Universal wing- and fin-beat frequency scaling.

We derive an equation that applies for the wing-beat frequency of flying animals and to the fin-stroke frequency of diving animals like penguins and whales. The equation states that the wing/fin-beat frequency is proportional to the square root of the animal's mass divided by the wing area. Data for birds, insects, bats, and even a robotic bird-supplemented by data for whales and penguins that must swim to stay submerged-show that the constant of proportionality is to a good approximation the same across all species; thus the equation is universal. The wing/fin-beat frequency equation is derived by dimensional analysis, which is a standard method of reasoning in physics. We finally demonstrate that a mathematically even simpler expression without the animal mass does not apply.

Solid-that-Flows Picture of Glass-Forming Liquids.

This perspective article reviews arguments that glass-forming liquids are different from those of standard liquid-state theory, which typically have a viscosity in the mPa·s range and relaxation times on the order of picoseconds. These numbers grow dramatically and become 1012 - 1015 times larger for liquids cooled toward the glass transition. This translates into a qualitative difference, and below the "solidity length" which is roughly one micron at the glass transition, a glass-forming liquid behaves much like a solid. Recent numerical evidence for the solidity of ultraviscous liquids is reviewed, and experimental consequences are discussed in relation to dynamic heterogeneity, frequency-dependent linear-response functions, and the temperature dependence of the average relaxation time.

Active-parameter polydispersity in the 2d ABP Yukawa model.

In experiments and simulations of passive as well as active matter the most commonly studied kind of parameter polydispersity is that of varying particles size. This paper investigates by simulations the effects of introducing polydispersity in other parameters for two-dimensional active Brownian particles with Yukawa pair interactions. Polydispersity is studied separately in the translational and rotational diffusion coefficients, as well as in the swim velocityv0. Uniform and binary parameter distributions are considered in the homogeneous and the motility-induced phase-separation (MIPS) phases. We find only minute changes in structure and dynamics upon the introduction of parameter polydispersity, even for situations involving 50% polydispersity. The reason for this is not clear. An exception is the case ofv0polydispersity for which the average radial distribution function with changing polydispersity shows significant variations in the MIPS phase. Even in this case, however, the dynamics is only modestly affected. As a possible application of our findings, we suggest that a temporary introduction of polydispersity into a single-component active-matter model characterized by a very long equilibration time, i.e. a glass-forming active system, may be used to equilibrate the system efficiently by particle swaps.

Active-matter isomorphs in the size-polydisperse Ornstein-Uhlenbeck Lennard-Jones model.

This paper studies size-polydisperse Lennard-Jones systems described by active Ornstein-Uhlenbeck particle (AOUP) dynamics. The focus is on the existence of isomorphs (curves of invariant structure and dynamics) in the model's three-dimensional phase diagram. Isomorphs are traced out from a single steady-state configuration by means of the configurational-temperature method. Good isomorph invariance of the reduced-unit radial distribution function and the mean-square displacement as a function of time is demonstrated for three uniform-distribution polydispersities,12%, 23%, and 29%. Comparing to active-matter isomorphs generated by the analytical direct-isomorph-check method, the latter have poorer invariance of the structure, but better invariance of the dynamics. We conclude that both methods can be used to quickly get an overview of the phase diagram of polydisperse AOUP models involving a potential-energy function obeying the hidden-scale-invariance property required for isomorph theory to apply.

Hidden scale invariance in the Gay-Berne model. II. Smectic-B phase.

This paper complements a previous study of the isotropic and nematic phases of the Gay-Berne liquid-crystal model [Mehri et al., Phys. Rev. E 105, 064703 (2022)2470-004510.1103/PhysRevE.105.064703] with a study of its smectic-B phase found at high density and low temperatures. We find also in this phase strong correlations between the virial and potential-energy thermal fluctuations, reflecting hidden scale invariance and implying the existence of isomorphs. The predicted approximate isomorph invariance of the physics is confirmed by simulations of the standard and orientational radial distribution functions, the mean-square displacement as a function of time, and the force, torque, velocity, angular velocity, and orientational time-autocorrelation functions. The regions of the Gay-Berne model that are relevant for liquid-crystal experiments can thus fully be simplified via the isomorph theory.

Scaling Properties of Liquid Dynamics Predicted from a Single Configuration: Small Rigid Molecules.

Isomorphs are curves in the thermodynamic phase diagram along which structure and dynamics are invariant to a good approximation. There are two main ways to trace out isomorphs, the configurational-adiabat method and the direct-isomorph-check method. Recently a new method based on the scaling properties of forces was introduced and shown to work very well for atomic systems [T. B. Schrøder, Phys. Rev. Lett. 2022, 129, 245501]. A unique feature of this method is that it only requires a single equilibrium configuration for tracing out an isomorph. We here test generalizations of this method to molecular systems and compare to simulations of three simple molecular models: the asymmetric dumbbell model of two Lennard-Jones spheres, the symmetric inverse-power-law dumbbell model, and the Lewis-Wahnström o-terphenyl model. We introduce and test two force-based and one torque-based methods, all of which require just a single configuration for tracing out an isomorph. Overall, the method based on requiring invariant center-of-mass reduced forces works best.

Comparing zero-parameter theories for the WCA and harmonic-repulsive melting lines.

The melting line of the Weeks-Chandler-Andersen (WCA) system was recently determined accurately and compared to the predictions of four analytical hard-sphere approximations [Attia et al., J. Chem. Phys. 157, 034502 (2022)]. Here, we study an alternative zero-parameter prediction based on the isomorph theory, the input of which are properties at a single reference state point on the melting line. The two central assumptions made are that the harmonic-repulsive potential approximates the WCA potential and that pair collisions are uncorrelated. The new approach gives excellent predictions at high temperatures, while the hard-sphere-theory based predictions are better at lower temperatures. Supplementing the WCA investigation, the face-centered-crystal to fluid coexistence line is determined for a system of harmonic-repulsive particles and compared to the zero-parameter theories. The results indicate that the excellent isomorph-theory predictions for the WCA potential at higher temperatures may be partly due to a cancellation of errors between the two above-mentioned assumptions.

Configurational temperature in active matter. II. Quantifying the deviation from thermal equilibrium.

This paper proposes using the configurational temperature T_{conf} for quantifying how far an active-matter system is from thermal equilibrium. We measure this "distance" by the ratio of the systemic temperature T_{s} to T_{conf}, where T_{s} is the canonical-ensemble temperature for which the average potential energy is equal to that of the active-matter system. T_{conf} is "local" in the sense that it is the average of a function, which depends only on how the potential energy varies in the vicinity of a given configuration. In contrast, T_{s} is a global quantity. The quantity T_{s}/T_{conf} is straightforward to evaluate in a computer simulation; equilibrium simulations in conjunction with a single steady-state active-matter configuration are enough to determine T_{s}/T_{conf}. We validate the suggestion that T_{s}/T_{conf} quantifies the deviation from thermal equilibrium by data for the radial distribution function of the 3D Kob-Andersen and 2D Yukawa active-matter models with active Ornstein-Uhlenbeck and active Brownian Particle dynamics. Moreover, we show that T_{s}/T_{conf}, structure, and dynamics of the homogeneous phase are all approximately invariant along the motility-induced phase separation boundary in the phase diagram of the 2D Yukawa model. The measure T_{s}/T_{conf} is not limited to active matter and can be used for quantifying how far any system involving a potential-energy function, e.g., a driven Hamiltonian system, is from thermal equilibrium.

Configurational temperature in active matter. I. Lines of invariant physics in the phase diagram of the Ornstein-Uhlenbeck model.

This paper shows that the configurational temperature of liquid-state theory, T_{conf}, defines an energy scale, which can be used for adjusting model parameters of active Ornstein-Uhlenbeck particle (AOUP) models in order to achieve approximately invariant structure and dynamics upon a density change. The required parameter changes are calculated from the variation of a single configuration's T_{conf} for a uniform scaling of all particle coordinates. The resulting equations are justified theoretically for models involving a potential-energy function with hidden scale invariance. The validity of the procedure is illustrated by computer simulations of the Kob-Andersen binary Lennard-Jones AOUP model, showing the existence of lines of approximate invariance of the reduced-unit radial distribution function and time-dependent mean-square displacement.

Even Strong Energy Polydispersity Does Not Affect the Average Structure and Dynamics of Simple Liquids.

Size-polydisperse liquids have become standard models for avoiding crystallization, thereby enabling studies of supercooled liquids and glasses formed, e.g., by colloidal systems. Purely energy-polydisperse liquids have been studied much less, but provide an interesting alternative. We here study numerically the difference in structure and dynamics obtained by introducing these two kinds of polydispersity into systems of particles interacting via the Lennard-Jones and EXP pair potentials. To a very good approximation, the average pair structure and dynamics are unchanged even for strong energy polydispersity, which is not the case for size-polydisperse systems. When the system at extreme energy polydispersity undergoes a continuous phase separation into lower and higher particle-energy regions whose structure and dynamics are different from the average, the average structure and dynamics are still virtually the same as for the monodisperse system. Our findings are consistent with the fact that the distribution of forces on the individual particles do not change when energy polydispersity is introduced, while they do change in the case of size polydispersity. A theoretical explanation remains to be found, however.

Distance-as-time in physical aging.

Although it has been known for half a century that the physical aging of glasses in experiments is described well by a linear thermal-history convolution integral over the so-called material time, the microscopic definition and interpretation of the material time remains a mystery. We propose that the material-time increase over a given time interval reflects the distance traveled by the system's particles. Different possible distance measures are discussed, starting from the standard mean-square displacement and its inherent-state version that excludes the vibrational contribution. The viewpoint adopted, which is inspired by and closely related to pioneering works of Cugliandolo and Kurchan from the 1990s, implies a "geometric reversibility" and a "unique-triangle property" characterizing the system's path in configuration space during aging. Both of these properties are inherited from equilibrium, and they are here confirmed by computer simulations of an aging binary Lennard-Jones system. Our simulations moreover show that the slow particles control the material time. This motivates a "dynamic-rigidity-percolation" picture of physical aging. The numerical data show that the material time is dominated by the slowest particles' inherent mean-square displacement, which is conveniently quantified by the inherent harmonic mean-square displacement. This distance measure collapses data for potential-energy aging well in the sense that the normalized relaxation functions following different temperature jumps are almost the same function of the material time. Finally, the standard Tool-Narayanaswamy linear material-time convolution-integral description of physical aging is derived from the assumption that when time is replaced by distance in the above sense, an aging system is described by the same expression as that of linear-response theory.

Hidden scale invariance in the Gay-Berne model.

This paper presents a numerical study of the Gay-Berne liquid crystal model with parameters corresponding to calamitic (rod-shaped) molecules. The focus is on the isotropic and nematic phases at temperatures above unity, where we find strong correlations between the virial and potential-energy thermal fluctuations, reflecting the hidden scale invariance symmetry. This implies the existence of isomorphs, which are curves in the thermodynamic phase diagram of approximately invariant physics. We study numerically one isomorph in the isotropic phase and one in the nematic phase. In both cases, good invariance of the dynamics is demonstrated via data for the mean-square displacement and the reduced-unit time-autocorrelation functions of the velocity, angular velocity, force, torque, and first- and second-order Legendre polynomial orientational order parameters. Deviations from isomorph invariance are observed at short times for the orientational time-autocorrelation functions, which reflects the fact that the moment of inertia is assumed to be constant and thus not isomorph-invariant in reduced units. Structural isomorph invariance is demonstrated from data for the radial distribution functions of the molecules and their orientations. For comparison, all quantities were also simulated along an isochore of similar temperature variation, in which case invariance is not observed. We conclude that the thermodynamic phase diagram of the calamitic Gay-Berne model is essentially one-dimensional in the studied regions as predicted by isomorph theory, a fact that potentially allows for simplifications of future theories and numerical studies.

Comparing four hard-sphere approximations for the low-temperature WCA melting line.

By combining interface-pinning simulations with numerical integration of the Clausius-Clapeyron equation, we accurately determine the melting-line coexistence pressure and fluid/crystal densities of the Weeks-Chandler-Andersen system, covering four decades of temperature. The data are used for comparing the melting-line predictions of the Boltzmann, Andersen-Weeks-Chandler, Barker-Henderson, and Stillinger hard-sphere approximations. The Andersen-Weeks-Chandler and Barker-Henderson theories give the most accurate predictions, and they both work excellently in the zero-temperature limit for which analytical expressions are derived here.

Rheological model for the alpha relaxation of glass-forming liquids and its comparison to data for DC704 and DC705.

Dynamic shear-modulus data are presented for two silicone oils DC704 and DC705 for frequencies between 1 mHz and 10 kHz at temperatures covering more than five decades of relaxation-time variation. Data are fitted to the alpha part of a phenomenological model previously shown to describe well the dynamic shear modulus of squalane, which has a large beta process [Hecksher et al., J. Chem. Phys. 146, 154504 (2017)]. That model is characterized by additivity of the alpha and beta shear compliance and by a high-frequency decay of the alpha process in proportion to ω-1/2, where ω is the angular frequency. The fits of the alpha part of this model to the DC704 and DC705 data are compared to fits by a Havriliak-Negami type model, a Barlow-Erginsav-Lamb model, and a Cole-Davidson type model. At all temperatures, the best fit is obtained by the alpha part of the squalane model. This strengthens the conjecture that so-called t-relaxation, leading to high-frequency loss decays proportional to ω-1/2, is generic to the alpha relaxation of supercooled liquids [J. C. Dyre, Phys. Rev. E 74, 021502 (2006); Nielsen et al., J. Chem. Phys. 130, 154508 (2009); and Pabst et al., J. Phys. Chem. Lett. 12, 3685-3690 (2021)].

Predicting nonlinear physical aging of glasses from equilibrium relaxation via the material time.

The noncrystalline glassy state of matter plays a role in virtually all fields of materials science and offers complementary properties to those of the crystalline counterpart. The caveat of the glassy state is that it is out of equilibrium and therefore exhibits physical aging, i.e., material properties change over time. For half a century, the physical aging of glasses has been known to be described well by the material-time concept, although the existence of a material time has never been directly validated. We do this here by successfully predicting the aging of the molecular glass 4-vinyl-1,3-dioxolan-2-one from its linear relaxation behavior. This establishes the defining property of the material time. Via the fluctuation-dissipation theorem, our results imply that physical aging can be predicted from thermal-equilibrium fluctuation data, which is confirmed by computer simulations of a binary liquid mixture.

Generalized hydrodynamics of the Lennard-Jones liquid in view of hidden scale invariance.

In recent years lines along which structure and dynamics are invariant to a good approximation, so-called isomorphs, have been identified in the thermodynamic phase diagrams of several model liquids and solids. This paper reports computer simulation data of the transverse and longitudinal collective dynamics at different length scales along an isomorph of the Lennard-Jones system. Our findings are compared to corresponding results along an isotherm and an isochore. Confirming the theoretical prediction, the reduced-unit dynamics of the transverse momentum density is invariant to a good approximation along the isomorph on all time and length scales. Likewise, the wave-vector dependent shear-stress autocorrelation function is found to be isomorph invariant (with minor deviations at very short times). A similar invariance is not seen along the isotherm or the isochore. Using a spatially nonlocal hydrodynamic model for the transverse momentum-density time-autocorrelation function, the macroscopic shear viscosity and its wave dependence are determined, demonstrating that the shear viscosity is isomorphic invariant on all length scales studied. This analysis implies the existence of a length scale that is isomorph invariant in reduced units, i.e., which characterizes each isomorph. The transverse sound-wave velocity, the Maxwell relaxation time, and the rigidity shear modulus are also isomorph invariant. In contrast to the isomorph invariance of all aspects of the transverse dynamics, the reduced-unit dynamics of the mass density is not invariant on length scales longer than the interparticle distance. By fitting to a generalized hydrodynamic model, we extract values for the wave-vector-dependent thermal diffusion coefficient, sound attenuation coefficient, and adiabatic sound velocity. The isomorph variation of these quantities in reduced units on long length scales can be eliminated by scaling with the density-scaling exponent, a fundamental quantity in the isomorph theory framework; this is an empirical observation that remains to be explained theoretically.

Isomorphs in nanoconfined liquids.

We study in this paper the possible existence of Roskilde-simple liquids and their isomorphs in a rough-wall nanoconfinement. Isomorphs are curves in the thermodynamic phase diagram along which structure and dynamics are invariant in suitable nondimensionalized units. Two model liquids using molecular dynamics computer simulations are considered: the single-component Lennard-Jones (LJ) liquid and the Kob-Andersen binary LJ mixture, both of which in the bulk phases are known to have good isomorphs. Nanoconfinement is implemented by adopting a slit-pore geometry with fcc crystalline walls; this implies inhomogenous density profiles both parallel and perpendicular to the confining walls. Despite this fact and consistent with an earlier study [Ingebrigtsen et al., Phys. Rev. Lett., 2013, 111, 235901] we find that these two nanoconfined liquids have isomorphs to a good approximation. More specifically, we show good invariance along the isomorphs of inhomogenous density profiles, mean-square displacements, and higher-order structures probed using the topological cluster classification algorithm. Our study thus provides an alternative framework for understanding nanoconfined liquids.

Does mesoscopic elasticity control viscous slowing down in glassforming liquids?

The dramatic slowing down of relaxation dynamics of liquids approaching the glass transition remains a highly debated problem, where the crux of the puzzle resides in the elusive increase in the activation barrier ΔE(T) with decreasing temperature T. A class of theoretical frameworks-known as elastic models-attribute this temperature dependence to the variations of the liquid's macroscopic elasticity, quantified by the high-frequency shear modulus G∞(T). While elastic models find some support in a number of experimental studies, these models do not take into account the spatial structures, length scales, and heterogeneity associated with structural relaxation in supercooled liquids. Here, we propose and test the possibility that viscous slowing down is controlled by a mesoscopic elastic stiffness κ(T), defined as the characteristic stiffness of response fields to local dipole forces in the liquid's underlying inherent structures. First, we show that κ(T)-which is intimately related to the energy and length scales characterizing quasilocalized, nonphononic excitations in glasses-increases more strongly with decreasing T than the macroscopic inherent structure shear modulus G(T) [the glass counterpart of liquids' G∞(T)] in several computer liquids. Second, we show that the simple relation ΔE(T) ∝ κ(T) holds remarkably well for some computer liquids, suggesting a direct connection between the liquid's underlying mesoscopic elasticity and enthalpic energy barriers. On the other hand, we show that for other computer liquids, the above relation fails. Finally, we provide strong evidence that what distinguishes computer liquids in which the ΔE(T) ∝ κ(T) relation holds from those in which it does not is that the latter feature highly fragmented/granular potential energy landscapes, where many sub-basins separated by low activation barriers exist. Under such conditions, it appears that the sub-basins do not properly represent the landscape properties relevant for structural relaxation.