Characterizing rare fluctuations in soft particulate flows

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.Soft particulate flows such as granular media are prone to fluctuations like jamming and avalanches. Here Rahbari et al. consider the statistics of rare fluctuations to identify an effective temperature which, unlike previous ones, is valid for packing fractions both near and far from the jamming point.

Fig. 1

Probability of rare fluctuations. a A typical PDF of the rescaled power $p∕\overline{p}$ for ϕ = 0.7, $\stackrel{°}{\gamma }=0.01$, and L = 30. The solid straight lines show the exponential decay of the PDF for large and small arguments. The area of the shaded region gives the probability, P(p < 0), that the local power takes a negative value, i.e., to encounter a negative power injection. b The probability to observe a negative power injection P(p < 0) as a function of packing fraction ϕ for different shear rates $\stackrel{°}{\gamma }=0.005,0.01,0.02,0.04,0.06,0.08,0.1$ and 0.2 from top to bottom, respectively, and system size L = 30. The vertical dashed line marks the critical packing fraction, ϕ _J, where the jamming transition occurs in the static limit. Error bars correspond to square root of variance

Fig. 2

Mutually exclusive fluctuations. Joint probabilities $P\left({\sigma }_{xy}^{-},\delta v^{+}\right)$ (left axis, filled symbols) and $P\left(\delta v^{-},{\sigma }_{xy}^{+}\right)$ (right axis, hollow symbols) as functions of packing fraction, ϕ, for various shear rates, $\stackrel{°}{\gamma }$. In the fluid state, ϕ < ϕ _J, the dominant mechanism of negative power injection is the reversal of the shear stress. In the jammed state, ϕ > ϕ _J, it is due to the reversion of the velocity gradient. Error bars correspond to square root of variance

Fig. 3

Verification of the instantaneous FR. Plot of $\ln \left[P\left(p\right)∕P\left(-p\right)\right]$ vs. p for two packing fractions a ϕ = 0.7 and, b ϕ = 0.9. The solid lines are linear fits of slope β _e = τ/T _e of the data for different shear rates. The slope decreases by increasing the shear rate $\stackrel{°}{\gamma }$, implying that the effective temperature T _e increases by $\stackrel{°}{\gamma }$. The slope has a weak dependence on $\stackrel{°}{\gamma }$ in the jammed state. For n ⁺ and n ⁻ representing number of positive (+p) and negative (−p) cases, the corresponding error bar of P(p)/P(−p) is equal to (1/n ⁺+1/n ⁻)^1/2

Fig. 4

Scaling of effective and kinetic temperatures. a When rescaled with the critical exponents q = 1.5(1) and y = 1.44(15) the effective and granular temperatures collapse onto a scaling function. However, to achieve a data collapse we had to adopt slightly different critical densities, ϕ _c = 0.83 and ϕ _c = ϕ _J = 0.84 for T _e and T _g, respectively. In the fluid state and in the critical state the temperatures match. For the fluid state they exhibit Bagnoldian scaling with exponent 2. In the critical state they still share same non-trivial scaling for $\stackrel{°}{\gamma }∕\delta {\varphi }^{y∕q}\succsim 10$. In the jammed state the temperatures segregate into two different branches; T _e approaches a constant and T _g follows a power-law behavior with exponent 1.5(1). Different system sizes are given by different symbols in which filled and hollow symbols refer to T _g and T _e, respectively. The color code corresponds to different shear rates $\stackrel{°}{\gamma }=0.02$ (purple), 0.04 (magenta), 0.06 (blue), 0.08 (golden), and 0.1 (yellow). b The collapse of all data presented in Fig. 4a when the vertical axis is multiplied by a factor of T _g/τ with τ = 0.28. In these data, we cover a large range of packing fractions around jamming, 0.7 < ϕ < 0.9. Error bars correspond to square root of variance