Fluctuating Nonlinear Hydrodynamics Research Articles

Long-time correlations in a binary mixture: analysis of the nonlinearities of fluctuating-hydrodynamic equations

The equations of fluctuating nonlinear hydrodynamics (FNH) following the proper conservation laws are considered for a binary mixture. We focus on the density (ρ) correlations’ renormalization due to the FNH equations’ nonlinearities. The consequence of density nonlinearities treated in simplest approximations gives rise to the well-studied form of the mode coupling theory (MCT) for a two-component system. The MCT predicts a sharp ergodicity–nonergodicity transition similar to the one-component fluid in this idealized form. In the first part of the present paper, we compare the predictions of the idealized MCT model with the computer simulation results for a hard sphere mixture. We show that there is clear disagreement in long-time dynamic behaviour. Next, we consider the full set of nonlinearities in the FNH equations using a Martin–Siggia–Rose field theory. From the time reversal properties of the correlation and response function of the associated field theory, a set of fluctuation–dissipation relations (FDR) are obtained. These FDRs impose constraints on the long-time behaviour of the correlation functions. Our non-perturbative analysis considers the viability of freezing the time correlations for the two-component fluid over the longest time scale. Due to the FDR constraints arising from the nonlinearity in the FNH equations, a sharp ergodicity–nonergodicity transition for the binary mixture is not supported. If the are replaced as in terms of the average density ρ 0, ad hoc, while the density-nonlinearities in the pressure term of the corresponding FNH equations are kept, the ideal transition model of the simplified MCT is recovered.

Ergodicity–nonergodicity transition in a binary mixture: role of the adiabatic approximation

The mode coupling theory (MCT) for glassy dynamics, in its simplest form, predicts an ergodicity–nonergodicity (ENE) transition in a binary mixture beyond a critical packing fraction ηc. The ENE transition is characterized in terms of the so called nonergodicity-parameters (NEP) which are long time limits of the respective set of density-correlation functions for the mixture. The NEP’s remain zero in the ergodic liquid state and jump to respective nonzero values at ηc. We demonstrate here that this discontinuous jump of the NEP’s which is typical in MCT, changes to continuous growth for a binary mixture having large disparities in mass and sizes of its two constituent species. The NEP’s are obtained as respective solutions of a set of nonlinear integral equations, which follow from the equations of fluctuating nonlinear hydrodynamics (FNH). These FNH equations are based on appropriate conservation laws for the two component system. The nonlocal and nonlinear effects of hydrodynamic fluctuations at high density are considered here in the so called adiabatic approximation which assumes fast relaxation of fluctuations in momentum-density compared to that of number-density. The NEP’s behaviours are analysed with using both static and dynamic approaches and they are found to be in agreement.

Dynamic transition in a Brownian fluid: role of fluctuation–dissipation constraints

In this paper we study equations of fluctuating non-linear hydrodynamics (FNH) for a liquid in which the constituent particles follow Brownian dynamics (BD). Here the microscopic-level dynamics are dissipative as compared to the case of fluids with reversible Newtonian dynamics (ND). The implications of non-linearities in FNH equations for an ND liquid on its long-time dynamics and the possibility of an ideal ergodicity–non-ergodicity (ENE) transition at high density have been widely investigated in literature. It is known that in an ND fluid, dynamics described by FNH equations do not support a sharp ENE transition. In the present paper we demonstrate that, as a consequence of the fluctuation–dissipation constraints, an ideal ENE transition is also not supported from the FNH equations for BD fluid.

Tagged particle correlations in a dense fluid: Role of microscopic dynamics

We study the self diffusion process in a dense fluid in two cases for which the individual fluid particles follow respectively deterministic Newtonian dynamics (ND) and stochastic Brownian dynamics (BD). The role of correlated motion of fluid particles on the single particle diffusion is taken in to account here using mode coupling models. We demonstrate the subtle difference present in the corresponding mode coupling corrections to diffusion constants for fluids with BD and ND by carefully analysing the equations of fluctuating nonlinear hydrodynamics (FNH) in the respective cases. Finally, we use recently developed model for dynamics of binary mixtures to model a BD fluid in terms of a ND mixture whose species have a very large mass ratio. Here the high inertia larger particles are identified as Brownian and a mechanical interpretation of the noise in a Brownian fluid is made in terms of the smaller particles randomly colliding with them.

Nonlinear Fluctuating Hydrodynamics for the Classical XXZ Spin Chain

Using the framework of nonlinear fluctuating hydrodynamics (NFH), we examine equilibrium spatio-temporal correlations in classical ferromagnetic spin chains with nearest neighbor interactions. In particular, we consider the classical XXZ-Heisenberg spin chain (also known as Lattice Landau Lifshitz or LLL model) evolving deterministically and chaotically via Hamiltonian dynamics, for which energy and $z$-magnetization are the only locally conserved fields. For the easy-plane case, this system has a low-temperature regime in which the difference between neighboring spin's angular orientations in the XY plane is an \textit{almost conserved} field. According to the predictions of NFH, the dynamic correlations in this regime exhibit a heat peak and propagating sound peaks, all with anomalous broadening. We present a detailed molecular dynamics test of these predictions and find a reasonably accurate verification. We find that, in a suitable intermediate temperature regime, the system shows two sound peaks with Kardar-Parisi-Zhang (KPZ) scaling and a heat peak where the expected anomalous broadening is less clear. In high temperature regimes of both easy plane and easy axis case of LLL, our numerics show clear diffusive spin and energy peaks and absence of any sound modes, as one would expect. We also simulate an integrable version of the XXZ-model, for which the ballistic component instead moves with a broad range of speeds rather than being concentrated in narrower peaks around the sound speed.

Dynamics of coupled modes for sliding particles on a fluctuating landscape.

The recently developed formalism of nonlinear fluctuating hydrodynamics (NLFH) has been instrumental in unraveling many new dynamical universality classes in coupled driven systems with multiple conserved quantities. In principle, this formalism requires knowledge of the exact expression of locally conserved current in terms of local density of the conserved components. However, for most nonequilibrium systems an exact expression is not available and it is important to know what happens to the predictions of NLFH in these cases. We address this question here in a system with coupled time evolution of sliding particles on a fluctuating energy landscape. In the disordered phase this system shows short-ranged correlations, the exact form of which is not known, and so the exact expression for current cannot be obtained. We use approximate expressions based on mean-field theory and corrections to it, to test the prediction of NLFH using numerical simulations. In this process we also discover important finite size effects and show how they affect the predictions of NLFH. We find that our system is rich enough to show a large variety of universality classes. From our analytics and simulations we have been able to find parameter values which lead to diffusive, Karder-Parisi-Zhang (KPZ), 5/3 Lévy, and modified KPZ universality classes. Interestingly, the scaling function in the modified KPZ case turns out to be close to the Prähofer-Spohn function, which is known to describe usual KPZ scaling. Our analytics also predict the golden mean and the 3/2 Lévy universality classes within our model but our simulations could not verify this, perhaps due to strong finite size effects.

Fluctuating nonlinear hydrodynamics of flocking.

Starting from a microscopic model, the continuum field theoretic description of the dynamics of a system of active ingredients or "particles" is presented. The equations of motion for the respective collective densities of mass and momentum follow exactly from that of a single element in the flock. The single-particle dynamics has noise and anomalous momentum dependence in its frictional terms. The equations for the collective densities are averaged over a local equilibrium distribution to obtain the corresponding coarse grained equations of fluctuating nonlinear hydrodynamics (FNH). The latter are the equations used frequently for describing active systems on the basis of intuitive arguments. The transport coefficients which appear in the macroscopic FNH equations are determined in terms of the parameters of the microscopic dynamics.

BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids

A chain of kinetic equations for non-equilibrium one-particle, two-particle and $ s $-particle distribution functions of particles which take into account nonlinear hydrodynamic fluctuations is proposed. The method of Zubarev non-equilibrium statistical operator with projection is used. Nonlinear hydrodynamic fluctuations are described with non-equilibrium distribution function of collective variables that satisfies generalized Fokker-Planck equation. On the basis of the method of collective variables, a scheme of calculation of non-equilibrium structural distribution function of collective variables and their hydrodynamic speeds (above Gaussian approximation) contained in the generalized Fokker-Planck equation for the non-equilibrium distribution function of collective variables is proposed. Contributions of short- and long-range interactions between particles are separated, so that the short-range interactions (for example, the model of hard spheres) are described in the coordinate space, while the long-range interactions --- in the space of collective variables. Short-ranged component is regarded as basic, and corresponds to the BBGKY chain of equations for the model of hard spheres.

Fragility index of a simple liquid from structural inputs

We make a first-principles calculation of the fragility index m of a simple liquid with known inter particle interaction. The latter determines the corresponding equilibrium structure factor which is treated as an input in the calculation. Using the density functional theory (DFT) of classical liquids, we determine the configurational entropy at moderate supercooling and extrapolate this data to estimate the Kauzmann temperature TK. The relaxation time for the simple liquid has been obtained from direct solutions of the equations of fluctuating nonlinear hydrodynamics (FNH). These equations also form the basis of the mode coupling theory (MCT) for glassy dynamics. Using the Adam–Gibbs relation, we link the configurational entropy to the relaxation time of the supercooled liquid. The fragility index for the supercooled liquid is estimated from analysis of the curves on the Angell plot.

Ergodicity and slow diffusion in a supercooled liquid

A model for the slow dynamics of the supercooled liquid is formulated in terms of the standard equations of fluctuating nonlinear hydrodynamics (FNH) with the inclusion of an extra diffusive mode for the collective density fluctuations. If the compressible nature of the liquid is completely ignored, this diffusive mode sets the longest relaxation times in the supercooled state and smooths off a possible sharp ergodicity–nonergodicity (ENE) transition predicted in a mode coupling theory. The scenario changes when the complete dynamics is considered with the inclusion of 1/ρ nonlinearities in the FNH equations, reflecting the compressible nature of the liquid. The latter primarily determines the extent of slowing down in the supercooled liquid. The presence of slow diffusive modes in the supercooled liquid do not give rise to very long relaxation times unless the role of couplings between density and currents in the compressible liquid is negligible.

Chain of kinetic equations for the distribution functions of particles in simple liquid taking into account nonlinear hydrodynamic fluctuations

Chain of kinetic equations for non-equilibrium single, double and s-particle distribution functions of particles is obtained taking into account nonlinear hydrodynamic fluctuations. Non-equilibrium distribution function of non-linear hydrodynamic fluctuations satisfies a generalized Fokker–Planck equation. The method of non-equilibrium statistical operator by Zubarev is applied. A way of calculating the structural distribution function of hydrodynamic collective variables and their hydrodynamic velocities (above Gaussian approximation) contained in the generalized Fokker–Planck equation for the non-equilibrium distribution function of hydrodynamic collective variables is proposed.

Nonequilibrium dynamics of a supercooled liquid using schematic and structural models

The glassy aging data of dielectric relaxation fits to a modified Kohlrausch–Williams-Watts (MKWW) form exp−tage/τageβage. The relaxation time τage(tage) and the stretching exponent βage are both independent of the frequency of the corresponding data. We discuss two theoretical models to study this non equilibrium dynamics. First we discuss a schematic soft-spin model in which the aging behavior follows the MKWW form similar to the above dielectric data. The nature of the modified fluctuation-dissipation relation (FDR) is also studied in the time as well as correlation windows. Second, we solve numerically the equations of fluctuating nonlinear hydrodynamics (FNH) generalized to short wave lengths for a simple Lennard–Jones liquid. The correlation of density fluctuations at two and four points level are studied using the same density fluctuation data. The respective equilibration of both two point and four point functions follow the MKWW form with relaxation times (dependent of tage) which are very different in the two cases.

Quantum transport equations for Bose systems taking into account nonlinear hydrodynamic processes

Using the method of nonequilibrium statistical operator by Zubarev, an approach is proposed for the description of kinetics which takes into account the nonlinear hydrodynamic fluctuations for a quantum Bose system. Non-equilibrium statistical operator is presented which consistently describes both the kinetic and nonlinear hydrodynamic processes. Both a kinetic equation for the nonequilibrium one-particle distribution function and a generalized Fokker-Planck equation for nonequilibrium distribution function of hydrodynamic variables (densities of momentum, energy and particle number) are obtained. A structure function of hydrodynamic fluctuations in cumulant representation is calculated, which makes it possible to analyse the generalized Fokker-Planck equation in Gaussian and higher approximations of the dynamic correlations of hydrodynamic variables which is important in describing the quantum turbulent processes.

TESTING POWER-LAW RELAXATION SCENARIOS IN A METASTABLE LIQUID

The renormalized dynamics described by the equations of nonlinear fluctuating hydrodynamics (NFH) treated at one loop order gives rise to the basic model of the mode coupling theory (MCT). We investigate here by analyzing the density correlation function, a crucial prediction of ideal MCT, namely the validity of the multi step relaxation scenario. The equilibrium density correlation function is calculated here from the direct solutions of NFH equations for a hard sphere system. We make first detailed investigation for the robustness of the correlation functions obtained from the numerical solutions by varying the size of the grid. For an optimum choice of grid size we analyze the decay of the density correlation function to identify the multi-step relaxation process. Weak signatures of two step power law relaxation is seen with exponents which do not match predictions from the one loop MCT. For the final relaxation stretched exponential (KWW) behavior is seen and the relaxation time grows with increase of density. But apparent power law divergences indicate a critical packing fraction much higher than the corresponding MCT predictions for a hard sphere fluid.

Metastable-state dynamics of a liquid: A free-energy landscape study

Using the time dependence of density fluctuations in a supercooled liquid obtained from the solutions of the equations of nonlinear fluctuating hydrodynamics (NFH), the evolution of the system in the free energy landscape is studied. A crossover from a continuous fluid type dynamics to that of hopping between different free energy minima is observed as the liquid is increasingly supercooled. We demonstrate that our results are also in agreement with equilibrium density functional analysis of the same system. The density field obtained in the numerical solution of the NFH equations are further analyzed to introduce complimentary density of voids in the supercooled liquid state and its static and dynamic correlations are computed. The nature of the relaxation of vacancy correlations are observed to be similar to that of the density fluctuations.

Time-dependent correlations in a supercooled liquid from nonlinear fluctuating hydrodynamics

We solve numerically the equations of nonlinear fluctuating hydrodynamics (NFH). A coarse graining of the density field is applied at each time step to avoid instabilities which otherwise plague the algorithm at long times. The equilibrium correlation of the density fluctuations at different times obtained directly from the solutions of the NFH equations are shown here to be in quantitative agreement with corresponding molecular dynamics simulation data. Low-order perturbative treatment of the these NFH equations obtains the mode coupling model. The latter has been widely studied for understanding the slow dynamics characteristic of the supercooled state. A crucial aspect of this theory is a rounded version of a possible ergodic-nonergodic transition in the supercooled liquid at a temperature T(c) between melting point T(m) and the glass transition temperature T(g). In the present work we demonstrate numerically the role of strongly coupled density fluctuations in giving rise to slow dynamics and how the 1/ρ nonlinearity in the NFH equations of motion is essential in restoring the ergodic behavior in the liquid. The relaxation data indicate that at moderate supercooling near T(c), the time temperature superposition holds. The relaxation gets increasingly stretched with increased supercooling. The relaxation time τ shows an initial power-law divergence approaching a transition temperature T(c) generally identified as the mode coupling temperature. From the direct solutions we obtain a value for T(c) much lower than that typically estimated from solution of low-order integral equations of mode coupling theory. This is in agreement with the trend seen in computer simulations.

Renormalized perturbation theory for a toy model of fluctuating nonlinear hydrodynamics of supercooled liquids

We study the field theoretic renormalized perturbation theory for a toy model of fluctuating nonlinear hydrodynamics (FNH) of compressible liquids. The toy model contains a density-like and a momentum-like variable without any spatial dependence. We present a detailed derivation of a set of coupled equations among correlation and response functions for these variables. In particular, we focus on how the static limit of the correlation and response functions can be achieved in the renormalized perturbation theory. Numerical methods of solving these equations at a given order of the loop expansion are explained and the results for the one-loop theory are given in detail. The simple nature of the toy model enables us to compare the static limit obtained from the exact solution with that of the one-loop order. This shows explicitly the range of validity of the one-loop theory in the field theoretic formulation.

Density nonlinearities in field theories for a toy model of fluctuating nonlinear hydrodynamics of supercooled liquids

We study a zero-dimensional version of the fluctuating nonlinear hydrodynamics (FNH) of supercooled liquids originally investigated by Das and Mazenko (DM) [Shankar P. Das and Gene F. Mazenko Phys. Rev. A 34, 2265 (1986)]. The time-dependent density-like and momentum-like variables are introduced with no spatial degrees of freedom in this toy model. The structure of nonlinearities takes the similar form to the original FNH, which allows one to study in a simpler setting the issues raised recently regarding the field theoretical approaches to glass forming liquids. We study the effects of density nonlinearities on the time evolution of correlation and response functions by developing field theoretic formulations in two different ways: first by following the original prescription of DM and then by constructing a dynamical action which possesses a linear time-reversal symmetry as proposed recently. We show explicitly that, at the one-loop order of the perturbation theory, the DM-type field theory does not support a sharp ergodic-nonergodic transition, while the other admits one. The simple nature of the toy model in the DM formulation allows us to develop numerical solutions to a complete set of coupled dynamical equations for the correlation and response functions at the one-loop order.

Fluctuating nonlinear hydrodynamics does not support an ergodic-nonergodic transition

Despite its appeal, real and simulated glass forming systems do not undergo an ergodic-nonergodic (ENE) transition. We reconsider whether the fluctuating nonlinear hydrodynamics (FNH) model for this system, introduced by us in 1986, supports an ENE transition. Using nonperturbative arguments, with no reference to the hydrodynamic regime, we show that the FNH model does not support an ENE transition. Our results support the findings in the original paper. Assertions in the literature questioning the validity of the original work are shown to be in error.

Dynamical field theory for glass-forming liquids, self-consistent resummations andtime-reversal symmetry

We analyse the symmetries and the self-consistent perturbative approaches of dynamicalfield theories for glass-forming liquids. In particular, we focus on the time-reversalsymmetry, which is crucial to obtain fluctuation–dissipation relations (FDRs). Previousfield theoretical treatment violated this symmetry, whereas others pointed out thatconstructing symmetry-preserving perturbation theories is a crucial and open issue. In thiswork we solve this problem and then apply our results to the mode-coupling theory of theglass transition (MCT).We show that in the context of dynamical field theories for glass-forming liquidstime-reversal symmetry is expressed as a nonlinear field transformation that leaves theaction invariant. Because of this nonlinearity, standard perturbation theories generically donot preserve time-reversal symmetry and in particular fluctuation–dissipationrelations. We show how one can cure this problem and set up symmetry preservingperturbation theories by introducing some auxiliary fields. As an outcome we obtainSchwinger–Dyson dynamical equations that automatically preserve FDR and that serveas a basis for carrying out symmetry-preserving approximations. We apply ourresults to the mode-coupling theory of the glass transition, revisiting previousfield theory derivations of MCT equations and showing that they genericallyviolate FDR. We obtain symmetry-preserving mode-coupling equations and discusstheir advantages and drawbacks. Furthermore, we show, contrary to previousworks, that the structure of the dynamic equations is such that the ideal glasstransition is not cut off at any finite order of perturbation theory, even in thepresence of coupling between current and density. The opposite results found inprevious field theoretical works, such as the ones based on nonlinear fluctuatinghydrodynamics, were only due to an incorrect treatment of time-reversal symmetry.