Collisional statistics and dynamics of two-dimensional hard-disk systems: From fluid to solid

Alessandro Taloni, Yasmine Meroz, and Adrián Huerta

Phys. Rev. E 92, 022131 – Published 20 August 2015

Abstract

We perform extensive MD simulations of two-dimensional systems of hard disks, focusing on the collisional statistical properties. We analyze the distribution functions of velocity, free flight time, and free path length for packing fractions ranging from the fluid to the solid phase. The behaviors of the mean free flight time and path length between subsequent collisions are found to drastically change in the coexistence phase. We show that single-particle dynamical properties behave analogously in collisional and continuous-time representations, exhibiting apparent crossovers between the fluid and the solid phases. We find that, both in collisional and continuous-time representation, the mean-squared displacement, velocity autocorrelation functions, intermediate scattering functions, and self-part of the van Hove function (propagator) closely reproduce the same behavior exhibited by the corresponding quantities in granular media, colloids, and supercooled liquids close to the glass or jamming transition.

11 More

Received 26 March 2015

DOI:https://doi.org/10.1103/PhysRevE.92.022131

Authors & Affiliations

Alessandro Taloni^1,2,3, Yasmine Meroz⁴, and Adrián Huerta⁵

¹CNR-IENI, Via R. Cozzi 53, 20125 Milano, Italy
²Institute for Scientific Interchange (ISI), Via Alassio 11c, 10126 Turin, Italy
³Center for Complexity & Biosystems, Physics Department, University of Milan “La Statale,” Via Giovanni Celoria, 16, 20133 Milano, Italy
⁴School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
⁵Facultad de Física, Universidad Veracruzana, Circuito Gonzálo Aguirre Beltrán s/n Zona Universitaria, Xalapa, Veracruz 91000, México

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 92, Iss. 2 — August 2015

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Comparison of the velocity distribution in continuous-time and collisional representations. (a) Maxwell-Boltzman distribution function of single-component velocity for the continuous-time representation, $φ_{MB} (v)$ , plotted with a blue solid line (squares), and the collisional $φ_{coll} (v)$ in a red solid line (circles). Black dashed lines represent, respectively, the usual Maxwell-Boltzmann expression and the theoretical estimate obtained by numerical integration of Eq. (2) over the $y$ component. (b) Speed distribution function in collisional representation $f_{coll} (|v|)$ : Numerics are shown as a solid red line, while the theoretical expression $f_{coll} (|v|)$ is reported in dashed black line. $k_{B} T = 1, σ = 1, L_{x}$ , and $L_{y}$ are measured in units of $σ$ . The distribution in both panels are calculated at $η = 0.3$ , although the same results are obtained for any $η$ , as both $φ_{MB}$ and $φ_{coll}$ are constant throughout the dynamics.
Reuse & Permissions
Figure 2
Dependence of free flight times on velocity. (a) The probability distribution $p (τ | v)$ for different $η$ with $v = 0.0 \pm 0.05$ (arbitrary units). The exponential expression $\frac{e^{- t / 〈 τ (v) 〉}}{〈 τ (v) 〉}$ (dashed lines) successfully describes the numerical outcomes only for the lower value of $η$ , i.e., $η = 0.3, 0.56$ . (b) Average free flight time as a function of $v, 〈 τ (v) 〉$ , for several $η$ . The dashed lines are obtained from the numerical evaluation of Eq. (8).
Reuse & Permissions
Figure 3
Free flight time (a) and free path (b) probability distribution functions for several values of $η$ . The exponential expression $\frac{e^{- t / 〈 τ 〉}}{〈 τ 〉}$ [dashed lines in panel (a)] do not capture the numerical outcomes in the low as well as in the $high- η$ regime. Dashed lines corresponding to $η = 0.713$ and $η = 0.72$ are superimposed.
Reuse & Permissions
Figure 4
Average free flight time (a) and average free path (b) shown as a function of $η$ . It is seen that they both exhibit a plateau in correspondence of the coexistence phase $0.69 \leq η \leq 0.716$ [17] (blurred regions), while they decay monotonically in the fluid and solid phases. Insets: Blue curves show the kinetic theory predictions [Eqs. (B1) and (B3), respectively], while red curves refer to the Enskog theory Eqs. (B6) and (B5).
Reuse & Permissions
Figure 5
Dependence of free path length on velocity. (a) The probability distribution $p (| ξ | | v)$ for different $η$ for $v = 0.0 \pm 0.05$ (arbitrary units). The exponential expression $\frac{e^{- | ξ (v) | / 〈 | ξ (v) | 〉}}{〈 | ξ (v) | 〉}$ (dashed lines) describes the numerical outcomes only for the lower value of $η$ , i.e., $η = 0.3, 0.56$ . (b) $〈 | ξ | (v) 〉$ as a function of the velocity for several $η$ . The dashed lines are obtained by the relation $〈 | ξ (v) | 〉 = | v | 〈 τ (v) 〉$ .
Reuse & Permissions
Figure 6
Free-path-length probability distribution [appearing in Fig. 3] rescaled by (a) the kinetic theory expression [Eq. (B3)], (b) the average free path [Fig. 4], and (c) the free path furnished by the Enskog theory [Eq. (B5)]. None of them shows good collapse of the curves.
Reuse & Permissions
Figure 7
Normalized velocity ACF. (a) Continuous time. The velocity ACF exhibits persistent memory effects in the low- $η$ limit and in the high- $η$ region. The predicted $1 / t$ behavior is evident in the inset, while the antipersistent tails characterize the caging effect for high $η$ . (b) Collisional representation. The velocity appears to be non-Markovian also in the collisional representation, showing a surprisingly analogy with the continuous-time counterpart. In the inset the $1 / n$ behavior appears to define the $low- η$ systems, while the antipersistent memory effects dominate the $high- η$ regime.
Reuse & Permissions
Figure 8
(a) MSD in continuous time for several $η$ . The solid lines refer to the Einstein-Helfand formula (15) while the dashed ones are obtained by the Green-Kubo expression (16). They both coincide yielding, after an initial ballistic regime, normal diffusion for low $η$ and an intermediate slowdown (plateau) for $η$ in the coexistence phase and beyond. (b) Time-dependent transport coefficient defined as $D (t) = \int_{0}^{t} d s 〈 v (0) v (s) 〉$ . For the lowest analyzed $η, η = 0.3, D (t)$ increases as $\sim \sqrt{ln t}$ consistent with the predicted $〈 v (0) v (t) 〉 \sim {(t \sqrt{ln t})}^{- 1}$ [49, 50, 51]. For the higher $η$ shown, $D (t)$ displays a “subdiffusive” decay in correspondence of the caging regime in panel (a) (see Ref. [91]).
Reuse & Permissions
Figure 9
(a) MSD in collisional representation for several $η$ . The solid lines refer to the Einstein-Hellfand formula (17) while the dashed ones are obtained by the Green-Kubo expression (19). They both coincide, displaying normal diffusion for low $η$ and an intermediate slowdown (plateau) for $η$ in the coexistence phase and beyond, in analogy with the results shown in Fig. 8. (b) Transport coefficient defined as $D_{n} = \sum_{m = 0}^{n - 1} C_{ξ ξ} (m)$ . $D_{n}$ exhibits a persistent increasing behavior consistent with $\sim \sqrt{n ln n}$ for $η = 0.3$ . This is in agreement with the continuous-time behavior displayed in Fig. 8 and with the power-law tails in Fig. 10. For the higher $η, D_{n}$ shows the same “subdiffusive” decay of $D (t)$ [Fig. 8], in correspondence of the caging regime in panel (b) (see the detailed analysis in Appendix pp3).
Reuse & Permissions
Figure 10
Normalized free path length ACF. The non-Markovianity of the free path length is apparent both at low and high $η$ . For low $η$ (inset) the $1 / n$ behavior is clearly shown, consistent with the velocity ACF in Fig. 7. The systems with high $η$ are displayed in the main panel, exhibiting a negative part, i.e., antipersistence due to the caging phenomenon.
Reuse & Permissions
Figure 11
Normalized free-flight-time ACF, $C_{τ τ} (n) = 〈 τ_{0} τ_{n} 〉 - {〈 τ 〉}^{2}$ . The non-Markovianity of the free-flight-time process $τ_{n}$ characterizes both low and high $η$ .
Reuse & Permissions
Figure 12
Poisson approximated expression for the MSD. (a) MSD in collisional representation for several $η$ calculated according to the Poisson approximation in Eq. (23) (dashed lines). The solid lines refer to the Green-Kubo expression from Eq. (19), coinciding with the Poissonian approximation only in the $low- η$ limit. (b) MSD in continuous-time representation using the first two terms of the Poisson formula appearing in Eq. (40) (dashed lines), and the Green-Kubo expression in Eq. (16) (solid lines). Here, too, the $low- η$ systems appear to be well captured by the approximated formula. The $high- η$ limit requires a complete determination of the correlations appearing in Eq. (39).
Reuse & Permissions
Figure 13
Normalized cross-correlation of free flight time and free path length.
Reuse & Permissions
Figure 14
Intermediate scattering function. (a) Continuous time $F_{S} (q, t)$ for several $η$ calculated according to Eq. (26). The passage from the exponential decay to the stretched exponential regime, particular to caging in granular and colloidal systems is evident. (b) Collisional representation $F_{S} (q, n)$ , Eq. (27). The exhibited trend traces that of panel (a).
Reuse & Permissions
Figure 15
Self-part of the van Hove function. (a) Continuous-time $G_{S} (x, t)$ for $η = 0.3$ , calculated according to Eq. (28), for two different times $t$ (solid lines). The dashed lines represent the Gaussian form $\frac{1}{\sqrt{2 π 〈 δ x^{2} (t) 〉}} exp (- x^{2} / 2 〈 δ x^{2} (t) 〉)$ , where the values of $〈 δ x^{2} (t) 〉$ have been drawn from Fig. 8. (b) Collisional representation $G_{S} (x, n)$ for $η = 0.3$ , calculated according to Eq. (29), for two different values of $n$ (solid lines). The dashed lines represent the Gaussian form $\frac{1}{\sqrt{2 π 〈 δ x_{n}^{2} 〉}} exp (- x^{2} / 2 〈 δ x_{n}^{2} 〉)$ , where the values of $〈 δ x_{n}^{2} 〉$ have been drawn from Fig. 9.
Reuse & Permissions
Figure 16
Self-part of the van Hove function. (a) Continuous-time $G_{S} (x, t)$ for $η = 0.695$ , calculated according to Eq. (28), for two different times $t$ (solid lines). The dashed lines represent the Gaussian form $\frac{1}{\sqrt{2 π 〈 δ x^{2} (t) 〉}} exp (- x^{2} / 2 〈 δ x^{2} (t) 〉)$ , where the values of $〈 δ x^{2} (t) 〉$ have been drawn from Fig. 8. Small but persistent deviations from the Gaussian expression in the tails of the distributions are displayed at both times. (b) Collisional representation $G_{S} (x, n)$ for $η = 0.3$ , calculated according to Eq. (29), for two different values of $n$ (solid lines). The dashed lines represent the Gaussian form $\frac{1}{\sqrt{2 π 〈 δ x_{n}^{2} 〉}} exp (- x^{2} / 2 〈 δ x_{n}^{2} 〉)$ , where the values of $〈 δ x_{n}^{2} 〉$ have been drawn from Fig. 9. Deviations similar to the ontinuous time case appear.
Reuse & Permissions
Figure 17
Self-part of the van Hove function. (a) Continuous-time $G_{S} (x, t)$ for $η = 0.713$ , calculated according to Eq. (28), for two different times $t$ (solid lines). The dashed lines represent the Gaussian form $\frac{1}{\sqrt{2 π 〈 δ x^{2} (t) 〉}} exp (- x^{2} / 2 〈 δ x^{2} (t) 〉)$ , where the values of $〈 δ x^{2} (t) 〉$ have been drawn from Fig. 8. Deviations from the Gaussian form are clearly displayed: In particular, for the longer time corresponding to the asymptotic linear regime in Fig. 8, non-Gaussian tails are reminiscent of what is found in granular gases and colloids. (b) Collisional representation $G_{S} (x, n)$ for $η = 0.3$ , calculated according to Eq. (29), for two different values of $n$ (solid lines). The dashed lines represent the Gaussian form $\frac{1}{\sqrt{2 π 〈 δ x_{n}^{2} 〉}} exp (- x^{2} / 2 〈 δ x_{n}^{2} 〉)$ , where the values of $〈 δ x_{n}^{2} 〉$ have been drawn from Fig. 9. Deviations from the Gaussian diffusive form appear. Moreover, non-Gaussian tails similar to those in panel (a) characterize the free path dynamics for $n$ well beyond the plateau in Fig. 9. We note that oscillations present in the continuous, representing spatial order due to high packing fractions, are not apparent in the collisional representation, due to the decoupling of space and time.
Reuse & Permissions
Figure 18
Average time after $n$ collisions, $〈 t_{n} 〉$ , for several $η η$ (solid lines). Dashed lines correspond to the Poissonian approximation in Eq. (37).
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics