its minimum and its derivative at W τ = 0 for Gaussian fluctuations. We test if thisrelation holds generally when the fluctuations are not Gaussian in Sec. 3.2 . 7 arge deviation statistics of non-equilibrium fluctuations in a sheared model-fluid The probability distribution for W τ in the various phases is shown in Fig. 4(a) .The distribution at large noise amplitudes (shown by red filled circles) is Gaussian. Thedistribution in the shear banding phase is nearly Gaussian but deviates from it far awayfrom the mean. It is non-Gaussian in the slip-plane and two phase or coexistence regime.The variation of average stress with strain rate for different T is shown in Fig. 4(b) .For small T , the average stress plateaus before increasing sharply with ˙ γ . This plateauregion indicates the shear banding regime which diminishes with increasing T . Forlarge T , the average stress increases very slowly for small ˙ γ before increasing sharplyfor ˙ γ > . s − . Beyond ˙ γ = 4 . s − curves for different T fall on each other in theuniform shear flow regime. The variation of T eff with strain rate ˙ γ for a fixed noise amplitude, σ = 0 . γ covered, the system goes through three different phases(boundaries shown by vertical dashed lines) as can be inferred from the phase diagram;the uniform shear regime up to ˙ γ = 0 . s − , shear banding between ˙ γ = 0 . − . s − ,and the solid-fluid co-existence at higher ˙ γ . Its variation with ˙ γ is different in eachphase. The effective temperature shows a very slow increase with ˙ γ in uniform shearflow. In the shear banded phase it increases as √ ˙ γ and much faster in the coexistencephase. At the phase boundaries T eff does not vary much.The effective temperature is plotted along with the average interaction energy, < P E > and the standard deviation of the total energy, δE , in Fig. 5(b) . Thevariation of T eff is qualitatively similar to δE . It is proportional to δE in each of thephases albeit with a different constant of proportionality.We present our results for the variation of the effective temperature with the noiseamplitude, σ , in terms of T = σ / σ . Fig. 5(c) shows how T eff varies with T for systems of size N = 10 and N = 100. At N = 10 , we find that T eff ≈ T asindicated by the black solid line corresponding to T eff = T . This is not true as thesystem size increases where it deviates considerably from T even when the strain rateis small. The weak dependence for small N suggests a crossover length ( L = N ) belowwhich nothing happens.An analysis of the system size dependence of T eff , for fixed values of σ and ˙ γ ,indicates that it increases linearly with system size (see Fig. 5(d) ). We speculate thatthis is due to the driving speed, υ = ˙ γN , which increases linearly with N . Plotted inthe inset is T eff /N vs. N , which is nearly constant.Except in the region corresponding to very low noise amplitudes ( σ < . arge deviation statistics of non-equilibrium fluctuations in a sheared model-fluid T e ff γ (s -1 ) T e ff . γ (s -1 ) . (a) -3 -2 -1 T e ff & δ E
10 100 N T e ff
10 100 N (d) T e ff / N Figure 5. (a) T eff as a function of ˙ γ for fixed noise amplitude σ =0.1 . In the shearbanding regime the curve is fitted to ˙ γ / as shown in inset. The vertical lines indicatethe boundary between different phases. (b) Standard deviation in energy, averagepotential energy and T eff as a function of ˙ γ . (c) Variation of effective temperature( T eff ) with ( T ) for a fixed strain rate ˙ γ =0.004 s − for N = 10 and 100. Blacksolid line indicates the corresponding equilibrium temperature T . (d) T eff increaseslinearly with the system size. Inset: T eff /N as a function of N . The effective temperature for Gaussian fluctuations can be calculated from eq.(6) ifthe mean and standard deviation of the distribution are known. Here we relate it toquantities in the large deviation function, its curvature and slope at W τ = 0. We definethe dimensionless quantity X τ = W τ / h W τ i . X τ has the distribution, P ( X τ ) = exp[ − ( X τ − / σ W τ / h W τ i ) ] , (7)where h W τ i and σ are the mean and standard deviation, respectively, of the distributionof W τ . The LDF of X τ is then I ( X τ ) = − lim τ →∞ τ ln P ( X τ ) = lim τ →∞ τ ( X τ − σ W τ / h W τ i ) (8)Therefore, I ′ ( X τ = 0) = − ( h W τ i /σ W τ ) /τ, I ′′ ( X τ = 1) = ( h W τ i /σ W τ ) /τ (9)9 arge deviation statistics of non-equilibrium fluctuations in a sheared model-fluid -5 0 5 10 X τ I ( X τ )
74 t c
80 t c
86 t c (a) -1 -0.5 0 0.5 1 1.5 2 2.5 X τ I ( X τ )
10 t c
11 t c
13 t c (b) Figure 6. (a) LDF of X τ for Gaussian fluctuations for σ = 0 .
15 and ˙ γ = 0 . s − .(b) LDF for non-Gaussian fluctuations for σ = 0 . γ = 0 . s − . Both the curvature at the minima and the slope at X τ = 0 of the LDF are thesame, except for a difference in sign, and give the ratio of the mean to the standarddeviation of the original distribution of W τ . Eq.(6) can be rewritten in terms of thesequantities as 2 T h W τ i = − I ′ ( X τ = 0) , (10)2 T h W τ i = 1 I ′′ ( X τ = 1) . (11)Using the LDF for X τ , we have used eqs. (10,11) to calculate T for the following twocases:(i) P ( X τ ) is Gaussian - ˙ γ = 0 . s − , σ = 0 .
15 , I ′ ( X τ = 0) = − .
002 = − I ′′ ( X τ = 1).The yields an effective temperature T eff = 1 . . P ( X τ ) is non-Gaussian - ˙ γ = 0 . s − , σ = 0 . I ′ ( X τ = 0) = − .
38 , I ′′ ( X τ = 0) = − .
79 . Eq.(10) gives T eff = 17 . .
09 . The LDFs for the above two cases are plotted in Fig. 6 . Wefind that the effective temperature can be obtained from the derivative of the LDF at W τ = 0 . The mean strain rate, ˙ γ , imposed at the boundaries is constant in time but the localstrain rate ˙ γ i between neighbouring rotors i and i + 1 is a fluctuating quantity. We findthe probability distribution P ( ˙ γ τ ) of ˙ γ i averaged over rotors and duration τ . We definethe large deviation function (LDF) for the strain rate, F ( ˙ γ τ ) ≡ lim τ →∞ − (1 /τ ) ln P ( ˙ γ τ )and show that it exists (Fig. 7(a) ). The antisymmetric part of the LDF (Fig. 7(b) )obeys a fluctuation relation i.e. , F ( ˙ γ τ ) − F ( − ˙ γ τ ) ∝ τ ˙ γ τ (Fig. 7(c) ). These plots are for σ = 0 .
13 , ˙ γ = 0 . s − . 10 arge deviation statistics of non-equilibrium fluctuations in a sheared model-fluid -0.2 -0.1 0 0.1 0.2 0.3 γ τ (s -1 ) F ( γ τ )
75 t c
94 t c
112 t c σ = 0.13, γ = 0.04 s -1 . . . (a) l n { P ( + γ τ ) / P (- γ τ ) }
46 t c
54 t c
69 t c . . . γ τ (s -1 ) (b) ( / τ ) l n { P ( + γ τ ) / P (- γ τ ) }
46 t c
54 t c
69 t c . . (s -1 ) . γ τ (c) γ τ (s -1 ) F ( γ τ )
163 t c
175 t c
188 t c
200 t c
213 t c .. σ = 0.117, γ = 0.02 s -1 . (d) -0.1 0 0.1 0.200.010.02 F ( γ τ )
131 t c
150 t c
168 t c σ = 0.117, γ =0.04 s -1 . . . γ τ (s -1 ) (e) -0.2 -0.1 0 0.1 0.2 γ τ (s -1 ) F ( γ τ )
109 t c
122 t c
170 t c
182 t c
207 t c σ = 0.15, γ = 0.004 s -1 . . . (f) Figure 7. (a) The LDF for the local strain rate for parameter values σ = 0 .
13 and˙ γ = 0 . s − . (b) Plot of ln[ P (+ ˙ γ τ ) /P ( − ˙ γ τ )] vs ˙ γ τ for different τ expressed in termsof t c for same set of parameter values. The solid lines are the straight line fits todata. (c) All these lines of ln[ P (+ ˙ γ τ ) /P ( − ˙ γ τ )] vs ˙ γ τ collapse into a single straightline passing through the origin on scaling by τ , shown by the fitted solid line. (d) TheLDF for parameter values σ = 0 .
117 and ˙ γ = 0 . s − . (e) The LDF for σ = 0 .
117 and˙ γ = 0 . s − . (f) The large deviation function for σ = 0 .
15 and ˙ γ = 0 . s − . t c ≈ s in all the plots above. LDFs are plotted for 3 other sets of parameters. Fig. 7(d) and Fig. 7(e) showLDFs for ˙ γ = 0 . s − and ˙ γ = 0 . s − , respectively, with σ kept constant at 0.117 .Larger the strain rate ˙ γ , more asymmetric the LDF because of the rarity of negative11 arge deviation statistics of non-equilibrium fluctuations in a sheared model-fluid fluctuations. The LDF for σ = 0 .
15 and ˙ γ = 0 . s − is plotted in Fig. 7(f) . This LDFis almost symmetric around zero strain rate because of the large noise amplitude. Thereis no distinctly visible kink at ˙ γ τ = 0 in any of the three LDFs. We speculate thatthis is because of inertia of the rotors. A kink in the LDF has been observed earlier forentropy production in [26], and for the velocity of a self-propelled polar particle in [27].In [26], the kink was attributed to a dynamical cross-over between a regime of highentropy production and a regime of low entropy production. In [27], the kink is visiblebecause the dynamics of particles is overdamped and inertia is completely ignored. Thehigher the strain rate, the greater the asymmetry of the LDF about the minimum. Amore detailed study of the LDF for the local strain rate is currently under way.
4. Conclusions
We show that the Gallavotti-Cohen FT is satisfied across all phases of the sheared model fluid which exhibits phases similar to real complex fluids under shear. We studythe dependence of the effective temperature T eff (defined by the FT), on the strainrate, noise amplitude and system size. T eff ≈ T , the equilibrium thermodynamictemperature, at small strain rates. The linear response regime, when this happens,depends on the noise amplitude and system size. The larger the noise amplitude, thesmaller the ˙ γ at which the linear response regime sets in. T eff deviates considerablyfrom T as the noise amplitude increases (for fixed strain rate and system size). Thisdeviation is negligible when the system size is small, suggesting that there is a crossoverlength (or system size) below which nothing happens. The dependence of T eff on ˙ γ is phase-dependent. It doesn’t change much at the phase boundaries. The effectivetemperature can also be determined from the derivative of the LDF for the energy fluxat W τ = 0 . The local strain rate statistics obeys the large deviation principle andsatisfies a fluctuation relation. It does not exhibit a distinct kink at zero strain rate,seen in other systems [26, 27], because of the inertia of rotors in our system. Acknowledgments
We thank R. M. L. Evans, Sriram Ramaswamy, Abhik Basu and S. Govindrajan foruseful inputs and comments. We also thank P. B. Sunil Kumar for providing computingfacilities.
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