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Urna Basu

Publications and source records attributed to Urna Basu.

At least 37 records · Page 2Linked to original sources

arXiv2023

Dichotomous acceleration process in one dimension: Position fluctuations

We study the motion of a one-dimensional particle which reverses its direction of acceleration stochastically. We focus on two contrasting scenarios, where the waiting-times between two consecutive acceleration reversals are drawn from (i) an exponential distribution and (ii) a power-law distribution $ρ(τ)\sim τ^{-(1+α)}$. We compute the mean, variance and short-time distribution of the position $x(t)$ using a trajectory-based approach. We show that, while for the exponential waiting-time, $\langle x^2(t)\rangle\sim t^3$ at long times, for the power-law case, a non-trivial algebraic growth $\langle x^2(t)\rangle \sim t^{2ϕ(α)}$ emerges, where $ϕ(α)=2$, $(5-α)/2,$ and $3/2$ for $α<1,~1<α\leq 2$ and $α>2$, respectively. Interestingly, we find that the long-time position distribution in case (ii) is a function of the scaled variable $x/t^{ϕ(α)}$ with an $α$-dependent scaling function, which has qualitatively very different shapes for $α<1$ and $α>1$. In contrast, for case (i), the typical long-time fluctuations of position are Gaussian.

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arXiv2023

Long time behavior of run-and-tumble particles in two dimensions

We study the long-time asymptotic behavior of the position distribution of a run-and-tumble particle (RTP) in two dimensions and show that the distribution at a time $t$ can be expressed as a perturbative series in $(γt)^{-1}$, where $γ^{-1}$ is the persistence time of the RTP. We show that the higher order corrections to the leading order Gaussian distribution generically satisfy an inhomogeneous diffusion equation where the source term depends on the previous order solutions. The explicit solution of the inhomogeneous equation requires the position moments, and we develop a recursive formalism to compute the same.

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arXiv2023

Chirality Reversing Active Brownian Motion in Two Dimensions

We study the dynamics of a chirality reversing active Brownian particle, which models the chirality reversing active motion common in many microorganisms and microswimmers. We show that, for such a motion, the presence of the two time-scales set by the chirality reversing rate $γ$ and rotational diffusion constant $D_R$ gives rise to four dynamical regimes, namely, (I) $t \ll \text{min}(γ^{-1}, D_R^{-1})$, (II) $γ^{-1} \ll t \ll D_R^{-1}$, (III) $D_R^{-1} \ll t \ll γ^{-1}$ and (IV) $t \gg \text{max}(γ^{-1}, D_R^{-1})$, each showing different behaviour. The short-time regime (I) is characterized by a strongly anisotropic and non-Gaussian position distribution, which crosses over to a diffusive Gaussian behaviour in the long-time regime (IV) via an intermediate regime (II) or (III), depending on the relative strength of $γ$ and $D_R$. In regime (II), the chirality reversing active Brownian motion reduces to that of an ordinary active Brownian particle, with an effective rotation diffusion coefficient which depends on the angular velocity. Finally, we find that, the regime (III) is characterized by an effective chiral active Brownian motion.

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arXiv2023

Stationary states of activity-driven harmonic chains

We study the stationary state of a chain of harmonic oscillators driven by two active reservoirs at the two ends. These reservoirs exert correlated stochastic forces on the boundary oscillators which eventually leads to a nonequilibrium stationary state of the system. We consider three most well-known dynamics for the active force, namely, active Ornstein-Uhlenbeck process, run-and-tumble process and active Brownian process, all of which have exponentially decaying two-point temporal correlations but very different higher order fluctuations. We show that irrespective of the specific dynamics of the drive, the stationary velocity fluctuations are Gaussian in nature with a kinetic temperature which remains uniform in the bulk. Moreover, we find the emergence of an `equipartition of energy' in the bulk of the system -- the bulk kinetic temperature equals the bulk potential temperature in the thermodynamic limit. We also calculate the stationary distribution of the instantaneous energy current in the bulk which always shows a logarithmic divergence near the origin and asymmetric exponential tails. The signatures of specific active driving become visible in the behavior of the oscillators near the boundary. This is most prominent for the RTP and ABP driven chains where the boundary velocity distributions become non-Gaussian and current distribution has a finite cutoff.

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arXiv2022

Symmetric Exclusion Process under Stochastic Power-law Resetting

We study the behaviour of a symmetric exclusion process in the presence of non-Markovian stochastic resetting, where the configuration of the system is reset to a step-like profile at power-law waiting times with an exponent $α$. We find that the power-law resetting leads to a rich behaviour for the currents, as well as density profile. We show that, for any finite system, for $α<1$, the density profile eventually becomes uniform while for $α>1$, an eventual non-trivial stationary profile is reached. We also find that, in the limit of thermodynamic system size, at late times, the average diffusive current grows $\sim t^θ$ with $θ= 1/2$ for $α\le 1/2$, $θ= α$ for $1/2 < α\le 1$ and $θ=1$ for $α> 1$. We also analytically characterize the distribution of the diffusive current in the short-time regime using a trajectory-based perturbative approach. Using numerical simulations, we show that in the long-time regime, the diffusive current distribution follows a scaling form with an $α-$dependent scaling function. We also characterise the behaviour of the total current using renewal approach. We find that the average total current also grows algebraically $\sim t^ϕ$ where $ϕ= 1/2$ for $α\le 1$, $ϕ=3/2-α$ for $1 < α\le 3/2$, while for $α> 3/2$ the average total current reaches a stationary value, which we compute exactly. The variance of the total current also shows an algebraic growth with an exponent $Δ=1$ for $α\le 1$, and $Δ=2-α$ for $1 < α\le 2$, whereas it approaches a constant value for $α>2$. The total current distribution remains non-stationary for $α<1$, while, for $α>1$, it reaches a non-trivial and strongly non-Gaussian stationary distribution, which we also compute using the renewal approach.

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arXiv2022

Direction reversing active Brownian particle in a harmonic potential

We study the two-dimensional motion of an active Brownian particle of speed $v_0$, with intermittent directional reversals in the presence of a harmonic trap of strength $μ$. The presence of the trap ensures that the position of the particle eventually reaches a steady state where it is bounded within a circular region of radius $v_0/μ$, centered at the minimum of the trap. Due to the interplay between the rotational diffusion constant $D_R$, reversal rate $γ$, and the trap strength $μ$, the steady state distribution shows four different types of shapes, which we refer to as active-I & II, and passive-I & II phases. In the active-I phase, the weight of the distribution is concentrated along an annular region close to the circular boundary, whereas in active-II, an additional central diverging peak appears giving rise to a Mexican hat-like shape of the distribution. The passive-I is marked by a single Boltzmann-like centrally peaked distribution in the large $D_R$ limit. On the other hand, while the passive-II phase also shows a single central peak, it is distinguished from passive-I by a non-Boltzmann like divergence near the origin. We characterize these phases by calculating the exact analytical forms of the distributions in various limiting cases. In particular, we show that for $D_R\llγ$, the shape transition of the two-dimensional position distribution from active-II to passive-II occurs at $μ=γ$. We compliment these analytical results with numerical simulations beyond the limiting cases and obtain a qualitative phase diagram in the $(D_R,γ,μ^{-1})$ space.

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arXiv2022

Universal framework for the long-time position distribution of free active particles

Active particles self-propel themselves with a stochastically evolving velocity, generating a persistent motion leading to a non-diffusive behavior of the position distribution. Nevertheless, an effective diffusive behavior emerges at times much larger than the persistence time. Here we develop a general framework for studying the long-time behaviour for a class of active particle dynamics and illustrate it using the examples of run-and-tumble particle, active Ornstein-Uhlenbeck particle, active Brownian particle, and direction reversing active Brownian particle. Treating the ratio of the persistence-time to the observation time as the small parameter, we show that the position distribution generically satisfies the diffusion equation at the leading order. We further show that the sub-leading contributions, at each order, satisfies an inhomogeneous diffusion equation, where the source term depends on the previous order solutions. We explicitly obtain a few sub-leading contributions to the Gaussian position distribution. As a part of our framework, we also prescribe a way to find the position moments recursively and compute the first few explicitly for each model.

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arXiv2022

Activity driven transport in harmonic chains

How the transport properties of an extended system is affected by coupling to active reservoirs is a significant, yet virtually unexplored question. Here we address this issue in the context of energy transport between two active reservoirs connected by a chain of harmonic oscillators. The couplings to the reservoirs, which exert correlated stochastic forces on the boundary oscillators, lead to fascinating behavior of the energy current and kinetic temperature profile, which we compute exactly in the thermodynamic limit. We 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. For the example of a dichotomous active force, we find the physical origin of the NDC using nonequilibrium response formalism. It turns out that the kinetic temperature profile remains uniform at the bulk, and can be expressed in a form similar to the thermally driven case. We show that despite this apparent similarity, 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.

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arXiv2022

Effect of stochastic resetting on Brownian motion with stochastic diffusion coefficient

We study the dynamics of a Brownian motion with a diffusion coefficient which evolves stochastically. We first study this process in arbitrary dimensions and find the scaling form and the corresponding scaling function of the position distribution. We find that the tails of the distribution have exponential tails with a ballistic scaling. We then introduce the resetting dynamics where, at a constant rate, both the position and the diffusion coefficient are reset to zero. This eventually leads to a nonequilibrium stationary state, which we study in arbitrary dimensions. In stark contrast to ordinary Brownian motion under resetting, the stationary position distribution in one dimension has a logarithmic divergence at the origin. For higher dimensions, however, the divergence disappears and the distribution attains a dimension-dependent constant value at the origin, which we compute exactly. The distribution has a generic stretched exponential tail in all dimensions. We also study the approach to the stationary state and find that, as time increases, an inner core region around the origin attains the stationary state, while the outside region still has a transient distribution -- this inner stationary region grows $\sim t^2$, i.e., with a constant acceleration, much faster than ordinary Brownian motion.

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arXiv2021

Active Brownian Motion with Directional Reversals

Active Brownian motion with intermittent direction reversals are common in a class of bacteria like {\it Myxococcus xanthus} and {\it Pseudomonas putida}. We show that, for such a motion in two dimensions, the presence of the two time scales set by the rotational diffusion constant $D_R$ and the reversal rate $γ$ gives rise to four distinct dynamical regimes: (I) $t\ll \min (γ^{-1}, D_R^{-1}),$ (II) $γ^{-1}\ll t\ll D_R^{-1}$, (III) $D_R^{-1} \ll t \ll γ^{-1}$, and (IV) $t\gg \max (γ^{-1}$, $D_R^{-1})$, showing distinct behaviors. We characterize these behaviors by analytically computing the position distribution and persistence exponents. The position distribution shows a crossover from a strongly non-diffusive and anisotropic behavior at short-times to a diffusive isotropic behavior via an intermediate regime (II) or (III). In regime (II), we show that, the position distribution along the direction orthogonal to the initial orientation is a function of the scaled variable $z\propto x_{\perp}/t$ with a non-trivial scaling function, $f(z)=(2π^3)^{-1/2}Γ(1/4+iz)Γ(1/4-iz)$. Furthermore, by computing the exact first-passage time distribution, we show that a novel persistence exponent $α=1$ emerges due to the direction reversal in this regime.

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arXiv2020

Coarse-grained Second Order Response Theory

While linear response theory, manifested by the fluctuation dissipation theorem, can be applied at any level of coarse graining, nonlinear response theory is fundamentally of microscopic nature. For perturbations of equilibrium systems, we develop an exact theoretical framework for analyzing the nonlinear (second order) response of coarse grained observables to time-dependent perturbations, using a path-integral formalism. The resulting expressions involve correlations of the observable with coarse grained path weights. The time symmetric part of these weights depends on the paths and perturbation protocol in a complex manner; in addition, the absence of Markovianity prevents slicing of the coarse-grained path integral. We show that these difficulties can be overcome and the response function can be expressed in terms of path weights corresponding to a single-step perturbation. This formalism thus leads to an extrapolation scheme where measuring linear responses of coarse-grained variables suffices to determine their second order response. We illustrate the validity of the formalism with an exactly solvable four-state model and the near-critical Ising model.

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arXiv2020

Run-and-Tumble particles in Two-dimensions under Stochastic Resetting

We study the effect of stochastic resetting on a run and tumble particle (RTP) in two spatial dimensions. We consider a resetting protocol which affects both the position and orientation of the RTP: with a constant rate the particle undergoes a positional resetting to a fixed point in space and orientation randomization. We compute the radial and $x$-marginal stationary state distributions and show that while the former approaches a constant value as $r \to 0$, the latter diverges logarithmically as $x \to 0$. On the other hand, both the marginal distributions decay exponentially with the same exponent far away from the origin. We also study the temporal relaxation of the RTP and show that the position distribution undergoes a dynamical transition to a stationary state. We also study the first passage properties of the RTP in the presence of the resetting and show that the optimization of the resetting rate can minimize the mean first passage time. We also give a brief discussion on the stationary states for resetting to the initial position with fixed orientation.

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arXiv2020

Active Brownian Motion in two-dimensions under Stochastic Resetting

We study the position distribution of an active Brownian particle (ABP) in the presence of stochastic resetting in two spatial dimensions. We consider three different resetting protocols : (I) where both position and orientation of the particle are reset, (II) where only the position is reset, and (III) where only the orientation is reset with a certain rate $r.$ We show that in the first two cases the ABP reaches a stationary state. Using a renewal approach, we calculate exactly the stationary marginal position distributions in the limiting cases when the resetting rate $r$ is much larger or much smaller than the rotational diffusion constant $D_R$ of the ABP. We find that, in some cases, for a large resetting rate, the position distribution diverges near the resetting point; the nature of the divergence depends on the specific protocol. For the orientation resetting, there is no stationary state, but the motion changes from a ballistic one at short-times to a diffusive one at late times. We characterize the short-time non-Gaussian marginal position distributions using a perturbative approach.

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arXiv2020

Active velocity processes with suprathermal stationary distributions and long-time tails

When a particle moves through a spatially-random force field, its momentum may change at a rate which grows with its speed. Suppose moreover that a thermal bath provides friction which gets weaker for large speeds, enabling high-energy localization. The result is a unifying framework for the emergence of heavy tails in the velocity distribution, relevant for understanding the power-law decay in the electron velocity distribution of space plasma or more generally for explaining non-Maxwellian behavior of driven gases. We also find long-time tails in the velocity autocorrelation, indicating persistence at large speeds for a wide range of parameters and implying superdiffusion of the position variable.

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arXiv2020

Run-and-tumble particles in two-dimensions : Marginal position distributions

We study a set of Run-and-tumble particle (RTP) dynamics in two spatial dimensions. In the first case of the orientation θ of the particle can assume a set of n possible discrete values while in the second case θ is a continuous variable. We calculate exactly the marginal position distributions for n = 3,4 and the continuous case and show that in all the cases the RTP shows a cross-over from a ballistic to diffusive regime. The ballistic regime is a typical signature of the active nature of the systems and is characterized by non-trivial position distributions which depends on the specific model. We also show that, the signature of activity at long-times can be found in the atypical fluctuations which we also characterize by computing the large deviation functions explicitly.

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arXiv2020

Active Brownian Motion in Two Dimensions

We study the dynamics of a single active Brownian particle (ABP) in two spatial dimensions. The ABP has an intrinsic time scale $D_R^{-1}$ set by the rotational diffusion constant $D_R$. We show that, at short-times $t \ll D_R^{-1}$, the presence of `activness' results in a strongly anisotropic and non-diffusive dynamics in the $(xy)$ plane. We compute exactly the marginal distributions of the $x$ and $y$ position coordinates along with the radial distribution, which are all shown to be non-Brownian. In addition, we show that, at early times, the ABP has anomalous first-passage properties, characterized by non-Brownian exponents.

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arXiv2020

Zero-current Nonequilibrium State in Symmetric Exclusion Process with Dichotomous Stochastic Resetting

We study the dynamics of symmetric exclusion process (SEP) in the presence of stochastic resetting to two possible specific configurations -- with rate $r_1$ (respectively, $r_2$) the system is reset to a step-like configuration where all the particles are clustered in the left (respectively, right) half of the system. We show that this dichotomous resetting leads to a range of rich behaviour, both dynamical and in the stationary state. We calculate the exact stationary profile in the presence of this dichotomous resetting and show that the diffusive current grows linearly in time, but unlike the resetting to a single configuration, the current can have negative average value in this case. For $r_1=r_2,$ the average current vanishes, and density profile becomes flat in the stationary state, similar to the equilibrium SEP. However, the system remains far from equilibrium and we characterize the nonequilibrium signatures of this `zero-current state'. We show that both the spatial and temporal density correlations in this zero-current state are radically different than in equilibrium SEP. We also study the behaviour of this zero-current state under an external perturbation and demonstrate that its response differs drastically from that of equilibrium SEP -- while a small driving field generates a current which grows as $\sqrt{t}$ in the absence of resetting, the zero-current state in the presence of dichotomous resetting shows a current $\sim t$ under the same perturbation.

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arXiv2019

Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap

We study the motion of a one-dimensional run-and-tumble particle with three discrete internal states in the presence of a harmonic trap of stiffness $μ.$ The three internal states, corresponding to positive, negative and zero velocities respectively, evolve following a jump process with rate $γ$. We compute the stationary position distribution exactly for arbitrary values of $μ$ and $γ$ which turns out to have a finite support on the real line. We show that the distribution undergoes a shape-transition as $β=γ/μ$ is changed. For $β<1,$ the distribution has a double-concave shape and shows algebraic divergences with an exponent $(β-1)$ both at the origin and at the boundaries. For $β>1,$ the position distribution becomes convex, vanishing at the boundaries and with a single, finite, peak at the origin. We also show that for the special case $β=1,$ the distribution shows a logarithmic divergence near the origin while saturating to a constant value at the boundaries.

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