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ABSTRACT Soft particulate media include a wide range of systems involving athermal dissipative particles both in non-living and biological materials. Characterization of flows of particulate

media is of great practical and theoretical importance. A fascinating feature of these systems is the existence of a critical rigidity transition in the dense regime dominated by highly

intermittent fluctuations that severely affects the flow properties. Here, we unveil the underlying mechanisms of rare fluctuations in soft particulate flows. We find that rare fluctuations

have different origins above and below the critical jamming density and become suppressed near the jamming transition. We then conjecture a time-independent local fluctuation relation, which

we verify numerically, and that gives rise to an effective temperature. We discuss similarities and differences between our proposed effective temperature with the conventional kinetic

temperature in the system by means of a universal scaling collapse. SIMILAR CONTENT BEING VIEWED BY OTHERS INTERMITTENCY IN THE _NOT-SO-SMOOTH_ ELASTIC TURBULENCE Article Open access 27 May

2024 MINIMALLY RIGID CLUSTERS IN DENSE SUSPENSION FLOW Article 24 January 2024 INTERPLAY BETWEEN PARTICLE TRAPPING AND HETEROGENEITY IN ANOMALOUS DIFFUSION Article Open access 08 September

2023 INTRODUCTION Large fluctuations are a distinguishing feature of soft particulate flows, like flows of granular media1, 2, bubbles and foams3, and in living matter such as biological

tissues4, 5. Very dense systems are in a jammed state. They only move in response to a strong external force. Less packed systems are in a fluid state. They flow in response to any finite

force. The flows are highly intermittent and involve rare, very large fluctuations6 that can trigger transitions between the jammed and the fluid state7. Landslides8 and avalanches9 are

transitions from a jammed to a fluid state. Clogging of hoppers10 and breakdown of silos11 involve the transition from a fluid to a jammed state. Predicting the frequency of appearance of

such fluctuations is a question of great practical and theoretical interest. Fluctuation relations (FRs) compare the probability of the forward progression of a dynamics and its reverse;

akin of watching a movie played in forward and reverse direction. They provide an exact symmetry property of the probability distribution function (PDF) characterizing the likelihood to

encounter a given course of states in an observation of the dynamics. Close to equilibrium this symmetry entails linear response. Far from equilibrium FRs have been adopted in

micro-biological systems to determine the free energy of a folding RNA12 and thermodynamic properties of other biomolecules13, 14. In contrast to the dynamics of the microscopic biological

systems the dynamics of most macroscopic systems do not move against an exerted force15. The strongly fluctuating and intermittent flows of soft particulate matter are a noticeable exception

to this rule. Here, we analyze the statistics of those very large fluctuations where the flow is moving against a driving force. We discuss rare fluctuations in flows of soft particulate

matter, where the injected power, _p_ = d_w_/d_t ∝ σ_ _xy_  ⋅ _δv_ takes negative values in a finite domain that is subjected to a velocity gradient _δv_ and that resists flow by a shear

stress _σ_ _xy_ (the shear stress is the force resisting the flow, see Supplementary Notes 1, 2 and 3). In a steady state the injected energy balances the energy dissipated by the viscosity

of the fluid. Hence, on average _p_ takes a positive value, and in the thermodynamic limit it does not fluctuate. When there is a finite number of particles in the considered domain there is

a small chance to encounter rare fluctuations where _p_ takes a negative value. This can either be due to the reversal of the shear stress _σ_ _xy_ or to the velocity gradient _δv_. While

one might naively expect that fluctuations in _σ_ _xy_ and _δv_ would equally contribute to such violations, our numerical simulations show an unexpected interplay of these two mechanisms of

rare fluctuations. Moreover, we establish a variation of FR for the statistics of the injected power driving the flow and use it to define an effective temperature for far-from-equilibrium

soft particulate flows. Our approach can be easily generalized to study negative power fluctuations and effective temperatures both in simulations and experiments in a wide range of problems

such as in the sheared foams, vibrated granular media, particles down an inclined plane, emulsions and other soft particulate media. RESULTS PDF OF INJECTED POWER _P_ In the Fig. 1a, we

show a typical example of the PDF, \({\cal P}(p/{\bar p})\), of the local power flux rescaled by the mean power, i.e., \(p/{\bar p}\). The PDF exhibits several remarkable features. The power

flux can take negative values with a rather high probability. The distribution is strongly skewed towards positive events. At the both sides the PDF decays exponentially to a good

approximations. It is very different from a Gaussian distribution. Still, the negative part of the PDF (the _shaded area_ in Fig. 1a) decreases rapidly with system size. In the following

this area will be denoted as \(P(p < 0)\equiv {\int }_{-\infty }^{0}{\cal P}(p)\,{\rm d}p\). The slopes of the exponential decay are roughly proportional to the number of particles in the

considered volume such that _P_(_p_ < 0) decays exponentially to zero with system size. PROBABILITY OF RARE FLUCTUATIONS In Fig. 1b we show _P_(_p_ < 0) as a function of packing

fraction _ϕ_. The _lines_ in _different color_ indicate data for different shear rates, \({\dot \gamma }\). For shear rates, \({\dot \gamma }\gtrapprox 0.2\) (olive), this probability is

very small, and it grows upon decreasing the shear rate. There is pronounced growth in the fluid and in the jammed states. However, close to the jamming point _ϕ_ _J_ (marked by the

_vertical dashed line_) it remains small for all shear rates. The critical point lies to the right of the minima. However, the minima converge towards _ϕ_ _J_ in the limit \({\dot \gamma

}\to 0\) and system size _L_→∞. Hence, the minimum will eventually approach _ϕ_ _J_ ; the discrepancy is due to finite-size effects. The emergence of the global minimum is remarkable because

fluctuations are expected to diverge close to a critical point16,17,18, i.e., one expects a larger probability to encounter fluctuations close to the critical point. The minimum is a

distinctive feature of the jamming transition. It has no counterpart in equilibrium thermodynamics. DECOMPOSITION OF THE PROBABILITY OF RARE FLUCTUATIONS In order to understand the origins

of the negative power injection, we recall that the power flux has two contributions: The local shear stress _σ_ _xy_ , and the local velocity gradient _δv_. Negative power injection arises

whenever either _σ_ _xy_  < 0 and _δv_ > 0, or _δv_ < 0 and _σ_ _xy_  > 0. The former events will be denoted as \(({\sigma }_{xy}^{-},\delta {v}^{+})\). In this case the velocity

profile remains monotonic and negative power injection arises from fluctuations of the shear stress. The latter events will be denoted as \((\delta {v}^{-},{\sigma }_{xy}^{+})\). In this

case the negative power injection is connected to rare fluctuations where the velocity profile is no longer monotonic (see Supplementary Fig. 1). The joint probability of these events sum up

to the probability to encounter negative power injection: \(P(p < 0)=P({\sigma }_{xy}^{-},\delta {v}^{+})+P(\delta {v}^{-},{\sigma }_{xy}^{+})\) (In Supplementary Note 5 and

Supplementary Fig. 3 we numerically prove this equality). Figure 2 presents the numerical results for these two joint probabilities, i.e., \(P({\sigma }_{xy}^{-},\delta {v}^{+})\) (left

axis, _filled symbols_) and \(P(\delta {v}^{-},{\sigma }_{xy}^{+})\) (right axis, _hollow symbols_) as function of packing fraction _ϕ_ for various shear rates. For a given shear rate (a

given color), the intersection point of the joint probabilities is very close to the jamming point. Hence, _ϕ_ _J_ splits the probability space into two disjoint regions. Accordingly, the

two types of mechanisms, shear-stress and velocity-gradient fluctuations, are mutually exclusive. The reason can be sketched as follows. The positive shear stress corresponds to head-to-head

collisions while the negative shear stress is associated to backup collisions during which the local angular momentum is in the same and in the opposite direction of the global induced

angular momentum of the flow, respectively (see Supplementary Fig. 2 and Supplementary Note 4). For _ϕ_ < _ϕ_ _J_ , since the average coordination number is relatively small, the negative

contacts can be observed with high probability although they are suppressed by increasing the shearing flow. This justifies the decreasing dependence of the probability of negative stress

as function of both shear rate and packing fraction. For _ϕ_ > _ϕ_ _J_ , since the coordination number jumps to a value ≥4, the negative contacts can in average be suppressed by the

positive ones since the global symmetries of the flow favor the positive contacts. This elucidates the behavior seen in Fig. 2 for the negative shear stress and since the negative power

includes exclusive contributions from the negative shear stress and negative velocity gradient, our argument naturally explains the observation of the mutually exclusive fluctuations.

FLUCTUATION RELATION At this point we identified qualitatively different physics underlying the fluctuations of fluid and jammed systems. In order to gain more insight into the parameter

dependence of the strength of fluctuations we establish now a FR for the flows. It will characterize the width of the PDFs \({\cal P}(p)\) by an effective temperature _T_ _e_ . We conjecture

that the relation \(\rm{ln}\,[P(p)/P(-p)]=\beta p\) holds where _β_ = _τ_/_T_ _e_ has inverse dimension of power. Here, the constant _τ_ is the relevant elastic time scale which represents

the typical time scale of a single collision. It is approximately independent of _ϕ_ and \(\dot{\gamma }\) (see Supplementary Note 6). Figure 3 presents the numerical verification of our

conjecture for two packing fractions _ϕ_ = 0.7 and 0.9 that lie below and above the critical point, _ϕ_ _J_  = 0.84, respectively. There is a linear dependence between

\(\rm{ln}\,[P(p)/P(-p)]\) and _p_ whose slope is a decreasing function of the shear rate \(\dot{\gamma }\). This implies that _T_ _e_ is an increasing function of the shear rate, in

accordance with the shear-rate dependence of the average kinetic temperature of particles in the flow. Surprisingly, the correspondence is not only qualitative. It even holds quantitatively.

All data shown in Fig. 3a collapse to a straight line when \(\rm{ln}\,[P(p)/P(-p)]\) is multiplied by the granular temperature, _T_ _g_ (Fig. 3b). The resulting straight line has a slope 1

with _τ_ = 0.28. (In Supplementary Note 8 and Supplementary Fig. 8, we numerically prove that our proposed FR is also satisfied for highly damped systems corresponding to non-Brownian

suspensions.) SCALING COLLAPSE OF EFFECTIVE AND KINETIC TEMPERATURES In the jammed state the effective temperature, _T_ _e_ , and the granular temperature, _T_ _g_ , differ: _T_ _e_ is

always larger than _T_ _g_ . We explore the parameter dependence of the two temperatures by exploring their scaling properties19. In Fig. 4a we demonstrate that the full \(\dot{\gamma }\)

and _ϕ_ dependence of the temperatures for different system sizes can be represented in terms of a master plot where _T_/|_δϕ_|_y_ is plotted as function of \(\dot{\gamma }/|\delta \phi

{|}^{y/q}\) with appropriate choice of the exponents _y_, _q_ and _δϕ_ = _ϕ_−_ϕ_ _c_ . For the fluid state we thus find the well-known Bagnoldian scaling with exponent 2. In the jammed

state, we find that _T_ _e_ and _T_ _g_ still collapse uniformly in the critical region, \(\dot{\gamma }/|\delta \phi {|}^{y/q}\gtrapprox 10\). For jammed flows, however, they segregate into

two different branches in accord with our earlier statement. In this limit _T_ _e_ approaches a constant—yield stress emerges. In contrast, _T_ _g_ shows a power-law behavior with exponent

∼ 1.5(1). We have checked consistency of all our exponents, i.e., exponents of granular temperature and components of stress tensor, in Supplementary Note 7 and Supplementary Figs 4–7. We

also show in Supplementary Note 8 that these exponents are universal in a sense that the same scaling collapse is achieved with the same critical exponents for highly dissipative regime, in

connection with the non-Brownian suspensions (See Supplementary Fig. 9). Heussinger _et al_.20, 21 have studied fluctuations of some observables in the flow of an assembly of frictionless,

soft discs at zero temperature, in the vicinity of and slightly above _ϕ_ _J_ . They have found that the contact-number fluctuations and relative fluctuations of the shear stress diverge

upon approaching _ϕ_ _J_ from above. They also report on strong finite-size effects when using _ϕ_ as control parameter. However, the effective temperature _T_ _e_ in our study, is the

product of thermodynamically forbidden fluctuations (negative stress below _ϕ_ _J_ and negative velocity gradient above _ϕ_ _J_ ) which are specific to small-size systems and vanish in the

vicinity of _ϕ_ _J_ . Our observation indicates that the scaling behavior of _T_ _e_ is not significantly altered by the finite-size effects below and above _ϕ_ _J_ for _L_ > 10 (see Fig.

4a). We find that to a very good extent, _T_ _e_ is independent of the system size. We only see small deviations for _L_ = 10. These deviations vanish as \(\dot{\gamma }\to 0\). DISCUSSION

The flow of particulate matter is similar to classical fluids in so far as it involves the motion of many particles that interact by short-range forces. As function of packing fraction the

flows undergo a phase transition from a fluid into a jammed state. Close to the critical point the materials obey scaling relations22, 23, reminiscent of critical phenomena. The data

collapse of the granular temperature, i.e., the kinetic energy per degree of freedom, is shown here in Fig. 4a. In the fluid state and in the critical region this temperature agrees with an

effective temperature that characterizes the probability to encounter different power injections. The proposed effective temperature is sensitive to the inherent properties of the systems,

and it potentially qualifies as the effective temperature that has been searched for recently with great urgency24, 25. The effective temperatures proposed in the past26 are based on

fluctuation–dissipation relations, i.e., they assume linear response. Our study goes beyond linear response by introducing a FR in order to define a shear-rate-dependent effective

temperature. This effective temperature is valid for packing fractions far from the jamming point, in contrast to the previous ones that are meaningful measure only near the transition

point26. Various types of fluctuation theorems have been extensively studied over the last two decades13, 27,28,29. Motivated by molecular dynamics simulations, Evans _et al_.30 proposed an

empirical FR for entropy production rate in a two-dimensional sheared Lennard–Jones fluid. Later, this empirical relation was rigorously proved by Gallavotti and Cohen31, 32. This is now

known as steady-state fluctuation theorem (SSFT). In a SSFT, the entropy production rate is time averaged over a single, randomly sampled interval of duration _τ_. In contrast, the transient

fluctuation theorem (TFT) of Evans and Searles33 applies to a system that evolves from an initial equilibrium state to a nonequilibrium steady state. TFTs are different from SSFTs from a

practical point of view. Whereas TFRs rely on ensemble averaging all starting from the same initial macro-state, SSFRs may be verified from steady-state evolution of a system over a

sufficiently long time34. As we have already stressed out, properties of soft particulate flows are predominated by the rare fluctuations which result to intermittent behavior of these

flows. It can be seen that FRs of type SSFTs are not suitable for soft particulate flows. The reason is that as a consequence of averaging process during the sampling time, the rare

fluctuations can be washed out. Therefore, we use an instantaneous, time-independent FR to characterize strength of rare fluctuations. Whether our postulated FR would enjoy a rigorous

treatment, will remain a theoretical challenge. In addition, we have shown here that fluctuations in soft particulate flows differ essentially from those of classical fluids. First of all,

they are very strong as demonstrated by the exponential decay of the PDFs of negative power injection (Fig. 1a) rather than the much faster decay of Gaussian distributions. Even for shear

rates as large as \(\dot{\gamma }=0.2\) we observe negative power injection (lowermost curve in Fig. 1b). Even more surprising, rare fluctuations are strongly suppressed close to the

critical point (minima of the curves in Fig. 1b). They behave exactly contrary to the strength of critical fluctuations that diverge at the critical point and die out rapidly outside the

critical region16. It will be challenging numerically, but extremely interesting from a conceptual point of view to explore how classical fluids behave in this respect. Finally, we have

shown that there are different physical mechanisms underlying rare fluctuations in fluid and jammed states: In fluid states negative power injection originates from fluctuations where the

shear stress takes a negative sign, in jammed states they arise in regions with negative shear rates. This dichonometry of mutually exclusive mechanisms of rare fluctuations above and below

a critical state is a distinctive feature of soft particulate flows that has no counterpart in equilibrium thermodynamics. Rheological properties of particulate flows are commonly

characterized in terms of hydrodynamic equations and constitutive relations35, 36. Fluctuations are not a part of the modeling. The present study takes a different approach to characterize

the systems: We focus on the fluctuations as an inherent property of the dynamics. Remarkably, the fluctuations obey a local, time-independent FR, and this relation can be used to define an

effective temperature of the system. In contrast to the hydrodynamic approaches the temperature is not a field variable in this setting. Rather it characterizes the variability of snapshots

of the flow. It is a scalar quantity that characterizes the ensemble of observations of the flows, taking full note of fluctuations. It neither requires to find appropriate heuristic

constitutive equations, nor does it rely on the scale separation at mesoscopic scales that is implicit to the definition of thermodynamic fields. Hence, it is less prone to pitfalls arising

from inapt choices of constitutive relations and applicable to a larger class of far-from-equilibrium flows. Thus, the present study opens a qualitatively new road to the description of

far-from-equilibrium particulate flows. In a recent study, Maloney37 investigated distribution of dissipation power \({\cal{P}}(\Gamma )\) in Durian’s bubble model. Whereas we find that

\({\cal{P}}(p)\) has always exponential tails, it is shown that above jamming density \({\cal{P}}(\Gamma )\) becomes power law for small shear rates. This implies that distributions of

injection and dissipation powers might not be equivalent. Stationarity condition implies that first moments of these distributions must be equal. But as one can see, higher moments of these

distributions, which refer to the characteristic of the tails, might be different. This suggests a new avenue of research for investigation of steady state properties of non-equilibrium

states. In this context, Maloney’s work37 together with our approach to calculate \({\cal{P}}(p)\) in shear flows provide a solid framework for investigation of similarities and differences

between distributions of injection and dissipation powers. METHODS SIMULATION DETAILS We perform molecular-dynamics simulations of two-dimensional frictionless bidisperse disks. Particles

interact via short range repulsive and dissipative forces. Two particles _i_ and _j_ of radii \({R}_{i}^{a}\) and \({R}_{j}^{b}\) (where _a_, _b_ = 0, 1 stand for two different radius of

bidisperse particles) at positions R _i_ and R _j_ interact when \({\xi }_{ij}={R}_{i}^{a}+{R}_{j}^{b}-{r}_{ij} >0\). Here _ξ_ _ij_ is called the mutual compression of particles _i_ and

_j_, _r_ _ij_  = |R _i_ −R _j_ |. The particles interact via a linear Dashpot model, _F_ _ij_  = _Yξ_ _ij_ +_γ_ \({{\rm d}\xi_{ij}\over {\rm d}t}\), where _Y_ and _γ_ are denoted as elastic

and dissipative constant, respectively. Throughout the study we adopt the values _Y_ = 100 and _γ_ = 0.315, respectively. In order to prevent crystallization we use a 1:1 binary mixture of

particles where the ratio of the radii of large and small particles is set to _R_ 1/_R_ 0 = 1.4. The equations of motion are non-dimensionalized by choosing the unit of the length to be _R_

0+_R_ 1 = 1, and setting the mass of each particle equal to its area, _m_ _a_  = _π_[_R_ _a_]2. Finally, the ratio of _Y_ and _γ_ provides the time scale \({t}^{\star }=\gamma /Y=3.15\times

{10}^{-3}\). Lees–Edwards boundary conditions are applied along _x_-direction. They create a uniform overall shear rate, \(\dot{\gamma }\). These equations of motion are integrated with a

5th-order predictor-corrector Gear algorithm with time step, d_t_ = 10−4. DATA AVAILABILITY The authors declare that the data supporting the findings of this study are available from the

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Nagler and M. Schröter are highly acknowledged. S.H.E.R. thanks the Korea Institute for Advanced Study for providing computing resources (KIAS Center for Advanced Computation—Linux cluster

system) for this work, and specially consultations from Hoyoung Kim. A.A.S. would like to acknowledge supports from the Alexander von Humboldt Foundation, and partial financial supports by

the research council of the University of Tehran. AUTHOR INFORMATION Author notes * S.H.E. Rahbari and A.A. Saberi equally contributed to this work. AUTHORS AND AFFILIATIONS * School of

Physics, Korea Institute for Advanced Study, Seoul, 130-722, South Korea S.H.E. Rahbari & Hyunggyu Park * Department of Physics, College of Science, University of Tehran, 14395-547,

Tehran, Iran A.A. Saberi * School of Physics and Accelerators, Institute for research in Fundamental Science (IPM), 19395-5531, Tehran, Iran A.A. Saberi * Institut für Theoretische Physik,

Universitat zu Köln, Zülpicher Strasse 77, 50937, Köln, Germany A.A. Saberi * Max Planck Institute for Dynamics and Self-Organization (MPI DS), Göttingen, 37077, Germany J. Vollmer * Faculty

of Physics, Georg August University Göttingen, Göttingen, 37077, Germany J. Vollmer * Faculty of Mathematics and Computer Sciences, Georg August University Göttingen, Göttingen, 37073,

Germany J. Vollmer * Politecnico di Torino, Department of Mathematical Sciences, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy J. Vollmer Authors * S.H.E. Rahbari View author

publications You can also search for this author inPubMed Google Scholar * A.A. Saberi View author publications You can also search for this author inPubMed Google Scholar * Hyunggyu Park

View author publications You can also search for this author inPubMed Google Scholar * J. Vollmer View author publications You can also search for this author inPubMed Google Scholar

CONTRIBUTIONS S.H.E.R. performed the simulations. All authors discussed the data, participated in the data analysis, and in writing the manuscript. S.H.E.R. and A.A.S. had equal

contributions in the development of the study. CORRESPONDING AUTHORS Correspondence to S.H.E. Rahbari or A.A. Saberi. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing

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