arXiv subjects

Anupam Kundu

Publications and source records attributed to Anupam Kundu.

At least 37 records · Page 2Linked to original sources

arXiv2024

Harmonically trapped inertial run-and-tumble particle in one dimension

We study the nonequilibrium stationary state of a one-dimensional inertial run-and-tumble particle (IRTP) trapped in a harmonic potential. We find that the presence of inertia leads to two distinct dynamical scenarios, namely, overdamped and underdamped, characterized by the relative strength of the viscous and the trap time-scales. We also find that inertial nature of the active dynamics leads to the particle being confined in specific regions of the phase plane in the overdamped and underdamped cases, which we compute analytically. Moreover, the interplay of the inertial and active time-scales gives rise to several sub-regimes, which are characterized by very different behaviour of position and velocity fluctuations of the IRTP. In particular, in the underdamped regime, both the position and velocity undergoes transitions from a novel multi-peaked structure in the strongly active limit to a single peaked Gaussian-like distribution in the passive limit. On the other hand, in the overdamped scenario, the position distribution shows a transition from a U-shape to a dome-shape, as activity is decreased. Interestingly, the velocity distribution in the overdamped scenario shows two transitions -- from a single-peaked shape with an algebraic divergence at the origin in the strongly active regime to a double peaked one in the moderately active regime to a dome-shaped one in the passive regime.

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arXiv2024

Full counting statistics of 1d short-range Riesz gases in confinement

We investigate the full counting statistics (FCS) of a harmonically confined 1d short-range Riesz gas consisting of $N$ particles in equilibrium at finite temperature. The particles interact with each other through a repulsive power-law interaction with an exponent $k>1$ which includes the Calogero-Moser model for $k=2$. We examine the probability distribution of the number of particles in a finite domain $[-W, W]$ called number distribution, denoted by $\mathcal{N}(W, N)$. We analyze the probability distribution of $\mathcal{N}(W, N)$ and show that it exhibits a large deviation form for large $N$ characterised by a speed $N^{\frac{3k+2}{k+2}}$ and by a large deviation function of the fraction $c = \mathcal{N}(W, N)/N$ of the particles inside the domain and $W$. We show that the density profiles that create the large deviations display interesting shape transitions as one varies $c$ and $W$. This is manifested by a third-order phase transition exhibited by the large deviation function that has discontinuous third derivatives. Monte-Carlo (MC) simulations show good agreement with our analytical expressions for the corresponding density profiles. We find that the typical fluctuations of $\mathcal{N}(W, N)$, obtained from our field theoretic calculations are Gaussian distributed with a variance that scales as $N^{ν_k}$, with $ν_k = (2-k)/(2+k)$. We also present some numerical findings on the mean and the variance. Furthermore, we adapt our formalism to study the index distribution (where the domain is semi-infinite $(-\infty, W])$, linear statistics (the variance), thermodynamic pressure and bulk modulus.

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arXiv2024

Universal distribution of the number of minima for random walks and Lévy flights

We compute exactly the full distribution of the number $m$ of local minima in a one-dimensional landscape generated by a random walk or a Lévy flight. We consider two different ensembles of landscapes, one with a fixed number of steps $N$ and the other till the first-passage time of the random walk to the origin. We show that the distribution of $m$ is drastically different in the two ensembles (Gaussian in the former case, while having a power-law tail in the latter $m^{-3/2}$ in the latter case). However, the most striking aspect of our results is that, in each case, the distribution is completely universal for all $m$ (and not just for large $m$), i.e., independent of the jump distribution in the random walk. This means that the distributions are exactly identical for Lévy flights and random walks with finite jump variance. Our analytical results are in excellent agreement with our numerical simulations.

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arXiv2024

Observation of multiple attractors and diffusive transport in a periodically driven Klein-Gordon chain

We consider a Klein-Gordon chain that is periodically driven at one end and has dissipation at one or both boundaries. An interesting numerical observation in a recent study~[arXiv:2209.03977] was that for driving frequency in the phonon band, there is a range of values of the driving amplitude $F_d\in (F_1, F_2)$ over which the energy current remains constant. In this range, the system exhibits a "resonant nonlinear wave" (RNW) mode of energy transmission which is a time and space periodic solution. It was noted that the range $(F_1,F_2)$, for which the RNW mode occurs, shrinks with increasing system size $N$ and disappears eventually. Remarkably, we find that the RNW mode is in fact a stable solution even for $F_d$ much larger than $F_2$ and quite large $N$ ($\approx 1000$). For $F_d>F_{2}$, there exists a second attractor which is chaotic. Both attractors have finite basins of attraction and can be reached by appropriate choice of initial conditions. Corresponding to the two attractors for large $F_d$, the system can now be in two nonequilibrium steady states. We improve the perturbative treatment of [arXiv:2209.03977] for the RNW mode by including the contributions of the third harmonics. We also consider the effect of thermal noise at the boundaries and find that the RNW mode is stable for small temperatures. Finally, we present results for a different driving protocol studied in [arXiv:2205.03839] where $F_d$ is taken to scale with system size as $N^{-1/2}$ and there is dissipation only at the non-driven end. We find that the steady state can be characterized by Fourier's law as in [arXiv:2205.03839] for a stochastic model. We point out interesting differences that occur since our dynamics is nonlinear and Hamiltonian. Our results suggest the intriguing possibility of observing the high current carrying RNW phase in experiments by careful preparation of initial conditions.

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arXiv2023

Thermalization and hydrodynamics in an interacting integrable system: the case of hard rods

We consider the relaxation of an initial non-equilibrium state in a one-dimensional fluid of hard rods. Since it is an interacting integrable system, we expect it to reach the Generalized Gibbs Ensemble (GGE) at long times for generic initial conditions. Here we show that there exist initial conditions for which the system does not reach GGE even at very long times and in the thermodynamic limit. In particular, we consider an initial condition of uniformly distributed hard-rods in a box with the left half having particles with a singular velocity distribution (all moving with unit velocity) and the right half particles in thermal equilibrium. We find that the density profile for the singular component does not spread to the full extent of the box and keeps moving with a fixed effective speed at long times. We show that such density profiles can be well described by the solution of the Euler equations almost everywhere except at the location of the shocks, where we observe slight discrepancies due to dissipation arising from the initial fluctuations of the thermal background. To demonstrate this effect of dissipation analytically, we consider a second initial condition with a single particle at the origin with unit velocity in a thermal background. We find that the probability distribution of the position of the unit velocity quasi-particle has diffusive spreading which can be understood from the solution of the Navier-Stokes equation of the hard rods. Finally, we consider an initial condition with a spread in velocity distribution for which we show convergence to GGE. Our conclusions are based on molecular dynamics simulations supported by analytical arguments.

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arXiv2023

Non-local linear response in anomalous transport

Anomalous heat transport observed in low dimensional classical systems is associated to super-diffusive spreading of space-time correlation of the conserved fields in the system. This leads to non-local linear response relation between the heat current and the local temperature gradient in non-equilibrium steady state. This relation provides a generalisation of Fourier's law of heat transfer and is characterised by a non-local kernel operator which is related to fractional operators describing super-diffusion. The kernel is essentially proportional, in appropriate hydrodynamic scaling limit, to the time integral of the space-time correlations of local currents in equilibrium. In finite size systems, the time integral of correlation of microscopic currents at different locations over infinite duration is independent of the locations. On the other hand the kernel operator is space-dependent. We demonstrate that the resolution of this apparent puzzle appears through taking appropriate combination of limits of large system size and large integration time duration. Our study shows the importance of taking the limits in proper way even for (open) systems connected to reservoirs. In particular we reveal how to extract the kernel operator from simulation data of microscopic current-current correlation. For two model systems exhibiting anomalous transport, we provide direct and detailed numerical verification of the kernel operators.

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arXiv2023

Unusual ergodic and chaotic properties of trapped hard rods

We investigate ergodicity, chaos and thermalization for a one-dimensional classical gas of hard rods confined to an external quadratic or quartic trap, which breaks microscopic integrability. To quantify the strength of chaos in this system, we compute its maximal Lyapunov exponent numerically. The approach to thermal equilibrium is studied by considering the time evolution of particle position and velocity distributions and comparing the late-time profiles with the Gibbs state. Remarkably, we find that quadratically trapped hard rods are highly non-ergodic and do not resemble a Gibbs state even at extremely long times, despite compelling evidence of chaos for four or more rods. On the other hand, our numerical results reveal that hard rods in a quartic trap exhibit both chaos and thermalization, and equilibrate to a Gibbs state as expected for a nonintegrable many-body system.

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arXiv2023

Super-diffusion and crossover from diffusive to anomalous transport in a one-dimensional system

We study transport in a one-dimensional lattice system with two conserved quantities -- `volume' and energy. Considering a slowly evolving local equilibrium state that is slightly deviated from an underlying global equilibrium, we estimate the correction to the local equilibrium distribution. This correction arises mainly through the space-time correlations of some local currents. In the continuum limit, we show that the local equilibrium distribution along with the correction yields drift-diffusion equation for the `volume' and super-diffusion equation for the energy in the linear response regime as macroscopic hydrodynamics as one would obtain from non-linear fluctuating hydrodynamic theory. We find explicit expression of the super-diffusion equation. Further, we find diffusive correction to the super-diffusive evolution. Such a correction allows us to study a crossover from diffusive to anomalous transport. We demonstrate this crossover numerically through the spreading of an initially localized heat pulse in equilibrium as well as through the system size scaling of the stationary current in non-equilibrium steady state.

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arXiv2023

Boltzmann entropy of a freely expanding quantum ideal gas

We study the time evolution of the Boltzmann entropy of a microstate during the non-equilibrium free expansion of a one-dimensional quantum ideal gas. This quantum Boltzmann entropy, $S_B$, essentially counts the "number" of independent wavefunctions (microstates) giving rise to a specified macrostate. It generally depends on the choice of macrovariables, such as the type and amount of coarse-graining, specifying a non-equilibrium macrostate of the system, but its extensive part agrees with the thermodynamic entropy in thermal equilibrium macrostates. We examine two choices of macrovariables: the $U$-macrovariables are local observables in position space, while the $f$-macrovariables also include structure in momentum space. For the quantum gas, we use a non-classical choice of the $f$-macrovariables. For both choices, the corresponding entropies $s_B^f$ and $s_B^U$ grow and eventually saturate. As in the classical case, the growth rate of $s_B^f$ depends on the momentum coarse-graining scale. If the gas is initially at equilibrium and is then released to expand to occupy twice the initial volume, the per-particle increase in the entropy for the $f$-macrostate, $Δs_B^f$, satisfies $\log{2}\leqΔs_B^f\leq 2\log{2}$ for fermions, and $0\leqΔs_B^f\leq\log{2}$ for bosons. For the same initial conditions, the change in the entropy $Δs_B^U$ for the $U$-macrostate is greater than $Δs_B^f$ when the gas is in the quantum regime where the final stationary state is not at thermal equilibrium.

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arXiv2022

Boltzmann's entropy during free expansion of an interacting ideal gas

In this work we study the evolution of Boltzmann's entropy in the context of free expansion of a one dimensional interacting gas inside a box. Boltzmann's entropy is defined for single microstates and is given by the phase-space volume occupied by microstates with the same value of macrovariables which are coarse-grained physical observables. We demonstrate the idea of typicality in the growth of the Boltzmann's entropy for two choices of macro-variables -- the single particle phase space distribution and the hydrodynamic fields. Due to the presence of interaction, the growth curves for both these entropies are observed to converge to a monotonically increasing limiting curve, on taking the appropriate order of limits, of large system size and small coarse graining scale. Moreover, we observe that the limiting growth curves for the two choices of entropies are identical as implied by local thermal equilibrium. We also discuss issues related to finite size and finite coarse gaining scale which lead interesting features such as oscillations in the entropy growth curve. We also discuss shocks observed in the hydrodynamic fields.

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arXiv2022

Finite temperature equilibrium density profiles of integrable systems in confining potentials

We study the equilibrium density profile of particles in two one-dimensional classical integrable models, namely hard rods and the hyperbolic Calogero model, placed in confining potentials. For both of these models the inter-particle repulsion is strong enough to prevent particle trajectories from intersecting. We use field theoretic techniques to compute the density profile and their scaling with system size and temperature, and compare them with results from Monte-Carlo simulations. In both cases we find good agreement between the field theory and simulations. We also consider the case of the Toda model in which inter-particle repulsion is weak and particle trajectories can cross. In this case, we find that a field theoretic description is ill-suited due to the lack of a thermodynamic length scale. The density profiles for the Toda model obtained from Monte-Carlo simulations can be understood by studying the analytically tractable harmonic chain model (Hessian approximation of the Toda model). For the harmonic chain model one can derive an exact expression for the density that shines light on some of the qualitative features of the Toda model in a quadratic trap. Our work provides an analytical approach towards understanding the equilibrium properties for interacting integrable systems in confining traps.

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arXiv2022

Filling an empty lattice by local injection of quantum particles

We study the quantum dynamics of filling an empty lattice of size $L$, by connecting it locally with an equilibrium thermal bath that injects non-interacting bosons or fermions. We adopt four different approaches, namely (i) direct exact numerics, (ii) Redfield equation, (iii) Lindblad equation, and (iv) quantum Langevin equation -- which are unique in their ways for solving the time dynamics and the steady-state. Our setup offers a simplistic platform to understand fundamental aspects of dynamics and approach to thermalization. The quantities of interest that we consider are the spatial density profile and the total number of bosons/fermions in the lattice. The spatial spread is ballistic in nature and the local occupation eventually settles down owing to equilibration. The ballistic spread of local density admits a universal scaling form. We show that this universality is only seen when the condition of detailed balance is satisfied by the baths. The difference between bosons and fermions shows up in the early time growth rate and the saturation values of the profile. The techniques developed here are applicable to systems in arbitrary dimensions and for arbitrary geometries.

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arXiv2022

Mass fluctuations in Random Average Transfer Process in open set-up

We define a new mass transport model on a one-dimensional lattice of size $N$ with continuous masses at each site. The lattice is connected to mass reservoirs of different `chemical potentials' at the two ends. The mass transfer dynamics in the bulk is equivalent to the dynamics of the gaps between particles in the Random Average Process. In the non-equilibrium steady state, we find that the multi-site arbitrary order cumulants of the masses can be expressed as an expansion in powers of $1/N$ where at each order the cumulants have a scaling form. We introduce a novel operator approach which allows us to compute these scaling functions at different orders of $1/N$. Moreover, this approach reveals that, to express the scaling functions for higher order cumulants completely one requires all lower order multi-site cumulants. This is in contrast to the Wick's theorem in which all higher order cumulants are expressed solely in terms of two-site cumulants. We support our results with evidence from Monte-Carlo simulations.

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arXiv2022

Entropy growth during free expansion of an ideal gas

To illustrate Boltzmann's construction of an entropy function that is defined for a microstate of a macroscopic system, we present here the simple example of the free expansion of a one dimensional gas of non-interacting point particles. The construction requires one to define macrostates, corresponding to macroscopic variables. We define a macrostate $M$ by specifying the fraction of particles in rectangular boxes $Δx Δv$ of the single particle position-velocity space $\{x,v\}$. We verify that when the number of particles is large the Boltzmann entropy, $S_B(t)$, of a typical microstate of a nonequilibrium ensemble coincides with the Gibbs entropy of the coarse-grained time-evolved one-particle distribution associated with this ensemble. $S_B(t)$ approaches its maximum possible value for the dynamical evolution of the given initial state. The rate of approach depends on the size of $Δv$ in the definition of the macrostate, going to zero at any fixed time $t$ when $Δv \to 0$. Surprisingly the different curves $S_B(t)$ collapse when time is scaled with $Δv$ as: $t \sim τ/Δv$. We find an explicit expression for $S_B(τ)$ in the limit $Δv \to 0$. We also consider a different, more hydrodynamical, definition of macrostates for which $S_B(t)$ is monotone increasing, unlike the previous one which has small decaying oscillations near its maximum value. Our system is non-ergodic, non-chaotic and non-interacting; our results thus illustrate that these concepts are not as relevant as sometimes claimed, for observing macroscopic irreversibility and entropy increase. Rather, the notions of initial conditions, typicality, large numbers and coarse-graining are the important factors. We demonstrate these ideas through extensive simulations as well as analytic results.

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arXiv2022

Extremal statistics of a one dimensional run and tumble particle with an absorbing wall

We study the extreme value statistics of a run and tumble particle (RTP) in one dimension till its first passage to the origin starting from the position $x_0~(>0)$. This model has recently drawn a lot of interest due to its biological application in modelling the motion of certain species of bacteria. Herein, we analytically study the exact time-dependent propagators for a single RTP in a finite interval with absorbing conditions at its two ends. By exploiting a path decomposition technique, we use these propagators appropriately to compute the joint distribution $\mathscr{P}(M,t_m)$ of the maximum displacement $M$ till first-passage and the time $t_m$ at which this maximum is achieved exactly. The corresponding marginal distributions $\mathbb{P}_M(M)$ and $P_M(t_m)$ are studied separately and verified numerically. In particular, we find that the marginal distribution $P_M(t_m)$ has interesting asymptotic forms for large and small $t_m$. While for small $t_m$, the distribution $P_M(t_m)$ depends sensitively on the initial velocity direction $σ_i$ and is completely different from the Brownian motion, the large $t_m$ decay of $P_M(t_m)$ is same as that of the Brownian motion although the amplitude crucially depends on the initial conditions $x_0$ and $σ_i$. We verify all our analytical results to high precision by numerical simulations.

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arXiv2022

Run-and-tumble particle in one-dimensional confining potential: Steady state, relaxation and first passage properties

We study the dynamics of a one-dimensional run and tumble particle subjected to confining potentials of the type $V(x) = α\, |x|^p$, with $p>0$. The noise that drives the particle dynamics is telegraphic and alternates between $\pm 1$ values. We show that the stationary probability density $P(x)$ has a rich behavior in the $(p, α)$-plane. For $p>1$, the distribution has a finite support in $[x_-,x_+]$ and there is a critical line $α_c(p)$ that separates an active-like phase for $α> α_c(p)$ where $P(x)$ diverges at $x_\pm$, from a passive-like phase for $α< α_c(p)$ where $P(x)$ vanishes at $x_\pm$. For $p<1$, the stationary density $P(x)$ collapses to a delta function at the origin, $P(x) = δ(x)$. In the marginal case $p=1$, we show that, for $α< α_c$, the stationary density $P(x)$ is a symmetric exponential, while for $α> α_c$, it again is a delta function $P(x) = δ(x)$. For the special cases $p=2$ and $p=1$, we obtain exactly the full time-dependent distribution $P(x,t)$, that allows us to study how the system relaxes to its stationary state. In addition, in these two cases, we also study analytically the full distribution of the first-passage time to the origin. Numerical simulations are in complete agreement with our analytical predictions.

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arXiv2022

Gap Statistics for Confined Particles with Power-Law Interactions

We consider the $N$ particle classical Riesz gas confined in a one-dimensional external harmonic potential with power law interaction of the form $1/r^k$ where $r$ is the separation between particles. As special limits it contains several systems such as Dyson's log-gas ($k\to 0^+$), Calogero-Moser model ($k=2$), 1d one component plasma ($k=-1$) and the hard-rod gas ($k\to \infty$). Despite its growing importance, only large-$N$ field theory and average density profile are known for general $k$. In this Letter, we study the fluctuations in the system by looking at the statistics of the gap between successive particles. This quantity is analogous to the well-known level spacing statistics which is ubiquitous in several branches of physics. We show that the variance goes as $N^{-b_k}$ and we find the $k$ dependence of $b_k$ via direct Monte Carlo simulations. We provide supporting arguments based on microscopic Hessian calculation and a quadratic field theory approach. We compute the gap distribution and study its system size scaling. Except in the range $-1 -2$ with both Gaussian and non-Gaussian scaling forms.

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arXiv2022

Crossover behaviours exhibited by fluctuations and correlations in a chain of active particles

We study motion of tagged particles in a harmonic chain of active particles. We consider three models of active particle dynamics - run and tumble particle, active Ornstein-Uhlenbeck particle and active Brownian particle. We investigate the variance, auto correlation, covariance and unequal time cross correlation of two tagged particles. For all three models, we observe that the mean squared displacement undergoes a crossover from the super-diffusive $\sim t^μ$ scaling for $t \ll τ_A$ ($τ_A$ being the time scale arising due to the activity) to the sub-diffusive $\sim \sqrt{t}$ scaling for $t \gg τ_A$, where $μ= \frac{3}{2}$ for RTP, $μ= \frac{5}{2}$ for AOUP. For the $x$ and $y$-coordinates of ABPs we get $μ=\frac{7}{2}$ and $μ=\frac{5}{2}$ respectively. We show that these crossover behaviours in each case can be described by appropriate crossover function that connects these two scaling regimes. We compute these crossover functions explicitly. In addition, we also find that the equal and unequal time auto and cross correlations obey interesting scaling forms in the appropriate limits of the observation time $t$. The associated scaling functions are rigorously derived in all cases. All our analytical results are well supported by the numerical simulations.

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