arXiv subjects

Satya N. Majumdar

Publications and source records attributed to Satya N. Majumdar.

At least 145 records · Page 8Linked to original sources

arXiv2020

Kernels for noninteracting fermions via a Green's function approach with applications to step potentials

The quantum correlations of $N$ noninteracting spinless fermions in their ground state can be expressed in terms of a two-point function called the kernel. Here we develop a general and compact method for computing the kernel in a general trapping potential in terms of the Green's function for the corresponding single particle Schrödinger equation. For smooth potentials the method allows a simple alternative derivation of the local density approximation for the density and of the sine kernel in the bulk part of the trap in the large $N$ limit. It also recovers the density and the kernel of the so-called {\em Airy gas} at the edge. This method allows to analyse the quantum correlations in the ground state when the potential has a singular part with a fast variation in space. For the square step barrier of height $V_0$, we derive explicit expressions for the density and for the kernel. For large Fermi energy $μ>V_0$ it describes the interpolation between two regions of different densities in a Fermi gas, each described by a different sine kernel. Of particular interest is the {\em critical point} of the square well potential when $μ=V_0$. In this critical case, while there is a macroscopic number of fermions in the lower part of the step potential, there is only a finite $O(1)$ number of fermions on the shoulder, and moreover this number is independent of $μ$. In particular, the density exhibits an algebraic decay $\sim 1/x^2$, where $x$ is the distance from the jump. Furthermore, we show that the critical behaviour around $μ= V_0$ exhibits universality with respect with the shape of the barrier. This is established (i) by an exact solution for a smooth barrier (the Woods-Saxon potential) and (ii) by establishing a general relation between the large distance behavior of the kernel and the scattering amplitudes of the single-particle wave-function.

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arXiv2020

Statistics of the Number of Records for Random Walks and Lévy Flights on a ${1D}$ Lattice

We study the statistics of the number of records $R_n$ for a symmetric, $n$-step, discrete jump process on a $1D$ lattice. At a given step, the walker can jump by arbitrary lattice units drawn from a given symmetric probability distribution. This process includes, as a special case, the standard nearest neighbor lattice random walk. We derive explicitly the generating function of the distribution $P(R_n)$ of the number of records, valid for arbitrary discrete jump distributions. As a byproduct, we provide a relatively simple proof of the generalized Sparre Andersen theorem for the survival probability of a random walk on a line, with discrete or continuous jump distributions. For the discrete jump process, we then derive the asymptotic large $n$ behavior of $P(R_n)$ as well as of the average number of records $E(R_n)$. We show that unlike the case of random walks with symmetric and continuous jump distributions where the record statistics is strongly universal (i.e., independent of the jump distribution for all $n$), the record statistics for lattice walks depends on the jump distribution for any fixed $n$. However, in the large $n$ limit, we show that the distribution of the scaled record number $R_n/E(R_n)$ approaches a universal, half-Gaussian form for any discrete jump process. The dependence on the jump distribution enters only through the scale factor $E(R_n)$, which we also compute in the large $n$ limit for arbitrary jump distributions. We present explicit results for a few examples and provide numerical checks of our analytical predictions.

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arXiv2020

Position distribution in a generalised run and tumble process

We study a class of stochastic processes of the type $\frac{d^n x}{dt^n}= v_0\, σ(t)$ where $n>0$ is a positive integer and $σ(t)=\pm 1$ represents an `active' telegraphic noise that flips from one state to the other with a constant rate $γ$. For $n=1$, it reduces to the standard run and tumble process for active particles in one dimension. This process can be analytically continued to any $n>0$ including non-integer values. We compute exactly the mean squared displacement at time $t$ for all $n>0$ and show that at late times while it grows as $\sim t^{2n-1}$ for $n>1/2$, it approaches a constant for $n<1/2$. In the marginal case $n=1/2$, it grows very slowly with time as $\sim \ln t$. Thus the process undergoes a {\em localisation} transition at $n=1/2$. We also show that the position distribution $p_n(x,t)$ remains time-dependent even at late times for $n\ge 1/2$, but approaches a stationary time-independent form for $n<1/2$. The tails of the position distribution at late times exhibit a large deviation form, $p_n(x,t)\sim \exp\left[-γ\, t\, Φ_n\left(\frac{x}{x^*(t)}\right)\right]$, where $x^*(t)= v_0\, t^n/Γ(n+1)$. We compute the rate function $Φ_n(z)$ analytically for all $n>0$ and also numerically using importance sampling methods, finding excellent agreement between them. For three special values $n=1$, $n=2$ and $n=1/2$ we compute the exact cumulant generating function of the position distribution at all times $t$.

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arXiv2020

Toward the full short-time statistics of an active Brownian particle on the plane

We study the position distribution of a single active Brownian particle (ABP) on the plane. We show that this distribution has a compact support, the boundary of which is an expanding circle. We focus on a short-time regime and employ the optimal fluctuation method (OFM) to study large deviations of the particle position coordinates $x$ and $y$. We determine the optimal paths of the ABP, conditioned on reaching specified values of $x$ and $y$, and the large deviation functions of the marginal distributions of $x$, and of $y$. These marginal distributions match continuously with "near tails" of the $x$ and $y$ distributions of typical fluctuations, studied earlier. We also calculate the large deviation function of the joint $x$ and $y$ distribution $P(x,y,t)$ in a vicinity of a special "zero-noise" point, and show that $\ln P(x,y,t)$ has a nontrivial self-similar structure as a function of $x$, $y$ and $t$. The joint distribution vanishes extremely fast at the expanding circle, exhibiting an essential singularity there. This singularity is inherited by the marginal $x$- and $y$-distributions. We argue that this fingerprint of the short-time dynamics remains there at all times.

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arXiv2020

Universal Properties of a Run-and-Tumble Particle in Arbitrary Dimension

We consider an active run-and-tumble particle (RTP) in $d$ dimensions, starting from the origin and evolving over a time interval $[0,t]$. We examine three different models for the dynamics of the RTP: the standard RTP model with instantaneous tumblings, a variant with instantaneous runs and a general model in which both the tumblings and the runs are non-instantaneous. For each of these models, we use the Sparre Andersen theorem for discrete-time random walks to compute exactly the probability that the $x$ component does not change sign up to time $t$, showing that it does not depend on $d$. As a consequence of this result, we compute exactly other $x$-component properties, namely the distribution of the time of the maximum and the record statistics, showing that they are universal, i.e. they do not depend on $d$. Moreover, we show that these universal results hold also if the speed $v$ of the particle after each tumbling is random, drawn from a generic probability distribution. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 124, 090603 (2020)].

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arXiv2020

Noninteracting trapped Fermions in double-well potentials: inverted parabola kernel

We study a system of $N$ noninteracting spinless fermions in a confining, double-well potential in one dimension. When the Fermi energy is close to the value of the potential at its local maximum we show that physical properties, such as the average density and the fermion position correlation functions, display a universal behavior that depends only on the local properties of the potential near its maximum. This behavior describes the merging of two Fermi gases, which are disjoint at sufficiently low Fermi energies. We describe this behavior in terms of a new correlation kernel that we compute analytically and we call it the inverted parabola kernel". As an application, we calculate the mean and variance of the number of particles in an interval of size $2L$ centered around the position of the local maximum, for sufficiently small $L$. Finally, we discuss the possibility of observing our results in experiments, as well as the extensions to nonzero temperature and to higher space dimensions.

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arXiv2020

Current fluctuations in non-interacting run-and-tumble particles in one-dimension

We present a general framework to study the distribution of the flux through the origin up to time $t$, in a non-interacting one-dimensional system of particles with a step initial condition with a fixed density $ρ$ of particles to the left of the origin. We focus principally on two cases: (i) when the particles undergo diffusive dynamics (passive case) and (ii) run-and-tumble dynamics for each particle (active case). In analogy with disordered systems, we consider the flux distribution both for the annealed and the quenched initial conditions, for the passive and active particles. In the annealed case, we show that, for arbitrary particle dynamics, the flux distribution is a Poissonian with a mean $μ(t)$ that we compute exactly in terms of the Green's function of the single particle dynamics. For the quenched case, we show that, for the run-and-tumble dynamics, the quenched flux distribution takes an anomalous large deviation form at large times $P_{\rm qu}(Q,t) \sim \exp\left[-ρ\, v_0\, γ\, t^2 ψ_{\rm RTP}\left(\frac{Q}{ρv_0\,t} \right) \right]$, where $γ$ is the rate of tumbling and $v_0$ is the ballistic speed between two successive tumblings. In this paper, we compute the rate function $ψ_{\rm RTP}(q)$ and show that it is nontrivial. Our method also gives access to the probability of the rare event that, at time $t$, there is no particle to the right of the origin. For diffusive and run-and-tumble dynamics, we find that this probability decays with time as a stretched exponential, $\sim \exp(-c\, \sqrt{t})$ where the constant $c$ can be computed exactly. We verify our results for these large deviations by using an importance sampling Monte-Carlo method.

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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

Optimal mean first-passage time for a Brownian searcher subjected to resetting: experimental and theoretical results

We study experimentally and theoretically the optimal mean time needed by a free diffusing Brownian particle to reach a target at a distance L from an initial position in the presence of resetting. Both the initial position and the resetting position are Gaussian distributed with width $σ$. We derived and tested two resetting protocols, one with a periodic and one with random (Poissonian) resetting times. We computed and measured the full first-passage probability distribution that displays spectacular spikes immediately after each resetting time for close targets. We study the optimal mean first-passage time as a function of the resetting period/rate for different values of the ratio b = L/$σ$ and find an interesting phase transtion at a critical value b = bc. For bc < b < $\infty$, there is a metastable optimum time which disappears for b < bc. The intrinsic diffculties in implementing these protocols in experiments are also discussed.

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arXiv2020

Distribution of the Time Between Maximum and Minimum of Random Walks

We consider a one-dimensional Brownian motion of fixed duration $T$. Using a path-integral technique, we compute exactly the probability distribution of the difference $τ=t_{\min}-t_{\max}$ between the time $t_{\min}$ of the global minimum and the time $t_{\max}$ of the global maximum. We extend this result to a Brownian bridge, i.e. a periodic Brownian motion of period $T$. In both cases, we compute analytically the first few moments of $τ$, as well as the covariance of $t_{\max}$ and $t_{\min}$, showing that these times are anti-correlated. We demonstrate that the distribution of $τ$ for Brownian motion is valid for discrete-time random walks with $n$ steps and with a finite jump variance, in the limit $n\to \infty$. In the case of Lévy flights, which have a divergent jump variance, we numerically verify that the distribution of $τ$ differs from the Brownian case. For random walks with continuous and symmetric jumps we numerically verify that the probability of the event "$τ= n$" is exactly $1/(2n)$ for any finite $n$, independently of the jump distribution. Our results can be also applied to describe the distance between the maximal and minimal height of $(1+1)$-dimensional stationary-state Kardar-Parisi-Zhang interfaces growing over a substrate of finite size $L$. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 123, 200201 (2019)].

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arXiv2020

Time between the maximum and the minimum of a stochastic process

We present an exact solution for the probability density function $P(τ=t_{\min}-t_{\max}|T)$ of the time-difference between the minimum and the maximum of a one-dimensional Brownian motion of duration $T$. We then generalise our results to a Brownian bridge, i.e. a periodic Brownian motion of period $T$. We demonstrate that these results can be directly applied to study the position-difference between the minimal and the maximal height of a fluctuating $(1+1)$-dimensional Kardar-Parisi-Zhang interface on a substrate of size $L$, in its stationary state. We show that the Brownian motion result is universal and, asymptotically, holds for any discrete-time random walk with a finite jump variance. We also compute this distribution numerically for Lévy flights and find that it differs from the Brownian motion result.

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arXiv2020

Velocity and diffusion constant of an active particle in a one dimensional force field

We consider a run an tumble particle with two velocity states $\pm v_0$, in an inhomogeneous force field $f(x)$ in one dimension. We obtain exact formulae for its velocity $V_L$ and diffusion constant $D_L$ for arbitrary periodic $f(x)$ of period $L$. They involve the "active potential" which allows to define a global bias. Upon varying parameters, such as an external force $F$, the dynamics undergoes transitions from non-ergodic trapped states, to various moving states, some with non analyticities in the $V_L$ versus $F$ curve. A random landscape in the presence of a bias leads, for large $L$, to anomalous diffusion $x \sim t^μ$, $μ<1$, or to a phase with a finite velocity that we calculate.

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arXiv2020

Stochastic Resetting and Applications

In this Topical Review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate $r$, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate $r$. We then generalise to an arbitrary stochastic process (e.g. Lévy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field.

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arXiv2020

Universal gap statistics for random walks for a class of jump densities

We study the order statistics of a random walk (RW) of $n$ steps whose jumps are distributed according to symmetric Erlang densities $f_p(η)\sim |η|^p \,e^{-|η|}$, parametrized by a non-negative integer $p$. Our main focus is on the statistics of the gaps $d_{k,n}$ between two successive maxima $d_{k,n}=M_{k,n}-M_{k+1,n}$ where $M_{k,n}$ is the $k$-th maximum of the RW between step 1 and step $n$. In the limit of large $n$, we show that the probability density function of the gaps $P_{k,n}(Δ) = \Pr(d_{k,n} = Δ)$ reaches a stationary density $P_{k,n}(Δ) \to p_k(Δ)$. For large $k$, we demonstrate that the typical fluctuations of the gap, for $d_{k,n}= O(1/\sqrt{k})$ (and $n \to \infty$), are described by a non-trivial scaling function that is independent of $k$ and of the jump probability density function $f_p(η)$, thus corroborating our conjecture about the universality of the regime of typical fluctuations (see G. Schehr, S. N. Majumdar, Phys. Rev. Lett. 108, 040601 (2012)). We also investigate the large fluctuations of the gap, for $d_{k,n} = O(1)$ (and $n \to \infty$), and show that these two regimes of typical and large fluctuations of the gaps match smoothly.

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arXiv2020

Ising model with stochastic resetting

We study the stationary properties of the Ising model that, while evolving towards its equilibrium state at temperature $T$ according to the Glauber dynamics, is stochastically reset to its fixed initial configuration with magnetisation $m_0$ at a constant rate $r$. Resetting breaks detailed balance and drives the system to a non-equilibrium stationary state where the magnetisation acquires a nontrivial distribution, leading to a rich phase diagram in the $(T,r)$ plane. We establish these results exactly in one-dimension and present scaling arguments supported by numerical simulations in two-dimensions. We show that resetting gives rise to a novel "pseudo-ferro" phase in the $(T,r)$ plane for $r > r^*(T)$ and $T>T_c$ where $r^*(T)$ is a crossover line separating the pseudo-ferro phase from a paramagnetic phase. This pseudo-ferro phase is characterised by a non-zero typical magnetisation and a vanishing gap near $m=0$ of the magnetisation distribution.

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arXiv2020

Quantum dynamics with stochastic reset

We study non-equilibrium dynamics of integrable and non-integrable closed quantum systems whose unitary evolution is interrupted with stochastic resets, characterized by a reset rate $r$, that project the system to its initial state. We show that the steady state density matrix of a non-integrable system, averaged over the reset distribution, retains its off-diagonal elements for any finite $r$. Consequently a generic observable $\hat O$, whose expectation value receives contribution from these off-diagonal elements, never thermalizes under such dynamics for any finite $r$. We demonstrate this phenomenon by exact numerical studies of experimentally realizable models of ultracold bosonic atoms in a tilted optical lattice. For integrable Dirac-like fermionic models driven periodically between such resets, the reset-averaged steady state is found to be described by a family of generalized Gibbs ensembles (GGE s) characterized by $r$. We also study the spread of particle density of a non-interacting one-dimensional fermionic chain, starting from an initial state where all fermions occupy the left half of the sample, while the right half is empty. When driven by resetting dynamics, the density profile approaches at long times to a nonequilibrium stationary profile that we compute exactly. We suggest concrete experiments that can possibly test our theory.

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arXiv2020

Anderson-like localization transition of random walks with resetting

We study several lattice random walk models with stochastic resetting to previously visited sites which exhibit a phase transition between an anomalous diffusive regime and a localization regime where diffusion is suppressed. The localized phase settles above a critical resetting rate, or rate of memory use, and the probability density asymptotically adopts in this regime a non-equilibrium steady state similar to that of the well known problem of diffusion with resetting to the origin. The transition occurs because of the presence of a single impurity site where the resetting rate is lower than on other sites, and around which the walker spontaneously localizes. Near criticality, the localization length diverges with a critical exponent that falls in the same class as the self-consistent theory of Anderson localization of waves in random media. The critical dimensions are also the same in both problems. Our study provides analytically tractable examples of localization transitions in path-dependent, reinforced stochastic processes, which can be also useful for understanding spatial learning by living organisms.

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arXiv2020

Universal survival probability for a $d$-dimensional run-and-tumble particle

We consider an active run-and-tumble particle (RTP) in $d$ dimensions and compute exactly the probability $S(t)$ that the $x$-component of the position of the RTP does not change sign up to time $t$. When the tumblings occur at a constant rate, we show that $S(t)$ is independent of $d$ for any finite time $t$ (and not just for large $t$), as a consequence of the celebrated Sparre Andersen theorem for discrete-time random walks in one dimension. Moreover, we show that this universal result holds for a much wider class of RTP models in which the speed $v$ of the particle after each tumbling is random, drawn from an arbitrary probability distribution. We further demonstrate, as a consequence, the universality of the record statistics in the RTP problem.

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