arXiv subjects

Satya N. Majumdar

Publications and source records attributed to Satya N. Majumdar.

At least 91 records · Page 5Linked to original sources

arXiv2023

A sluggish random walk with subdiffusive spread

We study a one-dimensional sluggish random walk with space-dependent transition probabilities between nearest-neighbour lattice sites. Motivated by trap models of slow dynamics, we consider a model in which the trap depth increases logarithmically with distance from the origin. This leads to a random walk which has symmetric transition probabilities that decrease with distance $|k|$ from the origin as $1/|k|$ for large $|k|$. We show that the typical position after time $t$ scales as $t^{1/3}$ with a nontrivial scaling function for the position distribution which has a trough (a cusp singularity) at the origin. Therefore an effective central bias away from the origin emerges even though the transition probabilities are symmetric. We also compute the survival probability of the walker in the presence of a sink at the origin and show that it decays as $t^{-1/3}$ at late times. Furthermore we compute the distribution of the maximum position, $M(t)$, to the right of the origin up to time $t$, and show that it has a nontrivial scaling function. Finally we provide a generalisation of this model where the transition probabilities decay as $1/|k|^α$ with $α>0$.

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arXiv2023

Current fluctuations in stochastically resetting particle systems

We consider a system of non-interacting particles on a line with initial positions distributed uniformly with density $ρ$ on the negative half-line. We consider two different models: (i) each particle performs independent Brownian motion with stochastic resetting to its initial position with rate $r$ and (ii) each particle performs run and tumble motion, and with rate $r$ its position gets reset to its initial value and simultaneously its velocity gets randomised. We study the effects of resetting on the distribution $P(Q,t)$ of the integrated particle current $Q$ up to time $t$ through the origin (from left to right). We study both the annealed and the quenched current distributions and in both cases, we find that resetting induces a stationary limiting distribution of the current at long times. However, we show that the approach to the stationary state of the current distribution in the annealed and the quenched cases are drastically different for both models. In the annealed case, the whole distribution $P_{\rm an}(Q,t)$ approaches its stationary limit uniformly for all $Q$. In contrast, the quenched distribution $P_{\rm qu}(Q,t)$ attains its stationary form for $Q Q_{\rm crit}(t)$. We show that $Q_{\rm crit}(t)$ increases linearly with $t$ for large $t$. On the scale where $Q \sim Q_{\rm crit}(t)$, we show that $P_{\rm qu}(Q,t)$ has an unusual large deviation form with a rate function that has a third-order phase transition at the critical point. We have computed the associated rate functions analytically for both models. Using an importance sampling method that allows to probe probabilities as tiny as $10^{-14000}$, we were able to compute numerically this non-analytic rate function for the resetting Brownian dynamics and found excellent agreement with our analytical prediction.

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arXiv2023

Out of equilibrium dynamics of repulsive ranked diffusions: the expanding crystal

We study the non-equilibrium Langevin dynamics of $N$ particles in one dimension with Coulomb repulsive linear interactions. This is a dynamical version of the so-called jellium model (without confinement) also known as ranked diffusion. Using a mapping to the Lieb-Liniger model of quantum bosons, we obtain an exact formula for the joint distribution of the positions of the $N$ particles at time $t$, all starting from the origin. A saddle point analysis shows that the system converges at large time to a linearly expanding crystal. Properly rescaled, this dynamical state resembles the equilibrium crystal in a time dependent effective quadratic potential. This analogy allows to study the fluctuations around the perfect crystal, which, to leading order, are Gaussian. There are however deviations from this Gaussian behavior, which embody long-range correlations of purely dynamical origin, characterized by the higher order cumulants of, e.g., the gaps between the particles, that we calculate exactly. We complement these results using a recent approach by one of us in terms of a noisy Burgers equation. In the large $N$ limit, the mean density of the gas can be obtained at any time from the solution of a deterministic viscous Burgers equation. This approach provides a quantitative description of the dense regime at shorter times. Our predictions are in good agreement with numerical simulations for finite and large $N$.

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arXiv2022

An exact formula for the variance of linear statistics in the one-dimensional jellium mode

We consider the jellium model of $N$ particles on a line confined in an external harmonic potential and with a pairwise one-dimensional Coulomb repulsion of strength $α> 0$. Using a Coulomb gas method, we study the statistics of $s = (1/N) \sum_{i=1}^N f(x_i)$ where $f(x)$, in principle, is an arbitrary smooth function. While the mean of $s$ is easy to compute, the variance is nontrivial due to the long-range Coulomb interactions. In this paper we demonstrate that the fluctuations around this mean are Gaussian with a variance ${\rm Var}(s) \approx b/N^3$ for large $N$. We provide an exact compact formula for the constant $b = 1/(4α) \int_{-2 α}^{2α} [f'(x)]^2\, dx$. In addition, we also calculate the full large deviation function characterising the tails of the full distribution ${\cal P}(s,N)$ for several different examples of $f(x)$. Our analytical predictions are confirmed by numerical simulations.

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arXiv2022

Exact position distribution of a harmonically-confined run-and-tumble particle in two dimensions

We consider an overdamped run-and-tumble particle in two dimensions, with self propulsion in an orientation that stochastically rotates by 90 degrees at a constant rate, clockwise or counter-clockwise with equal probabilities. In addition, the particle is confined by an external harmonic potential of stiffness $μ$, and possibly diffuses. We find the exact time-dependent distribution $P\left(x,y,t\right)$ of the particle's position, and in particular, the steady-state distribution $P_{\text{st}}\left(x,y\right)$ that is reached in the long-time limit. We also find $P\left(x,y,t\right)$ for a "free" particle, $μ=0$. We achieve this by showing that, under a proper change of coordinates, the problem decomposes into two statistically-independent one-dimensional problems, whose exact solution has recently been obtained. We then extend these results in several directions, to two such run-and-tumble particles with a harmonic interaction, to analogous systems of dimension three or higher, and by allowing stochastic resetting.

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arXiv2022

Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature

We study the average density of $N$ spinless noninteracting fermions in a $2d$ harmonic trap rotating with a constant frequency $Ω$ and in the presence of an additional repulsive central potential $γ/r^2$. The average density at zero temperature was recently studied in Phys. Rev. A $\textbf{103}$, 033321 (2021) and an interesting multi-layered "wedding cake" structure with a "hole" at the center was found for the density in the large $N$ limit. In this paper, we study the average density at finite temperature. We demonstrate how this "wedding-cake" structure is modified at finite temperature. These large $N$ results warrant going much beyond the standard Local Density Approximation. We also generalize our results to a wide variety of trapping potentials and demonstrate the universality of the associated scaling functions both in the bulk and at the edges of the "wedding-cake".

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arXiv2022

Reducing mean first passage times with intermittent confining potentials: a realization of resetting processes

During a random search, resetting the searcher's position from time to time to the starting point often reduces the mean completion time of the process. Although many different resetting models have been studied over the past ten years, only a few can be physically implemented. Here we study theoretically a protocol that can be realised experimentally and which exhibits unusual optimization properties. A Brownian particle is subject to an arbitrary confining potential $v(x)$ which is switched on and off intermittently at fixed rates. Motion is constrained between an absorbing wall located at the origin and a reflective wall. When the walls are sufficiently far apart, the interplay between free diffusion during the "off" phases and attraction toward the potential minimum during the "on" phases gives rise to rich behaviours, not observed in ideal resetting models. For potentials of the form $v(x)=k|x-x_0|^n/n$, with $n>0$, the switch-on and switch-off rates that minimise the mean first passage time (MFPT) to the origin undergo a continuous phase transition as the potential stiffness $k$ is varied. When $k$ is above a critical value $k_c$, potential intermittency enhances target encounter: the minimal MFPT is lower than the Kramer's time and is attained for a non-vanishing pair of switching rates. We focus on the harmonic case $n=2$, extending previous results for the piecewise linear potential ($n=1$) in unbounded domains. We also study the non-equilibrium stationary states emerging in this process.

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arXiv2022

Time to reach the maximum for a stationary stochastic process

We consider a one-dimensional stationary time series of fixed duration $T$. We investigate the time $t_{\rm m}$ at which the process reaches the global maximum within the time interval $[0,T]$. By using a path-decomposition technique, we compute the probability density function $P(t_{\rm m}|T)$ of $t_{\rm m}$ for several processes, that are either at equilibrium (such as the Ornstein-Uhlenbeck process) or out of equilibrium (such as Brownian motion with stochastic resetting). We show that for equilibrium processes the distribution of $P(t_{\rm m}|T)$ is always symmetric around the midpoint $t_{\rm m}=T/2$, as a consequence of the time-reversal symmetry. This property can be used to detect nonequilibrium fluctuations in stationary time series. Moreover, for a diffusive particle in a confining potential, we show that the scaled distribution $P(t_{\rm m}|T)$ becomes universal, i.e., independent of the details of the potential, at late times. This distribution $P(t_{\rm m}|T)$ becomes uniform in the "bulk" $1\ll t_{\rm m}\ll T$ and has a nontrivial universal shape in the "edge regimes" $t_{\rm m}\to0$ and $t_{\rm m} \to T$. Some of these results have been announced in a recent Letter [Europhys. Lett. {\bf 135}, 30003 (2021)].

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arXiv2022

Freezing transitions of Brownian particles in confining potentials

We study the mean first passage time (MFPT) to an absorbing target of a one-dimensional Brownian particle subject to an external potential $v(x)$ in a finite domain. We focus on the cases in which the external potential is confining, of the form $v(x)=k|x-x_0|^n/n$, and where the particle's initial position coincides with $x_0$. We first consider a particle between an absorbing target at $x=0$ and a reflective wall at $x=c$. At fixed $x_0$, we show that when the target distance $c$ exceeds a critical value, there exists a nonzero optimal stiffness $k_{\rm opt}$ that minimizes the MFPT to the target. However, when $c$ lies below the critical value, the optimal stiffness $k_{\rm opt}$ vanishes. Hence, for any value of $n$, the optimal potential stiffness undergoes a continuous "freezing" transition as the domain size is varied. On the other hand, when the reflective wall is replaced by a second absorbing target, the freezing transition in $k_{\rm opt}$ becomes discontinuous. The phase diagram in the $(x_0,n)$-plane then exhibits three dynamical phases and metastability, with a "triple" point at $(x_0/c\simeq 0.17185$, $n\simeq 0.39539)$. For harmonic or higher order potentials $(n\ge 2)$, the MFPT always increases with $k$ at small $k$, for any $x_0$ or domain size. These results are contrasted with problems of diffusion under optimal resetting in bounded domains.

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arXiv2022

Universal order statistics for random walks & Lévy flights

We consider one-dimensional discrete-time random walks (RWs) of $n$ steps, starting from $x_0=0$, with arbitrary symmetric and continuous jump distributions $f(η)$, including the important case of Lévy flights. We study the statistics of the gaps $Δ_{k,n}$ between the $k^\text{th}$ and $(k+1)^\text{th}$ maximum of the set of positions $\{x_1,\ldots,x_n\}$. We obtain an exact analytical expression for the probability distribution $P_{k,n}(Δ)$ valid for any $k$ and $n$, and jump distribution $f(η)$, which we then analyse in the large $n$ limit. For jump distributions whose Fourier transform behaves, for small $q$, as $\hat f (q) \sim 1 - |q|^μ$ with a Lévy index $0< μ\leq 2$, we find that, the distribution becomes stationary in the limit of $n\to \infty$, i.e. $\lim_{n\to \infty} P_{k,n}(Δ)=P_k(Δ)$. We obtain an explicit expression for its first moment $\mathbb{E}[Δ_{k}]$, valid for any $k$ and jump distribution $f(η)$ with $μ>1$, and show that it exhibits a universal algebraic decay $ \mathbb{E}[Δ_{k}]\sim k^{1/μ-1} Γ\left(1-1/μ\right)/π$ for large $k$. Furthermore, for $μ>1$, we show that in the limit of $k\to\infty$ the stationary distribution exhibits a universal scaling form $P_k(Δ) \sim k^{1-1/μ} \mathcal{P}_μ(k^{1-1/μ}Δ)$ which depends only on the Lévy index $μ$, but not on the details of the jump distribution. We compute explicitly the limiting scaling function $\mathcal{P}_μ(x)$ in terms of Mittag-Leffler functions. For $1< μ<2$, we show that, while this scaling function captures the distribution of the typical gaps on the scale $k^{1/μ-1}$, the atypical large gaps are not described by this scaling function since they occur at a larger scale of order $k^{1/μ}$.

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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

An exactly solvable predator prey model with resetting

We study a simple model of a diffusing particle (the prey) that on encounter with one of a swarm of diffusing predators can either perish or be reset to its original position at the origin. We show that the survival probability of the prey up to time $t$ decays algebraically as $\sim t^{-θ(p, γ)}$ where the exponent $θ$ depends continuously on two parameters of the model, with $p$ denoting the probability that a prey survives upon encounter with a predator and $γ= D_A/(D_A+D_B)$ where $D_A$ and $D_B$ are the diffusion constants of the prey and the predator respectively. We also compute exactly the probability distribution $P(N|t_c)$ of the total number of encounters till the capture time $t_c$ and show that it exhibits an anomalous large deviation form $P(N|t_c)\sim t_c^{- Φ\left(\frac{N}{\ln t_c}=z\right)}$ for large $t_c$. The rate function $Φ(z)$ is computed explicitly. Numerical simulations are in excellent agreement with our analytical results.

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arXiv2022

Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting

We study the fluctuations of the area $A(t)= \int_0^t x(τ)\, dτ$ under a self-similar Gaussian process (SGP) $x(τ)$ with Hurst exponent $H>0$ (e.g., standard or fractional Brownian motion, or the random acceleration process) that stochastically resets to the origin at rate $r$. Typical fluctuations of $A(t)$ scale as $\sim \sqrt{t}$ for large $t$ and on this scale the distribution is Gaussian, as one would expect from the central limit theorem. Here our main focus is on atypically large fluctuations of $A(t)$. In the long-time limit $t\to\infty$, we find that the full distribution of the area takes the form $P_{r}\left(A|t\right)\sim\exp\left[-t^αΦ\left(A/t^β\right)\right]$ with anomalous exponents $α=1/(2H+2)$ and $β= (2H+3)/(4H+4)$ in the regime of moderately large fluctuations, and a different anomalous scaling form $P_{r}\left(A|t\right)\sim\exp\left[-tΨ\left(A/t^{\left(2H+3\right)/2}\right)\right]$ in the regime of very large fluctuations. The associated rate functions $Φ(y)$ and $Ψ(w)$ depend on $H$ and are found exactly. Remarkably, $Φ(y)$ has a singularity that we interpret as a first-order dynamical condensation transition, while $Ψ(w)$ exhibits a second-order dynamical phase transition above which the number of resetting events ceases to be extensive. The parabolic behavior of $Φ(y)$ around the origin $y=0$ correctly describes the typical, Gaussian fluctuations of $A(t)$. Despite these anomalous scalings, we find that all of the cumulants of the distribution $P_{r}\left(A|t\right)$ grow linearly in time, $\langle A^n\rangle_c\approx c_n \, t$, in the long-time limit. For the case of reset Brownian motion (corresponding to $H=1/2$), we develop a recursive scheme to calculate the coefficients $c_n$ exactly and use it to calculate the first 6 nonvanishing cumulants.

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arXiv2022

Counting statistics for non-interacting fermions in a rotating trap

We study the ground state of $N \gg 1$ noninteracting fermions in a two-dimensional harmonic trap rotating at angular frequency $Ω>0$. The support of the density of the Fermi gas is a disk of radius $R_e$. We calculate the variance of the number of fermions ${\cal N}_R$ inside a disk of radius $R$ centered at the origin for $R$ in the bulk of the Fermi gas. We find rich and interesting behaviours in two different scaling regimes: (i) $Ω/ ω<1 $ and (ii) $1 - Ω/ ω= O(1/N)$, where $ω$ is the angular frequency of the oscillator. In the first regime (i) we find that ${\rm Var}\,{\cal N}_{R}\simeq\left(A\log N+B\right)\sqrt{N}$ and we calculate $A$ and $B$ as functions of $R/R_e$, $Ω$ and $ω$. We also predict the higher cumulants of ${\cal N}_{R}$ and the bipartite entanglement entropy of the disk with the rest of the system. In the second regime (ii), the mean fermion density exhibits a staircase form, with discrete plateaus corresponding to filling $k$ successive Landau levels, as found in previous studies. Here, we show that ${\rm Var}\,{\cal N}_{R}$ is a discontinuous piecewise linear function of $\sim (R/R_e) \sqrt{N}$ within each plateau, with coefficients that we calculate exactly, and with steps whose precise shape we obtain for any $k$. We argue that a similar piecewise linear behavior extends to all the cumulants of ${\cal N}_{R}$ and to the entanglement entropy. We show that these results match smoothly at large $k$ with the above results for $Ω/ω=O(1)$. These findings are nicely confirmed by numerical simulations. Finally, we uncover a universal behavior of ${\rm Var}\,{\cal N}_{R}$ near the fermionic edge. We extend our results to a three-dimensional geometry, where an additional confining potential is applied in the $z$ direction.

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arXiv2022

First-passage time of run-and-tumble particles with non-instantaneous resetting

We study the statistics of the first-passage time of a single run and tumble particle (RTP) in one spatial dimension, with or without resetting, to a fixed target located at $L>0$. First, we compute the first-passage time distribution of a free RTP, without resetting nor in a confining potential, but averaged over the initial position drawn from an arbitrary distribution $p(x)$. Recent experiments used a non-instantaneous resetting protocol that motivated us to study in particular the case where $p(x)$ corresponds to the stationary non-Boltzmann distribution of an RTP in the presence of a harmonic trap. This distribution $p(x)$ is characterized by a parameter $ν>0$, which depends on the microscopic parameters of the RTP dynamics. We show that the first-passage time distribution of the free RTP, drawn from this initial distribution, develops interesting singular behaviours, depending on the parameter $ν$. We then switch on resetting, mimicked by thermal relaxation of the RTP in the presence of a harmonic trap. Resetting leads to a finite mean first-passage time (MFPT) and we study this as a function of the resetting rate for different values of the parameters $ν$ and $b = L/c$ where $c$ is the right edge of the initial distribution $p(x)$. In the diffusive limit of the RTP dynamics, we find a rich phase diagram in the $(b,ν)$ plane, with an interesting re-entrance phase transition. Away from the diffusive limit, qualitatively similar rich behaviours emerge for the full RTP dynamics.

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arXiv2022

Gap probability and full counting statistics in the one dimensional one-component plasma

We consider the $1d$ one-component plasma (OCP) in thermal equilibrium, consisting of $N$ equally charged particles on a line, with pairwise Coulomb repulsion and confined by an external harmonic potential. We study two observables: (i) the distribution of the gap between two consecutive particles in the bulk and (ii) the distribution of the number of particles $N_I$ in a fixed interval $I=[-L,+L]$ inside the bulk, the so-called full-counting-statistics (FCS). For both observables, we compute, for large $N$, the distribution of the typical as well as atypical large fluctuations. We show that the distribution of the typical fluctuations of the gap are described by the scaling form ${\cal P}_{\rm gap, bulk}(g,N) \sim N H_α(g\,N)$, where $α$ is the interaction coupling and the scaling function $H_α(z)$ is computed explicitly. It has a faster than Gaussian tail for large $z$: $H_α(z) \sim e^{-z^3/(96 α)}$ as $z \to \infty$. Similarly, for the FCS, we show that the distribution of the typical fluctuations of $N_I$ is described by the scaling form ${\cal P}_{\rm FCS}(N_I,N) \sim 2α\, U_α[2 α(N_I - \bar{N}_I)]$, where $\bar{N}_I = L\,N/(2 α)$ is the average value of $N_I$ and the scaling function $U_α(z)$ is obtained explicitly. For both observables, we show that the probability of large fluctuations are described by large deviations forms with respective rate functions that we compute explicitly. Our numerical Monte-Carlo simulations are in good agreement with our analytical predictions.

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arXiv2022

Number of distinct sites visited by a resetting random walker

We investigate the number $V_p(n)$ of distinct sites visited by an $n$-step resetting random walker on a $d$-dimensional hypercubic lattice with resetting probability $p$. In the case $p=0$, we recover the well-known result that the average number of distinct sites grows for large $n$ as $\langle V_0(n)\rangle\sim n^{d/2}$ for $d<2$ and as $\langle V_0(n)\rangle\sim n$ for $d>2$. For $p>0$, we show that $\langle V_p(n)\rangle$ grows extremely slowly as $\sim \left[\log(n)\right]^d$. We observe that the recurrence-transience transition at $d=2$ for standard random walks (without resetting) disappears in the presence of resetting. In the limit $p\to 0$, we compute the exact crossover scaling function between the two regimes. In the one-dimensional case, we derive analytically the full distribution of $V_p(n)$ in the limit of large $n$. Moreover, for a one-dimensional random walker, we introduce a new observable, which we call imbalance, that measures how much the visited region is symmetric around the starting position. We analytically compute the full distribution of the imbalance both for $p=0$ and for $p>0$. Our theoretical results are verified by extensive numerical simulations.

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arXiv2022

Some considerations about reviewing and open-access in scientific publishing

Scientific research changed profoundly over the last 30 years, in all its aspects. Scientific publishing has changed as well, mainly because of the strong increased number of submitted papers and because of the appearance of Open Access journals and publishers. We propose some reflections on these issues.

cs.DL↗