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Satya N. Majumdar

Publications and source records attributed to Satya N. Majumdar.

At least 73 records · Page 4Linked to original sources
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.

cond-mat.stat-mech↗
arXiv2024

Optimal mean first-passage time of a run-and-tumble particle in a class of one-dimensional confining potentials

We consider a run-and-tumble particle (RTP) in one dimension, subjected to a telegraphic noise with a constant rate $γ$, and in the presence of an external confining potential $V(x) = α|x|^p$ with $p \geq 1$. We compute the mean first-passage time (MFPT) at the origin $τ_γ(x_0)$ for an RTP starting at $x_0$. We obtain a closed form expression for $τ_γ(x_0)$ for all $p \geq 1$, which becomes fully explicit in the case $p=1$, $p=2$ and in the limit $p \to \infty$. For generic $p>1$ we find that there exists an optimal rate $γ_{\rm opt}$ that minimizes the MFPT and we characterize in detail its dependence on $x_0$. We find that $γ_{\rm opt} \propto 1/x_0$ as $x_0 \to 0$, while $γ_{\rm opt}$ converges to a nontrivial constant as $x_0 \to \infty$. In contrast, for $p=1$, there is no finite optimum and $γ_{\rm opt} \to \infty$ in this case. These analytical results are confirmed by our numerical simulations.

cond-mat.stat-mech↗
arXiv2024

Active particle in a harmonic trap driven by a resetting noise: an approach via Kesten variables

We consider the statics and dynamics of a single particle trapped in a one-dimensional harmonic potential, and subjected to a driving noise with memory, that is represented by a resetting stochastic process. The finite memory of this driving noise makes the dynamics of this particle ``active''. At some chosen times (deterministic or random), the noise is reset to an arbitrary position and restarts its motion. We focus on two resetting protocols: periodic resetting, where the period is deterministic, and Poissonian resetting, where times between resets are exponentially distributed with a rate $r$. Between the different resetting epochs, we can express recursively the position of the particle. The random relation obtained takes a simple Kesten form that can be used to derive an integral equation for the stationary distribution of the position. We provide a detailed analysis of the distribution when the noise is a resetting Brownian motion. In this particular instance, we also derive a renewal equation for the full time dependent distribution of the position that we extensively study. These methods are quite general and can be used to study any process harmonically trapped when the noise is reset at random times.

cond-mat.stat-mech↗
arXiv2024

Optimizing the random search of a finite-lived target by a Lévy flight

In many random search processes of interest in chemistry, biology or during rescue operations, an entity must find a specific target site before the latter becomes inactive, no longer available for reaction or lost. We present exact results on a minimal model system, a one-dimensional searcher performing a discrete time random walk or Lévy flight. In contrast with the case of a permanent target, the capture probability and the conditional mean first passage time can be optimized. The optimal Lévy index takes a non-trivial value, even in the long lifetime limit, and exhibits an abrupt transition as the initial distance to the target is varied. Depending on the target lifetime, this transition is discontinuous or continuous, separated by a non-conventional tricritical point. These results pave the way to the optimization of search processes under time constraints.

cond-mat.stat-mech↗
arXiv2024

Exact extreme, order and sum statistics in a class of strongly correlated system

Even though strongly correlated systems are abundant, only a few exceptional cases admit analytical solutions. In this paper we present a large class of solvable systems with strong correlations.. We consider a set of $N$ independent and identically distributed (i.i.d) random variables $\{X_1,\, X_2,\ldots, X_N\}$ whose common distribution has a parameter $Y$ (or a set of parameters) which itself is random with its own distribution. For a fixed value of this parameter $Y$, the $X_i$ variables are independent and we call them conditionally independent and identically distributed (c.i.i.d). However, once integrated over the distribution of the parameter $Y$, the $X_i$ variables get strongly correlated, yet retaining a solvable structure for various observables, such as for the sum and the extremes of $X_i$'s. This provides a simple procedure to generate a class of solvable strongly correlated systems. We illustrate how this procedure works via three physical examples where $N$ particles on a line perform independent (i) Brownian motions, (ii) ballistic motions with random initial velocities, and (iii) Lévy flights, but they get strongly correlated via {\it simultaneous resetting} to the origin. Our results are verified in numerical simulations. This procedure can be used to generate an endless variety of solvable strongly correlated systems.

cond-mat.stat-mech↗
arXiv2023

Local time of a system of Brownian particles on the line with steplike initial condition

We consider a system of non-interacting Brownian particles on a line with a step-like initial condition, and we investigate the behavior of the local time at the origin at large times. We compute the mean and the variance of the local time, and we show that the memory effects are governed by the Fano factor associated with the initial condition. For the uniform initial condition, we show that the probability distribution of the local time admits a large deviation form, and we compute the corresponding large deviation functions for the annealed and quenched averaging schemes. The two resulting large deviation functions are very different. Our analytical results are supported by extensive numerical simulations.

cond-mat.stat-mech↗
arXiv2023

Dynamically emergent correlations between particles in a switching harmonic trap

We study a one dimensional gas of $N$ noninteracting diffusing particles in a harmonic trap, whose stiffness switches between two values $μ_1$ and $μ_2$ with constant rates $r_1$ and $r_2$ respectively. Despite the absence of direct interaction between the particles, we show that strong correlations between them emerge in the stationary state at long times, induced purely by the dynamics itself. We compute exactly the joint distribution of the positions of the particles in the stationary state, which allows us to compute several physical observables analytically. In particular, we show that the extreme value statistics (EVS), i.e., the distribution of the position of the rightmost particle has a nontrivial shape in the large $N$ limit. The scaling function characterizing this EVS has a finite support with a tunable shape (by varying the parameters). Remarkably, this scaling function turns out to be universal. First, it also describes the distribution of the position of the $k$-th rightmost particle in a $1d$ trap. Moreover, the distribution of the position of the particle farthest from the center of the harmonic trap in $d$ dimensions is also described by the same scaling function for all $d \geq 1$. Numerical simulations are in excellent agreement with our analytical predictions.

cond-mat.stat-mech↗
arXiv2023

Linear statistics for Coulomb gases: higher order cumulants

We consider $N$ classical particles interacting via the Coulomb potential in spatial dimension $d$ and in the presence of an external trap, at equilibrium at inverse temperature $β$. In the large $N$ limit, the particles are confined within a droplet of finite size. We study smooth linear statistics, i.e. the fluctuations of sums of the form ${\cal L}_N = \sum_{i=1}^N f({\bf x}_i)$, where ${\bf x}_i$'s are the positions of the particles and where $f({\bf x}_i)$ is a sufficiently regular function. There exists at present standard results for the first and second moments of ${\cal L}_N$ in the large $N$ limit, as well as associated Central Limit Theorems in general dimension and for a wide class of confining potentials. Here we obtain explicit expressions for the higher order cumulants of ${\cal L}_N$ at large $N$, when the function $f({\bf x})=f(|{\bf x}|)$ and the confining potential are both rotationnally invariant. A remarkable feature of our results is that these higher cumulants depend only on the value of $f'(|{\bf x}|)$ and its higher order derivatives evaluated exactly at the boundary of the droplet, which in this case is a $d$-dimensional sphere. In the particular two-dimensional case $d=2$ at the special value $β=2$, a connection to the Ginibre ensemble allows us to derive these results in an alternative way using the tools of determinantal point processes. Finally we also obtain the large deviation form of the full probability distribution function of ${\cal L}_N$.

math-ph↗
arXiv2023

Nonlinear-Cost Random Walk: exact statistics of the distance covered for fixed budget

We consider the Nonlinear-Cost Random Walk model in discrete time introduced in [Phys. Rev. Lett. 130, 237102 (2023)], where a fee is charged for each jump of the walker. The nonlinear cost function is such that slow/short jumps incur a flat fee, while for fast/long jumps the cost is proportional to the distance covered. In this paper we compute analytically the average and variance of the distance covered in $n$ steps when the total budget $C$ is fixed, as well as the statistics of the number of long/short jumps in a trajectory of length $n$, for the exponential jump distribution. These observables exhibit a very rich and non-monotonic scaling behavior as a function of the variable $C/n$, which is traced back to the makeup of a typical trajectory in terms of long/short jumps, and the resulting "entropy" thereof. As a byproduct, we compute the asymptotic behavior of ratios of Kummer hypergeometric functions when both the first and last arguments are large. All our analytical results are corroborated by numerical simulations.

cond-mat.stat-mech↗
arXiv2023

The cost of stochastic resetting

Resetting a stochastic process has been shown to expedite the completion time of some complex tasks, such as finding a target for the first time. Here we consider the cost of resetting by associating to each reset a cost, which is a function of the distance travelled during the reset event. We compute the Laplace transform of the joint probability of first passage time $t_f$, number of resets $N$ and total resetting cost $C$, and use this to study the statistics of the total cost and also the time to completion ${\mathcal T} = C + t_f$. We show that in the limit of zero resetting rate, the mean total cost is finite for a linear cost function, vanishes for a sub-linear cost function and diverges for a super-linear cost function. This result contrasts with the case of no resetting where the cost is always zero. We also find that the resetting rate which optimizes the mean time to completion may be increased or decreased with respect to the case of no resetting cost according to the choice of cost function. For the case of an exponentially increasing cost function, we show that the mean total cost diverges at a finite resetting rate. We explain this by showing that the distribution of the cost has a power-law tail with a continuously varying exponent that depends on the resetting rate.

cond-mat.stat-mech↗
arXiv2023

Generating Entanglement by Quantum Resetting

We consider a closed quantum system subjected to stochastic Poissonian resetting with rate $r$ to its initial state. Resetting drives the system to a nonequilibrium stationary state (NESS) with a mixed density matrix which has both classical and quantum correlations. We provide a general framework to study these NESS correlations for a closed quantum system with a general Hamiltonian $H$. We then apply this framework to a simple model of a pair of ferromagnetically coupled spins, starting from state $\mid\downarrow\downarrow \rangle$ and resetting to the same state with rate $r$. We compute exactly the NESS density matrix of the full system. This then provides access to three basic observables, namely (i) the von Neumann entropy of a subsystem (ii) the fidelity between the NESS and the initial density matrix and (iii) the concurrence in the NESS (that provides a measure of the quantum entanglement in a mixed state), as a function of the two parameters: the resetting rate and the interaction strength. One of our main conclusions is that a nonzero resetting rate and a nonzero interaction strength generates quantum entanglement in the NESS (quantified by a nonzero concurrence) and moreover this concurrence can be maximized by appropriately choosing the two parameters. Our results show that quantum resetting provides a simple and effective mechanism to enhance entanglement between two parts of an interacting quantum system.

quant-ph↗
arXiv2023

Cost of diffusion: nonlinearity and giant fluctuations

We introduce a simple model of diffusive jump process where a fee is charged for each jump. The nonlinear cost function is such that slow jumps incur a flat fee, while for fast jumps the cost is proportional to the velocity of the jump. The model -- inspired by the way taxi meters work -- exhibits a very rich behavior. The cost for trajectories of equal length and equal duration exhibits giant fluctuations at a critical value of the scaled distance travelled. Furthermore, the full distribution of the cost until the target is reached exhibits an interesting ``freezing'' transition in the large-deviation regime. All the analytical results are corroborated by numerical simulations. Our results also apply to elastic systems near the depinning transition, when driven by a random force.

cond-mat.stat-mech↗
arXiv2023

Striking universalities in stochastic resetting processes

Given a random process $x(τ)$ which undergoes stochastic resetting at a constant rate $r$ to a position drawn from a distribution ${\cal P}(x)$, we consider a sequence of dynamical observables $A_1, \dots, A_n$ associated to the intervals between resetting events. We calculate exactly the probabilities of various events related to this sequence: that the last element is larger than all previous ones, that the sequence is monotonically increasing, etc. Remarkably, we find that these probabilities are ``super-universal'', i.e., that they are independent of the particular process $x(τ)$, the observables $A_k$'s in question and also the resetting distribution ${\cal P}(x)$. For some of the events in question, the universality is valid provided certain mild assumptions on the process and observables hold (e.g., mirror symmetry).

cond-mat.stat-mech↗
arXiv2023

First detection probability in quantum resetting via random projective measurements

We provide a general framework to compute the probability distribution $F_r(t)$ of the first detection time of a 'state of interest' in a generic quantum system subjected to random projective measurements. In our 'quantum resetting' protocol, resetting of a state is not implemented by an additional classical stochastic move, but rather by the random projective measurement. We then apply this general framework to Poissoinian measurement protocol with a constant rate $r$ and demonstrate that exact results for $F_r(t)$ can be obtained for a generic two level system. Interestingly, the result depends crucially on the detection schemes involved and we have studied two complementary schemes, where the state of interest either coincides or differs from the initial state. We show that $F_r(t)$ at short times vanishes universally as $F_r(t)\sim t^2$ as $t\to 0$ in the first scheme, while it approaches a constant as $t\to 0$ in the second scheme. The mean first detection time, as a function of the measurement rate $r$, also shows rather different behaviors in the two schemes. In the former, the mean detection time is a nonmonotonic function of $r$ with a single minimum at an optimal value $r^*$, while in the later, it is a monotonically decreasing function of $r$, signalling the absence of a finite optimal value. These general predictions for arbitrary two level systems are then verified via explicit computation in the Jaynes-Cummings model of light-matter interaction. We also generalise our results to non-Poissonian measurement protocols with a renewal structure where the intervals between successive independent measurements are distributed via a general distribution $p(τ)$ and show that the short time behavior of $F_r(t)\sim p(0)\, t^2$ is universal as long as $p(0)\ne 0$. This universal $t^2$ law emerges from purely quantum dynamics that dominates at early times.

quant-ph↗
arXiv2023

Metropolis Monte Carlo sampling: convergence, localization transition and optimality

Among random sampling methods, Markov Chain Monte Carlo algorithms are foremost. Using a combination of analytical and numerical approaches, we study their convergence properties towards the steady state, within a random walk Metropolis scheme. Analysing the relaxation properties of some model algorithms sufficiently simple to enable analytic progress, we show that the deviations from the target steady-state distribution can feature a localization transition as a function of the characteristic length of the attempted jumps defining the random walk. While the iteration of the Monte Carlo algorithm converges to equilibrium for all choices of jump parameters, the localization transition changes drastically the asymptotic shape of the difference between the probability distribution reached after a finite number of steps of the algorithm and the target equilibrium distribution. We argue that the relaxation before and after the localisation transition is respectively limited by diffusion and rejection rates.

cond-mat.stat-mech↗
arXiv2023

Critical number of walkers for diffusive search processes with resetting

We consider $N$ Brownian motions diffusing independently on a line, starting at $x_0>0$, in the presence of an absorbing target at the origin. The walkers undergo stochastic resetting under two protocols: (A) each walker resets independently to $x_0$ with rate $r$ and (B) all walkers reset simultaneously to $x_0$ with rate $r$. We compute analytically the mean first-passage time to the origin and show that, as a function of $r$ and for fixed $x_0$, it has a minimum at an optimal value $r^*>0$ as long as $N N_c$, the optimal value occurs at $r^*=0$ indicating that resetting hinders search processes. Continuing our results analytically to real $N$, we show that $N_c=7.3264773\ldots$ for protocol A and $N_c=6.3555864\ldots$ for protocol B, independently of $x_0$. Our theoretical predictions are verified in numerical Langevin simulations.

cond-mat.stat-mech↗
arXiv2023

Effusion of stochastic processes on a line

We consider the problem of leakage or effusion of an ensemble of independent stochastic processes from a region where they are initially randomly distributed. The case of Brownian motion, initially confined to the left half line with uniform density and leaking into the positive half line is an example which has been extensively studied in the literature. Here we derive new results for the average number and variance of the number of leaked particles for arbitrary Gaussian processes initially confined to the negative half line and also derive its joint two-time probability distribution, both for the annealed and the quenched initial conditions. For the annealed case, we show that the two-time joint distribution is a bivariate Poisson distribution. We also discuss the role of correlations in the initial particle positions on the statistics of the number of particles on the positive half line. We show that the strong memory effects in the variance of the particle number on the positive real axis for Brownian particles, seen in recent studies, persist for arbitrary Gaussian processes and also at the level of two-time correlation functions.

cond-mat.stat-mech↗
arXiv2023

Singular relaxation of a random walk in a box with a Metropolis Monte Carlo dynamics

We study analytically the relaxation eigenmodes of a simple Monte Carlo algorithm, corresponding to a particle in a box which moves by uniform random jumps. Moves outside of the box are rejected. At long times, the system approaches the equilibrium probability density, which is uniform inside the box. We show that the relaxation towards this equilibrium is unusual: for a jump length comparable to the size of the box, the number of relaxation eigenmodes can be surprisingly small, one or two. We provide a complete analytic description of the transition between these two regimes. When only a single relaxation eigenmode is present, a suitable choice of the symmetry of the initial conditions gives a localizing decay to equilibrium. In this case, the deviation from equilibrium concentrates at the edges of the box where the rejection probability is maximal. Finally, in addition to the relaxation analysis of the master equation, we also describe the full eigen-spectrum of the master equation including its sub-leading eigen-modes.

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