Prof. Shankar Prasad Das

Coarse grained descriptions of flocking

Generic examples of active matter are swimming microbes, schools of fish, and swarms of birds. Such systems are generally in states out of equilibrium. As a first step to understanding the non-equilibrium dynamics, the steady state of such a self-driven system is treated [1,2] by considering a small driving force acting on a thermal equilibrium state. A typical example of this approach is the model of a flocking system of active ingredients or "particles" in terms of the equations of fluctuating nonlinear hydrodynamics (FNH). The analysis of the corresponding nonlinear dynamics for the time evolution of the active polar particles predicts a non-equilibrium phase transition [3] from a disordered state to a state of long-range order in terms of particle velocities [4]. The FNH equations for an active system have been built primarily from simple, intuitive arguments applied to continuum mechanics. In recent years we have demonstrated how these equations are obtained from the stochastic equations of Brownian dynamics [5].

Rate of Entropy Production

For the active systems following the FNH equations of dynamics, the corresponding entropy S is obtained in terms of the logarithm of the local equilibrium distribution function. For fully reversible dynamics in terms of Euler equations, the entropy density s(r,t) follows a continuity equation, and the net entropy production vanishes. When dissipation is included, the constraint to maintain a positive entropy production rate inequality which links the correlations of the noise and strength of the self-propelling term in the generalized equation for the current density. When the driving term is absent, the natural fluctuation-dissipation relations which hold for the passive system are recovered. Our primary concern is how the entropy production rate depends on the parameters appearing in the FNH equations. The friction coefficients, the noise correlations, and the equal time velocity correlations of the active particles are considered within the constraints of positive entropy production. The strength of fluctuations or noise correlations is restricted to maintain positive entropy production in the system.

Galilean Invariance

The FNH equations of hydrodynamics for a dry-active system is affected under the Galilean transformation. The advective term of the hydrodynamic equation for the time evolution of the momentum g(x, t) is modified with factor λ(ρ), where ρ is the average density of the fluid. λ(ρ), being different from unity, implies breaking of invariance of the hydrodynamic equations under Galilean transformations. We proposed a microscopic-level interpretation for the factor λ(ρ), in terms of a set of activity indices {fi} introduced for the particles. The constant of each particle i. The indices define velocity transformation in the comoving frame in which the fluid is locally at rest. If f_i=1, the Galilean rule holds for all i= 1,…N, and λ(ρ), =1. The factor λ(ρ), differs from unity if the {fi} 's are treated as stochastic variables with finite variance. The rate of entropy productions for the system following the Euler-dynamics is calculated keeping the indices {fi}s introduced above. It is demonstrated that the net entropy production is zero when all fi=1, i=1, N. This case signifies reversible dynamics and is consistent with the non-GI factor λ(ρ), =1. The breaking of Galilean invariance is linked to the dissipative dynamics in the fluid. This observation agrees with a more microscopic approach of using the Boltzmann equation for calculating the GI factor λ(ρ) by other workers [6].

Active Brownian motion: Role of momentum fluctuations

Role of microscopic dynamics in long-time behaviour is a topic of much interest. Generally, the single-particle dynamics in the active matter have been described by a stochastic partial differential equation in terms of the configuration coordinates only. This equation is obtained in the approximation that momentum fluctuations are over-damped so the that the momentum variable is dropped from the formulation. In many situations, however, this is inappropriate, and one needs to keep the momentum variable in the dynamic description. This formulation involving the position and momentum variables is generally referred to as the inertial or underdamped case since the term involving the rate of change of momentum involving the inertia is retained. From a descriptive viewpoint, the trajectories are smooth, as for chemically propelled rods, birds, and some large swimmers with high inertia that move steadily. Underdamped dynamics sometimes provide a fast transfer of information through the system. Our interest is to understand how the underdamped approximation affects the dynamics at the level of collective asymptotic properties. In the case of passive fluids, this has been widely investigated, and dynamics considered without overdamping approximation bring in momentum density g(x,t) in the coarse-grained equations of the hydrodynamic model. Special nonlinearities in the reversible and dissipative parts of the hydrodynamic equations in the underdamped case change the behaviour of asymptotic correlations in the system. We are interested in analysing the consequences of the dynamics, respectively, for over-damped and underdamped cases of active Brownian motion [7] and the implications for the Galilean invariance in the coarse-grained description of active matter.

Coarse grained models

The correlation of hydrodynamic density fluctuations of various conserved properties (like mass and momentum) plays a key role in determining the time-dependent properties of a many-particle system. The equations of hydrodynamics are in terms of coarse-grained densities, which reflect microscopic conservation laws for the fluid. For the case in which the microscopic dynamics is a stochastic equation with noise, the corresponding equations of motion for the coarse-grained equations of fluctuating nonlinear hydrodynamics (FNH) with multiplicative noise [8]. In the case of underdamped motion, the microscopic equations of motion involve both the position and momentum {x,p} and the corresponding field equations of FNH involve collective mass density and momentum density {ρ,g} signifying mass and momentum conservations. In the case of over-damped motion, the microscopic equation involves only position coordinates and the corresponding exact balance equation [9] involving the bare potential of interaction between the particles. This microscopic balance equation has been averaged over the local equilibrium ensemble to obtain the coarse-grained equations of fluctuating hydrodynamics involving the free energy functional F[ρ] of classical DFT in terms of direct correlation functions. These equations are then useful precursors for calculating correlation functions at large lengths and time scales by averaging over the noise. More recently, we have included the rotational degrees of freedom for the particles in terms of angular velocities and angular momentums. The corresponding coarse-grained equations involve smoothly varying local fields following stochastic PDE. The unified description also reaches the form of the corresponding driving free energy functional, which controls the equilibrium state.

Fluctuation-Dissipation relations

The nonequilibrium behaviour of fluid close to glass transition is studied with the models for the nonlinear dynamics [10] of the correlation and response functions. The deterministic equations for the time-dependent correlation functions involve the generalized memory function. The latter is expressed in terms of the functions, giving rise to a nonlinear feedback mechanism, which is key to producing slow dynamics. Using the adaptive step integration method, we numerically solved the set of two nonlinear and coupled integrodifferential equations satisfied by the correlation and the response functions. We studied how the validity of the Fluctuation-dissipation the relation which holds for the equilibrium state evolves in time. The resulting analysis is useful for understanding the ageing dynamics of a supercooled system under thermal noise.

Numerical solutions of Field Equations

Field theoretic models for studying the dynamics of a classical many-particle system are in terms of stochastic partial differential equations (PDE)s or the so-called generalized Langevin equations of motion for a chosen set of slow modes of the fluid. The latter are collective variables reflecting either underlying conservation laws, broken symmetry or, in some cases, some intrinsic property of the fluid. We have worked on the numerical solution of the discretized equations of fluctuating nonlinear hydrodynamics, and computed the two-point and higher-order correlation functions in order to study aspects of dynamic heterogeneities in a supercooled liquid. The role of multiplicative noise in this basic equation for complex fluids is being further analyzed with numerical methods.

Dissipative Particle Dynamics

Several new numerical methods [11-12] for computer simulations on hydrodynamic time scales have been invented in recent decades. For studying the dynamical and rheological properties of complex fluids, the Dissipative Particle Dynamics (DPD) method is applied to a variety of systems. Theoretical analysis and explicit calculation of transport and thermodynamic properties in terms of model parameters [13-14] help us understand these techniques' construction and improvements. Our interest is in the transport mechanisms and exploring new algorithms for studying the dissipative dynamics of complex system.

Mode Coupling Models

Building a microscopic theory for understanding the physics of the high-density liquid transforming into an amorphous solid-like state have been our longstanding interest. The (Boltzmann-Enskog type) models dealing with the uncorrelated binary collision of the constituent particles provided the only microscopic theory of fluids over many years [15,16] and are successful and expected to be normal at low densities. However, they fail to explain the effects of strong correlation and collective behaviour that build up in a high-density liquid. Understanding the implications of these correlated dynamics from a basic statistical mechanical approach is particularly relevant in the case of glass formation phenomena seen universally in all liquids. The corresponding theoretical development is termed the mode coupling theory (MCT).

Entropy and Diffusion : Emperical relation

We have studied the standard MCT to compute the higher order cumulants of displacement and obtained the so-called non-gaussian parameter from the single-particle dynamics. The MCT has been used to link the transport property, like the self-diffusion coefficient, with the entropy [17] of the liquid and this improved results, which were predicted from empirical considerations.

The Binary Mixture and MCT

The equations of FNH for a two-component mixture are obtained with a proper choice of slow variables corresponding to the system's conservation laws. Analyzing those equations, we demonstrated that the dynamics of the mixture depend on the mass ratio of the species. This mass ratio dependence of the ideal transition was also subsequently verified in simulations.

Asymptotic behaviour of density correlations

The ergodicity nonergodicity (ENE) transition characteristic of standard MCT is marked by a jump in the long time limit of the density correlation function from zero to a finite value. The renormalization of the nonlinear dynamics corresponding to the coarse-grained field equations of FNH for a fluid, involving both density and momentum density {ρ,g}, has been done using an MSR field theoretic [18] formulation. However, a complete analysis shows that this sharp transition is only supported under ad hoc assumptions. The associated analysis of the constraints from the available fluctuation-dissipation relations [19] shows that the transition is not supported. The FNH equations only support the ENE transition if certain density nonlinearities are ignored, a result which is followed by a nonperturbative analysis of the (fully self-consistent formulation) of the renormalized model. An important observation here is that under such an ad hoc approximation, the current correlations behave similarly to density correlations in the compressible liquid. Their respective long-time limits are nonzero in the nonergodic state. Furthermore, in the corresponding MCT dynamics of the single-particle density-correlations, it becomes completely "slaved" to that of collective density-correlation. Beyond the ideal ergodicity-nonergodicity (ENE) transition point, the viscosity diverges, and the single particle diffusion constant vanishes. However, without any ad hoc assumptions, the self-correlation and the collective correlation behave differently in the complete model. Such difference in the behaviour of the two correlations was observed in experiments. In a companion paper we demonstrated a mechanism by which a sharp fall in self-diffusion coefficient may occur in the MCT. This overdamping, or the so-called adiabatic approximation for the supercooled state, does not maintain microscopic momentum conservation. We are considering the evolution of the correlation and response functions in the corresponding Non-Equilibrium dynamics involving both mass and momentum current densities and study consequences on the relevant Fluctuation-dissipation relations.
For the model in terms of coarse-grained density field {ρ} only, we have analysed the implications of the nonlinearities in the basic equation of FNH ( the Dean-Kawasaki equation) using a Martin-Siggia-Rose [18] field theory. We studied the long-time behaviour of the time correlation of density fluctuations. We showed that the ENE transition is not supported in this case due to nonlinear fluctuation-dissipation constraints, similar to the {ρ,g} formulation.

Complexity using Replica approach

In the free energy landscape (FEL) description of a supercooled liquid, the corresponding metastable states of the many-particle system are depicted as local minima of an appropriate free energy functional F[ψ], where the field ψ(x) constitutes a continuum field theoretic description of the many-particle system. In Classical density functional theory (DFT), a useful tool for understanding the freezing transition of an ergodic liquid into a crystalline state, the inhomogeneous density field ρ(x) is identified with ψ(x). The metastable liquid, close to vitrification, is caught in one of the many possible basins into which the FEL splits. In a structural glass, this fragmentation of the FEL occurs without any quenched disorder. We have used such a field-theoretic model for the spontaneous breaking of ergodicity. From a theoretical approach, the configurational entropy is estimated from the logarithm of the number of metastable minima which develop in the FEL on supercooling. We have calculated the configurational entropy of the supercooled liquid using the same density functional description. The free energy of the system without any quenched disorder is obtained by using the Replica approach and taking the physical limit of the replica index going to unity [20]. The extrapolated configurational entropy Sc tends to vanish at a finite temperature Tk at deep supercooling. The study of configurational entropy demonstrated the effects of multi-particle correlations in the loss of fluidity in the supercooled state. The formulation of the renormalized model with a perturbation theory in terms of the bare interaction potential is also being attempted. Further to this, we plan to study the dynamic correlations in the liquid using the replica approach.

Boson Peak and vibrational states

The extra density of vibrational states found in the amorphous solid state , generally termed the Boson Peak, has been linked to the anharmonicities in such systems. We have recently shown [21] that departure from the description of the glassy state in terms of purely Gaussian peaks, which signify vibrating particles around respective parent sites, can account for the extra density of states of the Boson peak. This result is consistent with our earlier works where we showed, using schematic MCT, that appearance of the peak is crucially linked to the development of transverse sound modes in the amorphous solids in the presence of defects. Such a scenario was also subsequently validated by Tanaka et al. simulation studies.

Chemotaxis: Stochastic Model in a coarsae-grained description

In biological systems, chemotaxis refers to a process in which somatic cells, bacteria, other single-celled organisms and multicellular organisms are made to move in a particular direction in response to a chemical stimulus. This phenomenon is attributed to the ability of bacteria and cells to detect the changes in the concentrations of some specific chemical molecules in their surrounding media, resulting in an adjustment in their direction of motion or an intrinsic polarity. From a wider perspective it has been conjectured that this phenomena of self-organization of chemotactic species resembles the structure formation in the universe or vortices appearing in two-dimensional turbulence at large length scales.

As a simple model for the chemotactic response of a single cell in a medium, it is proposed that the cell acquires a drift velocity equal to the gradient of a concentration field field Φ(x, t), termed as the Keller-Segel (KS) potential [22,23]. The effect can also depend on the cell's unit vector n, called the cell polarity, which characterizes the possible anisotropy in the cell's response to the chemical gradients [24]. Stochastic field description for the dynamics of a system of chemotactic particles are being formulated extending on the Keller-Segel approach with potential functions field Φ(x, t), and director n. The chemical field field Φ(x, t) at the position x at time t is continuously sustained by the diffusing chemical molecules that the particles release in the case of a self-chemotactic system. A key variable in this formulation is a coarse-grained density function for the particles that produce and consume chemicals, which they also chemotactically respond to in the chemotaxis process. The diffusion constant of the chemical molecules is often orders of magnitude larger than that of the particles secreting them, making the assumption of a steady state for the potential field field Φ(x, t) justified. Closing the set of field equations with self-consistent form is a major step in such models for the dynamics of living organisms. Another important consideration in formulations of the continuum equations of large-scale dynamics is the underlying conservation laws. In these systems, the balance equations reflecting specific conservation laws often require modification with additional terms. The corresponding stochastic partial differential equations with nonlinearities involving the coarse-grained fields [25,26]. These equations describing the steady state are used to calculate dynamic correlations and phase behaviours of such systems. An essential step here is to understand the statistical correlations neglected in a mean-field type KS formulation by introducing noise in the model.

Replica Approach to Structural Glasses

Replica approaches have been used in different contexts for glassy systems. The limiting values of the replica index n chosen differ for spin glasses and structural systems. In the context of spin glasses with quenched impurities for calculating the logarithm of the partition function averaged over an ensemble of quenched impurities, the replica index n is finally set to zero [27]. For structural glasses without any extrinsic or quenched impurities, the corresponding partition function for the system is computed using a field-theoretic approach in terms of an order parameter field like density, for example. By introducing an external field ψ(x) coupled to the density field ρ(x) in a suitable form to pick up the saddle-point contribution from the local minima of the free energy for the system when ψ(x)=ρ(x) [20]. This model allows the calculation of the configurational entropy or the complexity of the metastable liquid approaching the glass transition. The partition function is calculated for a hybrid system of n identical replicas of the system. Finally, the replica index is set to the physical limit unity. The transition's order parameter is the density field's inter-replica correlation. In contrast, the intra-replica correlation is identified as the static structure factor for the uniform liquid. In such a formulation, higher-order correlations (of the density field) present in the free energy functional used for the field-theoretic model play a key role in calculating the spontaneous ergodicity breaking point. However, in other theoretical approaches, the overlap function between two identical replicas was used to study the glassy behaviour, and the calculation of the order parameter of inter-replica overlap is obtained as the static correlations for a binary system with specially chosen interactions, and finally the interaction potential between the two species of the mixture is set to zero [28]. Understanding these relations and linking Replica models to MCT(the other microscopic model for glassy behaviour) and the mosaic picture of the glassy state [29] is an area of our research.

Hyperuniformity

The behaviour of long-wavelength density fluctuations in a disordered system and the characterization of hyperuniformity in density distribution have been a subject of recent interest. For different systems, the density distributions can be linked with the issue of hyperuniformity, e.g., fluctuations at long distances are suppressed in granular systems and biological tissues. Generally, the density of a system of many particle systems is expressed as the average density coarse-grained over a length scale of R (say). The density fluctuations' standard deviation σ should approach zero as the coarse-graining length scale R increases. The term hyperuniformity has been linked to the fall of σ with R. In our earlier studies, we have identified local minima of the free energy for heterogeneous density distributions. The hyperuniformity in such density distributions corresponding to these amorphous density profiles is a topic of our current research interest.

For glassy systems a topic of much interest was how with time a peak in the four point correlations of density fluctuation peaks, in the small wave number peaks. The width of the peak in the small wave number limit is linked to growing dynamic length scale in the system. This length scale has also been linked to a more familiar terminology of the dynamic heterogeneities in glassy systems. Calculation of the four-point correlations theoretically starting from microscopic description of a many particle system in the small wave number limit is needed to obtain the associated dynamic correlation length. It growth with increase of average density is an important cursor to describing the glassy state.

Kinetic Field theory

In recent years we have been involved in a new field-theoretic formulation of the kinetic theory of evolution of a classical many-particle system. This approach provides a renormalized perturbation theory in terms of the interaction potential of the many-particle system. This method is an alternative to the Mori-Zwanzig scheme of expressing the dynamics in terms of slow modes, in which the lowest-order theory represents uncorrelated dynamics. With this formulation, the location of the ideal ENE transition for a hard sphere system shifts closer to the random close packing value, improving the agreement between theoretical model predictions and simulations. An extension of these methods is expected to be useful for studying cosmic structure formation [30], starting from a particle distribution with Gaussian correlations in phase space, believed to be set by inflation. The application of basic statistical mechanics for a classical system of particles can facilitate such studies by formulating an analytical model, and such efforts are currently underway.