Frontiers in Statistical Physics Talks

I will first discuss the equilibrium properties of a gas of N interacting particles on a line. I will then introduce a simple model of $N$ independent Brownian particles that are subjected to simultaneous stochastic resetting with rate r. The simultaneous resetting generates an effective dynamical all-to-all attractions between particles that persist even at long times in its nonequilibrium stationary state (NESS). Despite the presence of strong correlations, many physical observables such as the average density, extreme statistics, order and gap/spacing statistics, full counting statistics etc. can be computed exactly in the NESS and they exhibit rich and interesting behaviors. The physical mechanism built in this simple model allows it to generalise and invent a whole class of solvable strongly correlated gases, some of which are experimentally realisable in optical trap systems.

Linear diffusions are used to model many stochastic processes in physics, including small mechanical and electrical systems perturbed by thermal noise, as well as Brownian particles controlled by electrical and optical forces. In this talk, I will present an exact solution for the large deviations of linear diffusions in any dimension, considering three classes of observables relevant for nonequilibrium systems that involve linear or quadratic integrals of the state in time. The solution shows that fluctuations arise in linear diffusions in terms of effective forces that remain linear in the state or, alternatively, in terms of fluctuating densities and currents that solve Riccati-type equations. I will illustrate these results for the stochastic area of 2D diffusions, introduced originally by Levy in 1951 and re-discovered more recently in physics as a measure of irreversibility.

This is joint work with Johan du Buisson and Thamu Mnyulwa.

We study the effect of spatially-varying potential and diffusivity on the dispersion of a tracer particle in single-file diffusion. Non-interacting particles in such a system exhibit normal diffusion at late times, which is characterised by an effective diffusion constant Deff. Here we demonstrate the physically appealing result that the dispersion of single-file tracers in this system has the same long-time behavior as that for Brownian particles in a spatially-homogeneous system with constant diffusivity Deff. Our results are based on a late-time analysis of the Fokker-Planck equation, motivated by the mathematical theory of homogenization. The findings are confirmed by numerical simulations for both annealed and quenched initial conditions.

We address the issue of what happens when the unitary evolution of a generic closed quantum system is interrupted at random times with non-unitary evolution due to interactions with either the external environment or a measuring apparatus. We adduce a general theoretical framework to obtain the average density operator of the system at any time during the dynamical evolution, which is applicable to any form of non-unitary interaction. We provide two explicit applications of the formalism in the context of the so-called tight-binding model relevant in various contexts in solid-state physics, for two representative forms of interactions: (i) stochastic resets, whereby the density operator is at random times reset to its initial form, and (ii) projective measurements at random times. For (i), we demonstrate with our exact results how the particle is localized on the sites at long times, leading to a time-independent mean-squared displacement of the particle about its initial location. For (ii), we show that repeated projection to the initial state of the particle results in an effective suppression of the temporal decay in the probability of the particle to be found on the initial state. The amount of suppression is comparable to the one in conventional Zeno effect scenarios, but which does not require to perform measurements at exactly regular intervals that are hallmarks of such scenarios.

We study self-gravitating bosonic systems, candidates for dark-matter halos, by carrying out a suite of direct numerical simulations designed to investigate the formation of finite-temperature, compact objects in the three-dimensional (3D) Fourier-truncated Gross-Pitaevskii-Poisson equation (GPPE). This truncation allows us to explore the collapse and fluctuations of compact objects and show the following:

1) The statistically steady state of the GPPE, in the large-time limit and for the system sizes we study, can also be obtained efficiently by tuning the temperature in an auxiliary stochastic Ginzburg-Landau-Poisson equation.

2) Over a wide range of model parameters, this system undergoes a thermally driven first-order transition from a collapsed, compact, Bose-Einstein condensate to a tenuous Bose gas (that is not gravitationally condensed).

3) By a suitable choice of initial conditions in the GPPE, we also obtain a binary condensate that comprises a pair of collapsed objects rotating around their center of mass.

4) We use a generalised GPPE to study the collapse of an axion star.

5) By introducing a solid-crust potential and rotation in the GPPE, we develop a minimal model for pulsars and their glitches.

References

1. A.K. Verma, R. Pandit, and M.E. Brachet, The formation of compact objects at finite temperatures in a dark-matter-candidate self-gravitating bosonic system, Phys. Rev. Research 3 (2), L022016 (2021).

2. A.K. Verma, R. Pandit, and M.E. Brachet, Rotating self-gravitating Bose-Einstein condensates with a crust: a minimal model for pulsar glitches, Physical Review Research 4 (1), 013026 (2022).

3. S. Shukla, A.K. Verma, M.E. Brachet, and R. Pandit, Gravity- and temperature-driven phase transitions in a model for collapsed axionic condensates, submitted for publication (November 2023).

The Mpemba effect has gained recent attention due to its prevalence in the context of non-equilibrium evolution of a variety of unrelated physical systems. The key idea here is that, a system initially farther from a final, time-independent state, for certain initial conditions, tends to reach the final state sooner compared to those started from initial states which are nearer.  It has been shown  that granular systems show such effects. In the talk I will briefly review these results and our  latest observations.

Random searches find applications in a broad range of problems in biology, chemistry or rescue operations.  In a typical set-up, a searcher follows a random motion and reacts with a fixed target site upon first encounter. Often, however, the target has a finite lifetime after which it becomes inactive, lost or no longer available for reaction.  To be successful, the searcher must therefore find the target site before the latter becomes inactive. We present exact results on a simple model, namely, a one-dimensional searcher performing a discrete-time Lévy flight while the target has an exponentially distributed lifetime. In contrast with the case of a permanent target, it is possible to optimise the capture probability and the conditional mean first passage time at the target. The optimal Lévy index takes a non-trivial value, even by taking the infinite lifetime limit, and exhibits a "phase transition" as the initial distance to the target site is varied. This transition can be discontinuous or continuous depending on the target lifetime, with a non-conventional tri-critical point in-between. We outline connections between this problem and search processes based on resetting.

The autocorrelation function in several complex systems shows a crossover in the form of its decay, from a stretched exponential at short times to a power-law at long times. In this talk we discuss two 1D models with kinetically constrained dynamics which show such a crossover, though the reasons are quite different in the two cases.

In the first model [1],  the constraint is that a spin can only flip when two of its nearest neighbours are opposite, which ensures that the number of domain walls (DWs) is conserved. A nonzero magnetic field leads to bound pairs of DWs, which we term doublons. The short-time behaviour is governed by individual DWs, leading to a stretched exponential decay of the spin-spin autocorrelation function. At long times, the dynamic behaviour is determined by doublons, which behave as extended particles with hard core repulsion,  leading ultimately to a power-law decay of the autocorrelation function.

In the second model, we study the effect of a rapid quench to zero temperature in a model with competing interactions, evolving through conserved spin dynamics. In a certain regime of model parameters, the resulting dynamics is kinetically constrained, and characterized by a conserved, sub-extensive number of DWs. The system shows stretched exponential relaxation with the unusual feature that the relaxation time diverges as a power of the system size. Interestingly, the autocorrelation function of a single typical sample in the steady state differs markedly from that averaged over an ensemble of initial conditions resulting from different quenches; the latter shows stretched exponential decay at short times and a  slow power-law decay at large times, and is described by a  Mittag-Leffler function.

[1] S. Mukherjee, P. Pareek, M. Barma,  and S. K. Nandi, arXiv:2307.01801

[2] V. Gupta, S. K. Nandi, and M. Barma, Phys. Rev. E 102, 022103 (2020).

We develop a formalism to calculate the probabilities of rare events in cluster-cluster aggregation for arbitrary collision kernels. By rewriting these probabilities in terms of minimizing an effective action, we establish a large deviation principle with the total mass M being the rate. The large deviation function is calculated exactly for the constant kernel, sum kernel and multiplicative kernel. The rate function for the multiplicative kernel is shown to be singular.

Biological evolution can be conceptualized as a search process in the space of gene sequences guided by the fitness landscape, a mapping that assigns a measure of reproductive value to each genotype. Here we discuss probabilistic models of fitness landscapes with a focus on their evolutionary accessibility, where a path in a fitness landscape is said to be accessible if the fitness values encountered along the path increase monotonically. For uncorrelated (random) landscapes with independent and identically distributed fitness values, the probability of existence of accessible paths between genotypes at a distance linear in the sequence length L becomes nonzero at a nontrivial threshold value of the fitness difference between the initial and final genotype, which can be explicitly computed for large classes of genotype graphs.

The behaviour in uncorrelated random landscapes is contrasted with landscape models that display additional, biologically motivated structural features. In particular, landscapes defined by a tradeoff between adaptation to environmental extremes have been found to display a combinatorially large number of accessible paths to all local fitness maxima. We show that this property is characteristic of a broad class of models that satisfy a certain global constraint, and provide further examples from this class.

The mean sojourn time plays an important role in understanding the genetic diversity in a biological population. I will describe our results for the sojourn time and genetic diversity when the environment is changing continually.

Resetting a stochastic process has been shown to expedite the completion time of some complex tasks, such as finding a target for the first time. In this talk we consider the cost of resetting by associating a cost to each reset, which is a function of the distance traveled during the reset event. We show some unexpected results. First, in the limit of a vanishing resetting rate, the mean total cost is finite for a linear cost function and diverges for a super-linear cost function. This result contrasts with the case of no resetting where the cost is always zero. Second, the resetting rate that minimises the time to completion, including the total resetting cost, is reduced from the case of no cost for a linear cost function, remains unchanged for a quadratic cost function but may be increased for a super-quadratic cost function. In the latter case an increased rate of resetting reduces the chance of costly resets.

In the past few years, resetting has become a subject of immense interest cutting across disciplines. Most of the theoretical studies so far focused on instantaneous resetting or teleportation which is, however, a major impediment to any practical realisation. Indeed in the real world, taking a particle or an object from one place to another requires finite time which tells us that a generalization of the existing theory to incorporate non-instantaneous resetting is very much in need. Such observations are also an integral part of the home range search where return to the home following a search is a finite time process or the optical trap experiments where it takes time for the laser to manipulate the colloids. In this talk, I will first recap the classical resetting problem and then briefly review some of these recent developments in the field.

The first half of my talk (with P. Popli and A. Maitra) will be on defect interactions in vision-cone XY models, related to flocking. The second half (with N. Bhatt and S. Mukerjee) will discuss a kind of thermalisation, and emergent hydrodynamics, in left-right nonreciprocal Heisenberg chains.

Measuring entropy production of a system directly from the experimental data is highly desirable since it gives a quantifiable measure of the time-irreversibility for non-equilibrium systems and can be used as a cost function to optimize the performance of the system. In this talk, I will discuss about a method which enables us to infer the mean entropy production for time-dependent Markov systems in terms of some experimentally accessible quantities.

We present some reflections on the nature and working of biological engines.

Stochastic systems that undergo random restarts to their initial state have been widely investigated in recent years, both theoretically and in experiments. But the thermodynamic costs for implementing such resets has been less well studied.  In this talk I will present some of our recent results on the thermodynamic costs that a resetting entails for some  variations of the resetting process, one of which has also been recently investigated in experiments.

It is an ensemble of paper in collaboration with Satya, Gregory, Urna and also some recent work with Kabir and Stephy.

The transport properties of an extended system driven by active reservoirs is an issue of paramount importance, which remains virtually unexplored. We address this issue in the context of energy transport between two active reservoirs connected by a chain of harmonic oscillators. The active reservoirs are modeled by a harmonic chain of overdamped run-and-tumble particles, the tumbling time-scales being a measure of the activity of the reservoir. These reservoirs satisfy a modified fluctuation-dissipation relation, which we illustrate by exactly computing the effective noise and dissipation kernels.  We study the energy transport through a chain of harmonic oscillators driven by two such active reservoirs of different activity and show that the stationary energy current shows remarkable features like negative differential conductivity and a non-trivial direction reversal. We also find nontrivial spatiotemporal velocity correlations in the stationary state, which distinguish this activity driven nonequibrium state from the usual thermally driven systems.  

We study the evolution of a two-state system that is monitored continuously but with interactions with the detector tuned so as to avoid the Zeno affect. The system is allowed to interact with a sequence of prepared probes. The post-interaction probe states are measured and this leads to a stochastic evolution of the system's state vector, which can be described by a single angle variable. We show that the system's effective evolution consists of a deterministic drift and a stochastic resetting to a fixed state at a rate that depends on the instantaneous state vector. The detector readout is a counting process. We obtain analytic results for the distribution of number of detector events and the time-evolution of the probability distribution.  

I will review analytical results that we have obtained in recent years for run and tumble particles (RTPs) in the presence of an external one-dimensional confining potential. In the large time limit, such systems generically reach a non-equilibrium stationary state (NESS) which is markedly different from the equilibrium Boltzmann-Gibbs result. I will start with the case of noninteracting RTPs, for which the NESS already exhibits quite riche behaviours, in particular transitions from passive-like to active-like phases. I will then present more recent results obtained for an interacting gas of RTPs, namely the "active Dyson Brownian motion", where the particles interact via a pairwise repulsive logarithmic potential.

In recent years, the behaviour of passive inclusions in dry scalar active systems has received a lot of attention. It is known that asymmetric objects generate long-range density modulation and slowly decaying currents of the active particles. This is turn leads to a stronger sensitivity to disorder of the phase behaviour compared to equilibrium. In this work, we seek to understand the fate of these results in the context of wet active matter, where the individual active particles are swimmers immersed in a momentum conserving viscous fluid. There, long-range hydrodynamic interactions, mediated by the fluid, emerge between the active particles and between the particles and the inclusion. We study the far-field density modulations and fluid flow generated by a passive asymmetric inclusion. The deviations from the dry case are found to be the most salient when the inclusion behaves as a momentum source, for example by being held at a fixed position externally. We show that the decay of the density field is characterized by an anomalous dimension that subtly depends on the competition between diffusive and convective transport.

(Joint work with Sriram Ramaswamy and Yariv Kafri)

XY models with randomly oriented fields have been studied in the context of disordered superconductors, whereas XY models with random crystal fields have been used widely to model and study physical systems such as amorphous magnets. In this talk we shall consider: 1) Infinite range XY model with random fields (XY-RF), and 2) Infinite range XY model with random crystal fields (XY-RCF). In both the cases we restrict to the scenario where the (crystal) fields are of equal magnitude but are oriented along random directions, distributed uniformly over a circle.

In particular, we will examine the distribution of spins in the zero temperature limit for both these models. In the case of the XY-RF, there is a first order phase transition as the strength of the random field is varied across the critical value, and the spins are distributed within a cone in the ordered phase and over a circle in the disordered phase. Whereas, in the case of the XY-RCF, there is no phase transition, and the spins are distributed within a cone which widens with the strength of the crystal field.

We consider a linear chain of run-and-tumble particles (RTP) interacting harmonically with nearest neighbors. For bulk tagged particles we  obtained a closed form expression for the mean-squared-displacement (MSD). Different scaling forms were extracted to show the   cross-over from  ballistic to diffusive to  sub-diffusive behavior, and  finite size effects were also studied.  The activity in this system is characterized by  the persistence time, defined as the inverse of the tumbling rate of a particle,  the passive limit corresponds to .  We also compute the full distribution of tagged particle displacement and find that it evolves from highly  non-Gaussian forms at early times to a Gaussian one at large times.  Finally we computed  static and dynamic correlations in the steady state for  the "stretch" variables, defined as relative displacements between nearest neighbors,   and pointed out interesting differences between the active and passive systems. For example, we show that the dynamical correlation shows very different scaling in the active case (small )  compared to the diffusive  scaling  for  the passive case.   Our results   easily extend to  other active particle models such as active Brownian particles and active Ornstein-Uhlenbeck particles.

Integrable many-particle systems are fine-tuned, but still arise widely. To illustrate their generalized hydrodynamics, the Calogero fluid will be used as prime example. The fluid consists of classical particles moving on the line and interacting through the repulsive 1/sinh^2 pair potential. Discussed are generalized Gibbs ensembles, the corresponding random Lax matrix, its density of states, and GGE averaged currents. The distinction between particle-based and soliton-based hydrodynamics is highlighted.

I shall present our exact result for the large deviation function of the density profile and of the current in the non-equilibrium stationary state of a one-dimensional symmetric exclusion process coupled to boundary reservoirs with varied coupling strength. These new results extend the earlier seminal works of Derrida and collaborators for the same model where rates at the boundaries are comparable to the bulk ones, to regimes where boundary rates are significantly slower or faster.

I shall then show how these new results can be reproduced using the fluctuating hydrodynamics description of the macroscopic fluctuation theory. In describing this hydrodynamic approach I shall present a derivation of the fluctuating hydrodynamics for the model and an exact solution of the variational problem for the large deviations of the density and of the current. An advantage of the hydrodynamics formulation is its generality. I shall conclude by presenting the results of large deviations for a class of diffusive systems, including those whose microscopic dynamics are non-integrable.

In several biological systems, such as bacterial cytoplasm, cytoskeleton-motor complexes and epithelial sheets of cells, self-propulsion or activity is found to fluidize states that exhibit characteristic glassy behaviour and jamming in the absence of activity. I will discuss some of the results of our recent studies of jamming in athermal models of dense active matter. We have used Langevin dynamics simulations to study the effects of activity in a two-dimensional dense glass-forming system of Lennard-Jones particles at zero temperature. In this model, the self-propulsion force is characterized by two parameters: its magnitude and the persistence time associated with the decorrelation of its direction. In our studies, we consider the limit of infinite persistence time. In this limit, the system exhibits a liquid state for large values of the self-propulsion force and a force-balanced jammed state if the self-propulsion force is smaller than a threshold value. The average kinetic energy in the liquid state increases with system size, suggesting the presence of long-range correlations. Each particle is found to have a non-zero average velocity in the direction of the self-propulsion force acting on it. A length scale extracted from spatial correlations of the velocity field increases with system size as a power law with exponent close to one. Spatial correlations of the self-propulsion forces also exhibit a similar length scale, indicating that these forces with randomly assigned directions at the beginning of the simulation self-organize to form a steady state in which particles with similar directions of self-propulsion forces come close to one another and move together. We also investigate how this active liquid approaches a force-balanced jammed state when the self-propulsion force is reduced to a small value. The jamming proceeds via a three-stage relaxation process whose timescale grows with the magnitude of the active force and the system size. We relate the dependence on the system size to the large correlation length observed in the liquid state. Properties of the jammed state obtained for small active forces depend strongly on the protocol used in its preparation. For a protocol designed to eliminate the presence of particles with two contacts in the jammed state, the distribution of contact forces exhibits scaling with the magnitude of the active force with a scaling function that has a gap at small values of the contact force. 

The investigation of the dependence of properties on spatial dimensionality has a rich history in statistical physics, most notably in the context of critical phenomena.  There has been an interest recently to investigate the glass transition and jamming behaviour over a range of spatial dimensions, to make comparisons with predictions of the mean field theory of the glass transition and attempts to extend it to finite dimensions. We investigate the dynamics of fluids of soft spheres for a wide range of temperatures and densities, using computer simulations, in spatial dimensions from 3 to 8. We specifically investigate the relationship between the glass transition and jamming densities in the hard sphere limit. Employing a scaling of density-temperature-dependent relaxation times, we precisely identify the density that is the ideal glass transition in the hard sphere limit, and compare it with the corresponding jamming densities. The glass transition occurs at progressively larger densities than the jamming point with increasing spatial dimensions, clearly distinguishing the two transition points.

(Monoj Adhikari, Smarajit Karmakar, Srikanth Sastry)

Recently developed generalised hydrodynamic theory has been quite successful in understanding non-equilibrium evolution in integrable systems like hard rods which do not relax to Gibbs state. For hard rod gas, I will discuss hydrodynamic evolution from certain non-equilibrium states and demonstrate if and when they approach generalised Gibbs state. I will also discuss how to see the effect of dissipation in such evolution. Finally, if time permits, I will discuss what happens when the microscopic integrability is broken by trapping the hard rods inside a confining potential.

We study the fluctuations of the integrated density current across the origin up to time T in one dimensional systems of non-interacting as well as interacting active particles. For non-interacting particles, we focus on the case of zero diffusion and study the differences between annealed and quenched initial conditions. We show that for the case of particles initiated with an initial bias in the positive direction, the fluctuations of the current at short times display a surprising difference: T versus T² behaviours respectively, with a √T behaviour emerging at large times. For the interacting case, we explore a lattice model of active particles with hard-core interactions that is amenable to an exact description within a fluctuating hydrodynamics framework. For the case of uniform initial profiles, we show that the second cumulant of the integrated current displays three regimes: an initial √T rise with a coefficient given by the symmetric simple exclusion process, a cross-over regime where the effects of activity increase the fluctuations, and a large time √T behavior. In the limit of zero diffusion for the interacting system, we show that the fluctuations once again exhibit a T² behavior at short times. Finally, we show that the results for non-interacting active particles are recovered for low densities.

I will discuss collective behaviour of N particles repelling each other via pairwise interaction potential and confined by an external trap. These family of models contain in them various well known systems of interest both in physics and mathematics such as one component plasma, Dyson’s log-gas, integrable Calogero-Moser model, hard rods to name a few. I will also present results in the presence of barriers and discuss our findings on edge fluctuations. I will then discuss gap statistics (analogous to the well known level spacing statistics) and some aspects of time dynamics and spatiotemporal spread of perturbations.

Shape asymmetry is the most abundant in nature and attracted great interest in recent research. The phenomenon is widely recognized: a free asymmetric Brownian particle displays anisotropic diffusion during short time intervals, which subsequently transitions to an isotropic diffusion pattern over longer time scales. We have further expanded this concept to incorporate active asymmetric particle characterized by a self-propelled velocity. We have also studied the approximations of different results of mean-square displacement in time-scale. We investigated diffusion dynamics for the free particle as well as the particle in a harmonic trap. Furthermore, we have studied the persistence probability p(t) of an active Brownian particle with shape asymmetry in two dimensions. The persistence probability is defined as the probability of a stochastic variable that has not changed its sign in a given fixed time interval. We have investigated two cases: (1) diffusion of a free active particle and (2) that of a harmonically trapped particle.

We report on reentrance in the infinite range random field Ising and Blume-Capel models, induced by an asymmetric bimodal random field distribution. The conventional continuous line of transitions between the paramagnetic and ferromagnetic phases, the lambda-line, is wiped away by the asymmetry. The phase diagram, then, consists of only first order transition lines that always end at ordered critical points. We will show that while for symmetric random field distributions there was no reentrance, the asymmetry in the random field results in a range of temperatures for which magnetisation shows reentrance.

Active turbulence — the spatio-temporally complex motion of a dense suspension of microorganisms such as bacteria — has gathered great traction recently as an intriguing class of emergent, complex flows, occurring in several living systems at the mesoscale, whose understanding lies at the interface of non-equilibrium physics and biology. However, are these low Reynolds number living flows really turbulent or just chaotic with structural, or even superficial, similarities with high Reynolds number (classical) inanimate turbulence? This is a vital question as the fingerprints of classical turbulence — universality, intermittency and chaos — makes it unique amongst the many different driven-dissipative systems. In this talk we address these questions with a focus on the issues of (approximate) scale-invariance, intermittency and maximally chaotic states and how they lead to anomalous diffusion in bacterial suspensions. In particular, we show the existing of a critical level of activity beyond which the physics of bacterial flows become universal, accompanied by maximally chaotic states which allow for efficient, Levy-walk mediated foraging strategies.