Enhanced Long Wavelength Mermin-Wagner Fluctuations in Active Crystals and Glasses

Abstract

In two-dimensions (2D), the Mermin-Wagner-Hohenberg (MWH) fluctuation plays a significant role, giving rise to striking dimensionality effects marked by long-range density fluctuations leading to the singularities of various dynamical properties. According to the MWH theorem, a 2D equilibrium system with continuous degrees of freedom cannot achieve long-range crystalline order at non-zero temperatures. Recently, MWH fluctuations have been observed in glass-forming liquids, evidenced by the logarithmic divergence in the plateau value of mean squared displacement (MSD). Our research investigates long-wavelength fluctuations in crystalline and glassy systems influenced by non-equilibrium active noises. Active systems serve as a minimal model for understanding diverse non-equilibrium dynamics, such as those in biological systems and self-propelled colloids. We demonstrate that fluctuations from active forces can strongly couple with long-wavelength density fluctuations, altering the lower critical dimension ($d_l$) from $2$ to $3$ and leading to a novel logarithmic divergence of the MSD plateau with system size in 3D.

Figures (9)

• Figure 1: Mean Square Displacement and Mermin-Wagner-Hohenberg Fluctuations. (a) Mean square displacement (MSD) as a function of time for activity $f_0=2.0$ at temperature T=0.221 with system size ranging from $10^3$ to $10^5$, here it shows increase in MSD plateau much faster than the passive system (Inset) at temperature T=0.430. (b) Two point density correlation Q(t) shows faster relaxation for the case of active system than the passive system (Inset) for large system size. (c) MSD($\tau$) plateau diverges faster than log(L) increasing activity. Inset: At larger activity it starts to power-law behavior unlike log(L) behavior of its passive counter-part. For $f_0=2.0$ the power exponent is $\zeta=0.86$. (a), (b) & (c) are for 2dmKA system. (d) MSD as function of time for active polycrystalline system for $f_0=2.0$ at temperature T=0.01 for system ranging from $10^2$ to $10^4$, it shows faster increase in MSD plateau than passive system (Inset) at same temperature similar to (a). (e) For active polycrystal we can get similar power-law of MSD plateau divergence, here for activity $f_0=2.0$ the exponent is $\zeta=1.65$. (f) Diffusivity as a function of relaxation time shows a power-law exponent $\kappa=1.35$. this breakdown of Stokes-Einstein relation ($\kappa>1.0$) is possible due to the presence of long wavelength phonon fluctuation in 2D, which is also valid for active system as well. We again get back the Stokes-Einstein relation with $\kappa\simeq1.0$ when we do cage-relative diffusivity and relaxation time calculations, again showing the presence of phonon like excitations in active liquids as well. Error bars in the figure panels are measured by computing the standard deviation (SD) of fluctuations in various statistically independent simulations.
• Figure 2: Cage-relative Correlation Function. Cage-relative MSD for (a) passive and (b) active system. CR-MSD plateau plotted in log-log (c) & lin-log (d) clearly show no changes with increasing system size. This can be considered as a proof that the increase in MSD plateau is due to long wavelength modes in the system, which is absent in the cage-relative measurements. Error bars in the figure panels are measured by computing the standard deviation (SD) of fluctuations in various statistically independent simulations.
• Figure 3: Giant Number fluctuation (GNF): (a) In equilibrium, the fluctuations of the number of particles, $\Delta N$ and average number of particles, $\left<N\right>$ in a given sub-volume of linear size $l$ ($<L$) is related to each other as $\Delta N \propto \left<N\right>^{\delta}$ with $\delta=0.5$ in accordance with the central limit theorem. Interestingly, in disordered system this exponent $\delta$ tends to vary from $0.3$ to $0.46$ with increasing activity in 2dmKA model. (b) GNF for the 3DKA in the densed limit with changing activity, which shows the exponent is limiting towards the passive normal liquid case of $0.5$. (c) Shows the variation of $\delta$ from $0.65$ to $0.55$ with increasing activity in 2dR10 model with only repulsive inter-particle interactions at $\rho = 0.5$. The curves are shifted by a scale factor of ($1.5$) with respect to each other for better readability. (d) Shows that GNFs are absent in high density even with activity (see text for detailed discussion). (e) Shows for 2dR10 model, one sees the exponent $\delta$ to increase beyond $0.5$ with decreasing density indicating some presence of GNFs. But at very high density and low temperature the exponent ($\delta$) reaches $0.3$ for a disordered glassy system similar to 2dmKA model. Error bars in the figure panels are measured by computing the standard deviation (SD) of fluctuations in various statistically independent simulations.
• Figure 4: Effective Vibrational Density of States (vDoS) and Phonons: (a) vDoS computed for passive systems for $N = 1000$ particles using exact diagonalization of the Hessian matrix (black circle), averaged displacement-displacement correlation matrix (orange square) and corrected via random matrix procedure (red diamond) (see text for details). The close agreement between these measurements suggests that the displacement-displacement correlation matrix method with random matrix correction is a robust method for the computation of vDoS. (b & c) vDoS is computed using the displacement correlation matrix method for active systems with increasing activity. The clear appearance of small frequency ($\omega$) peaks in the vDoS with increasing activity signals the increasing dominance of phonon-like modes. The increasing weight of vDoS at small $\omega$ also indicates a jamming to unjamming scenario. (d) & (e) vDoS computed for passive and active polycrystalline samples, respectively. (f) The measured MSD was compared with computed MSD from the correlation matrix. The excellent agreement suggests the validity of the effective dynamical matrix description of these active systems even at a significant degree of activity. (g) shows the same comparison for amorphous solids. Inset (f) and (g) highlight the importance of small $\omega$ modes in determining the plateau value of the MSD for all activities. (h) and (i) Comparison of plateau values vs system size, $L$, as obtained from MD simulations and from effective dynamical matrix description. This proves that dramatic Mermin-Wagner-Hohenberg (MWH) fluctuations in the active matter are due to the phonons, and thus, deviation of MWH theorem has to come from the details of the phonon dispersion relation. (f) & (h) is for polycrystalline system, and (g) & (i) is for amorphous solid. Error bars in the figure panels are measured by computing the standard deviation (SD) of fluctuations in various statistically independent simulations.
• Figure 5: Effective phonon dispersion: (a) Heat map of the $\omega$ vs $q$ for longitudinal spectrum of the polycrystalline samples with activity $f_0=1.0$. (b) shows the peak of heat map giving us the phonon dispersion relation of $\omega(q)$. The linear phonon dispersion relation for passive system is very clear and increasing non-linearity in active systems is also very evident, where for activity $f_0=1.0$ exponent is $\alpha \simeq 1.8$. Inset: log-log plot of the same. (c) the transverse spectrum of the polycrystalline samples for activity $f_0=1.0$ shows the dispersion exponent $\alpha \simeq 1.74$. (d) we show the heat map of the $\omega$ vs $q$ for longitudinal spectrum of the amorphous solid samples with activity f=1.0. (e) for amorphous solids samples with varying degree of activities. The spectrum is obtained for the longitudinal phonons and the inset shows the results in log-log plot. The exponent of the power law relation for activity $f_0=1.0$, $\alpha \simeq 1.47$. (f) shows the similar results but for transverse phonons with activity $f_0=1.0$ the exponent value is $\alpha \simeq 2.0$. Error bars in the figure panels are measured by computing the standard deviation (SD) of fluctuations in various statistically independent simulations.