ICTS

The existence of a phase transition between two distinct liquid phases in a single-component substance was first proposed for water and has been explored in other network forming liquids such as silica and silicon, and subsequently as a more generic phenomenon in a broader context. Because such a transition is expected under supercooled conditions, early evidence from numerical simulations have been questioned, with misinterpretation of slow crystallization being offered as an alternate explanation. Recent free energy calculations in computer simulations and new experimental results support the scenario of a liquid-liquid transition in the case of water. However, similar results have not so far been available for silicon, which forms an extreme case among network forming liquids that may be expected to exhibit such a transition. I present results from investigations of crystal nucleation barriers and free energies of a model of silicon that unambiguously demonstrate the existence of a liquid-liquid transition in silicon.

A prominent inequality for Non-equilibrium steady states, known as the “thermodynamic uncertainty relation” [A.C. Barato and U. Seifert, Phys. Rev. Lett. 114, 158101 (2015)] has evoked a lot of interest. I will present a rather introductory talk on  this relation as well as it's many variations and applications.

I will discuss the entropy of a set of identical hard objects, of general shape, with each object pivoted on the vertices of a d -dimensional regular lattice, but can have arbitrary orientations. Such a system shows multiple geometrical phase transitions as the spacing of lattice is varied. It is a model for the multiple crytalline phases found in many molecular cystals. I will consider in some detail three illustrative cases: hard linear rods, pivoted at one end, linear rods pivoted at mid point, and circular discs with pivot off-center. For asymmetrically placed pivots, in the range of lattice spacings, the problem reduces to a dimer model at finite negative fugacity. The orientation distribution shows multiple cusp-like singularities as a function of orientations. I will discuss an approximate theory that gives the correct positions, and qualitative behavior of these singularities.

The long-range nature of the effect of a pump or a battery on an interacting diffusive fluid is discussed. It is shown that off criticality the pump generates long-range modulation in the density profile of the form of a dipolar electric potential and a current profile in the form of a dipolar electric field. The density profile is drastically modified when the fluid is at its critical point: here, in addition to the long-rage influence of the current generated by the battery, the fluid is dominated by its intrinsic long- range critical correlations. It is demonstrated that the resulting density profile is of the same form as that of a fluid in equilibrium but under the influence of dipolar ordering field. As a result, the density profile at criticality can be expressed in terms of the equilibrium critical exponents of the fluid. In contrast, the current is shown to retain it off critical dipolar field form.

1. Tridib Sadhu, Satya Majumdar and David Mukamel, PRE 84, 051136 (2011)
2. Tridib Sadhu, Satya Majumdar and David Mukamel, PRE 90, 012107 (2014)
3. Ydan Ben Dor, Yariv Kafri, David Mukamel and Ari M Turner, PRL 128, 154501 (2022).
 

The transport properties of an extended system driven by active reservoirs is an issue of paramount importance, which remains virtually unexplored. Here we address this issue, for the first time, in the context of energy transport between two active reservoirs connected by a chain of harmonic oscillators. The couplings to the active reservoirs, which exert correlated stochastic forces on the boundary oscillators, lead to fascinating behavior of the energy current and kinetic temperature profile even for this linear system. We analytically show that the stationary active current (i) changes non-monotonically as the activity of the reservoirs are changed, leading to a negative differential conductivity (NDC), and (ii) exhibits an unexpected direction reversal at some finite value of the activity drive. The origin of this NDC is traced back to the Lorentzian frequency spectrum of the active reservoirs. We provide another physical insight to the NDC using nonequilibrium linear response formalism for the example of a dichotomous active force. We also show that despite an apparent similarity of the kinetic temperature profile to the thermally driven scenario, no effective thermal picture can be consistently built in general. However, such a picture emerges in the small activity limit, where many of the well-known results are recovered.

I will present results for the local fluctuations in the one-dimensional Coulomb gas in the presence of a confining harmonic potential – the so-called “jellium model” – with N particles. I will discuss three distinct observables: the position of the rightmost particle, the gap between the positions of two consecutive particles and the number of particles in a given interval [−L,L] centred around the origin. In all cases, for large N, I will discuss both the typical and the atypical fluctuations, characterized respectively by scaling and large deviation functions, that can be computed explicitly.

I will discuss the entropy of a set of identical hard objects, of general shape, with each object pivoted on the vertices of a d -dimensional regular lattice, but can have arbitrary orientations. Such a system shows multiple geometrical phase transitions as the spacing of lattice is varied. It is a model for the multiple crytalline phases found in many molecular cystals. I will consider in some detail three illustrative cases: hard linear rods, pivoted at one end, linear rods pivoted at mid point, and circular discs with pivot off-center. For asymmetrically placed pivots, in the range of lattice spacings, the problem reduces to a dimer model at finite negative fugacity. The orientation distribution shows multiple cusp-like singularities as a function of orientations. I will discuss an approximate theory that gives the correct positions, and qualitative behavior of these singularities.

In presence of long range dispersal, epidemics spread in spatially disconnected regions known as clusters. Similarly, depinning avalanches with long range elasticity are collections of disconnected clusters. In this talk I will discuss the statistical properties of clusters (their number, their size….) in two different models.

We consider the problem of approximating a covariance matrix with a low-rank matrix in a manner that is differentially private. A well-known approach towards this is to add symmetric Gaussian noise to the input matrix and compute the best low-rank approximation to the perturbed matrix. We present a new analysis of this ``Gaussian'' mechanism by viewing the addition of Gaussian noise as a continuous-time matrix Brownian motion. This viewpoint allows us to track the evolution of eigenvalues and eigenvectors of the matrix, which are governed by stochastic differential equations discovered by Dyson, and allows us to obtain new bounds on the Frobenius norm of the change in the low-rank approximation as a result of the Gaussian perturbation.

Based on joint work with Oren Mangoubi (https://arxiv.org/abs/2211.06418).
 

A monolayer of granular spheres in a cylindrical vial, driven continuously by an orbital shaker and subjected to a symmetric confining centrifugal potential, self-organizes to form a distinctively asymmetric structure which occupies only the rear half-space. Imaging shows that the regulation of motion of individual spheres occurs via toggling between two types of motion, namely, rolling and sliding. Experiments demonstrate and simulations confirm that global features of the structure are maintained robustly by an auto-tuning of the effective friction through internal dynamical states of rolling and sliding which provides a protocol-insensitive route to self-organization of a driven many-body system. Recent results show that restricting the motion of the system to a quasi-2 D space leads to efficient crystallization. Relation of two forms of locomotion to more general scenarios of autotuning of friction, as in chemotaxis of bacteria and prevention of stampede in crowd dynamics, will be speculated upon.

*Work done at the Tata Institute of Fundamental Research in collaboration with Deepak Kumar, Anit Sane, Soham Bhattacharya, Nitin Nitsure and Shankar Ghosh.

Activated Random Walk is a particle system displaying avalanches on all scales. How universal are these avalanches? I’ll narrate five interlocking conjectures aimed at different aspects of this question: infinite-volume limits, cutoff, incompressibility, rotational symmetry, and hyperuniformity.

Joint work with Feng Liang and with Vittoria Silvestri.

We study the fluctuations of the integrated density current and integrated magnetization current in a lattice model of interacting active particles. This model is amenable to an exact description within a fluctuating hydrodynamics framework. We focus on quenched initial conditions for both the density as well as magnetization fields, and derive expressions for the cumulants of the currents, which can be matched with direct numerical simulations of the microscopic lattice model. We show that the fluctuations of the integrated density current displays a large time T^{1/2} scaling which is sensitive to the initial conditions, the Peclet number and the density of particles. For the case of uniform initial profiles, we show that the second cumulant of the integrated density current displays three regimes: an initial rise described by the symmetric simple exclusion process, a linear regime where the effects of activity drive large fluctuations, and a large time diffusive behaviour that is governed by both the effects of activity and exclusion.

Uniformly biased random walks in a disordered medium show an interesting competition between drift and trapping. If the medium has an exponential distribution of branch lengths, the drift velocity vanishes beyond a certain threshold value of bias.

Interest in this problem has revived in recent years, and in this talk I will discuss two directions. The first concerns the order of the transition at the threshold, and the existence of a different threshold for the onset of anomalous fluctuations heralded by a divergence of the standard deviation of transit times. The second problem has to do with the effect of mutual exclusion between particles. Exclusion leads to screening of deep traps, leading to a finite drift velocity throughout. However, rare encounters with an unscreened trap leads to ultra-strong trapping in such traps and thus to strong fluctuations of transit times.
 

We characterize collective motion of strongly persistent interacting self-propelled particles (SPPs) and offer a generic mechanism that accounts for the “early-time” anomalous relaxations observed in such systems. For small tumbling rates, a suitably scaled bulk- (collective-) diffusion coefficient is found to vary as a power law over a wide range of density. As a result, the density relaxation is governed by a nonlinear diffusion equation, exhibiting anomalous spatio-temporal scaling, which, in certain regimes, is ballistic. We rationalize these findings through a scaling theory and substantiate our claims by directly calculating the bulk-diffusion coefficient in two idealized versions of interacting SPPs for arbitrary density and tumbling rate. We demonstrate that, for small tumbling rates, the scaled bulk-diffusion coefficient is a function of a single scaling variable, quantifying the fascinating interplay between persistence and interactions. Our arguments are rather generic and could be applicable to a broad class of SPPs.

One measure of the degree of order in a translation invariant point process in Rd is the variance in the number of points, N⇤, in a region ⇤, such as a sphere of radius R. When Var(N⇤)/|⇤| ! 0, as the volume |⇤| of ⇤, increases, the system is called hyperuniform  or superhomogeneous). This occurs when the Fourier transform of the truncated full pair correlation function S(k) vanishes when k = 0. When S(k) vanishes in an open set M in k-space the system is transparent to waves with wave vector k in M. It also has the property of “maximal rigidity”: the exact position of points in ⇤ is determined by the configuration of points outside ⇤. I will review some old results about charge fluctuations in Coulomb systems and describe some recent ones (joint with Subhro) about such systems.

We develop a Fourier-Chebyshev pseudospectral direct numerical simulation (DNS) to examine a potentially singular solution of the radially bounded, three-dimensional (3D), axisymmetric Euler equations [cf., G. Luo and T.Y. Hou, Proc. Natl. Acad. Sci. USA 111, 12968 (2014)]. We demonstrate that (a) the time of singularity is pre-ceded, in any spectrally truncated DNS, by the formation of oscillatory structures called tygers, first investigated in the one-dimensional (1D) Burgers and two-dimensional (2D) Euler equations; (b) the analyticity-strip method can be generalized to obtain an estimate for the (potential) singularity time.

This work has been done with Sai Swetha Venkata Kolluru and Puneet Sharma [Ref.: PHYSICAL REVIEW E 105, 065107 (2022)].

Incompressible Euler equation has zero viscosity and no external forcing, hence, it can be treated as an isolated system. In this presentation, I show that two-dimensional (2D) Euler turbulence is out of equilibrium and it exhibits evolution from disorder to order, even though the system is an isolated one. The small wavenumber modes exhibit nonzero energy energy transfers, hence, detailed balance is broken.

Since the constant thermodynamic entropy of Euler turbulence cannot capture the variable order of the flow, we propose “hydrodynamic entropy” for describing the disorder in Euler turbulence. The hydrodynamic entropy decreases with time for a significant period.

Ref: M. K. Verma and S. Chatterjee, Hydrodynamic entropy and emergence of order in two-dimensional Euler turbulence, in press, Phys. Rev. Fluids (2022); arXiv:2210.06445

Branching random walk is a system of growing particles that starts with one particle. This particle splits into a number of particles, and each new particle makes a displacement independently of each other. The same dynamics goes on and on giving rise to a branching random walk. It is important because it has connections to various other models in statistical physics and probability theory. In this overview talk, we shall mainly try to address the following question: if we run a branching random walk for a very very long time and take a snapshot of the particles, what would the entire system look like? In this context, we shall discuss how the predictions of Éric Brunet and Bernard Derrida are verified when the displacements follow a power law distribution.

This talk is based on joint work with Ayan Bhattacharya, Rajat Subhra Hazra, Krishanu Maulik, Zbigniew Palmowski, Souvik Ray and Philippe Soulier.