arXiv subjects

Satya N. Majumdar

Publications and source records attributed to Satya N. Majumdar.

At least 37 records · Page 2Linked to original sources

arXiv2026

Dynamically Emergent Correlations

In this perspective article, we discuss the scenario of dynamically emergent correlation (DEC) arising in classical and quantum noninteracting systems when they are subjected to a common fluctuating stochastic environment. The key property of such systems is that the strong correlations between different particles emerge from the dynamics and not from built-in interactions. In many cases, these strong correlations persist even at long times in the stationary state. Computing observables explicitly for such strongly correlated states in general is very hard. Remarkably, the stationary states in several models of DEC exhibit an interesting analytical structure that allows to compute physical observables, despite being strongly correlated. Recent experiments on trapped colloidal particles have established that these DEC in the stationary state can in fact be measured. DEC is a rapidly emerging domain of strongly correlated out-of-equilibrium statistical physics, with both theoretical and experimental, as well as classical and quantum, components.

cond-mat.stat-mech↗

arXiv2026

Equilibrium measures for higher dimensional rotationally symmetric Riesz gases

We study equilibrium measures for Riesz gases in dimension $d$ with pairwise interaction kernel $|x-y|^{-s}$, subject to radially symmetric external fields. We characterise broad classes of confining potentials for which the equilibrium measure is supported on the unit ball and admits an explicit density. Our main contribution is a converse construction: starting from a prescribed radially symmetric equilibrium density given as a power series in the squared radius, we determine the associated external potential and establish the corresponding Euler-Lagrange variational conditions. A key ingredient in the proof is an identity between two ${}_3F_2$ hypergeometric functions evaluated at unit argument, which is of independent interest. As applications, we identify the external potentials corresponding to equilibrium densities proportional to $(1-|x|^2)^α$, $α>-1$, and show that these potentials can be expressed in terms of Gauss hypergeometric functions ${}_2F_1$, reducing to polynomials for special values of $α$. We also determine the equilibrium measure associated with purely power-type external potentials, often referred to as Freud or Mittag--Leffler potentials in the context of log gases, for which the equilibrium density admits an explicit ${}_2F_1$ representation. Furthermore, we apply our framework to a Coulomb gas in dimension $d+1$ confined by a harmonic potential to the half-space. We derive a necessary condition under which the equilibrium measure is fully supported on the boundary hyperplane of dimension $d$, with the induced density corresponding to that of a Riesz gas with exponent $s=d-1$.

math-ph↗

arXiv2026

Dynamically emergent correlations in Brownian particles subject to simultaneous non-Poissonian resetting protocols

We consider a one-dimensional gas of $N$ independent Brownian particles subject to simultaneous stochastic resetting, with inter-reset times drawn from a general waiting-time distribution $ψ(τ)$. This includes the well-known Poissonian case, where $ψ(τ)=re^{-rτ}$, and extends to more general classes of resetting, such as heavy-tailed and bounded distributions. We show that the simultaneous resetting generates correlations between particles dynamically. These correlations grow with time and eventually drive the system into a strongly correlated non-equilibrium stationary state (NESS). Exploiting the renewal structure of the resetting dynamics, we derive explicit analytical expressions for the joint distribution of the positions of the particles in the NESS. We show that the NESS has a conditionally independent and identically distributed (CIID) structure that enables us to compute various physical observables exactly for arbitrary $ψ(τ)$. These observables include the average density, extreme value and order statistics, the spacing distribution between consecutive particles and the full counting statistics, i.e., the distribution of the number of particles in a given interval centered at the origin. We discuss the universal features of the large $N$ scaling behaviors of these observables for different choices of the resetting protocol $ψ(τ)$. Our results provide an interesting example of a stochastic control whereby, by tuning the inter-reset distribution $ψ(τ)$, one can generate a class of tunable, and yet solvable, strongly correlated NESS in a many-body system.

cond-mat.stat-mech↗

arXiv2026

The distribution of the maximum of independent resetting Brownian motions

The probability distribution of the maximum $M_t$ of a single resetting Brownian motion (RBM) of duration $t$ and resetting rate $r$, properly centred and scaled, is known to converge to the standard Gumbel distribution of the classical extreme value theory. This Gumbel law describes the typical fluctuations of $M_t$ around its average $\sim \ln (r t)$ for large $t$ on a scale of $O(1)$. Here we compute the large-deviation tails of this distribution when $M_t = O(t)$ and show that the large-deviation function has a singularity where the second derivative is discontinuous, signalling a dynamical phase transition. Then we consider a collection of independent RBMs with initial (and resetting) positions uniformly distributed with a density $ρ$ over the negative half-line. We show that the fluctuations in the initial positions of the particles modify the distribution of $M_t$. The average over the initial conditions can be performed in two different ways, in analogy with disordered systems: (i) the annealed case where one averages over all possible initial conditions and (ii) the quenched case where one considers only the contributions coming from typical initial configurations. We show that in the annealed case, the limiting distribution of the maximum is characterized by a new scaling function, different from the Gumbel law but the large-deviation function remains the same as in the single particle case. In contrast, for the quenched case, the limiting (typical) distribution remains Gumbel but the large-deviation behaviors are new and nontrivial. Our analytical results, both for the typical as well as for the large-deviation regime of $M_t$, are verified numerically with extremely high precision, down to $10^{-250}$ for the probability density of $M_t$.

cond-mat.stat-mech↗

arXiv2025

Dynamical phase transitions in certain non-ergodic stochastic processes

We present a class of stochastic processes in which the large deviation functions of time-integrated observables exhibit singularities that relate to dynamical phase transitions of trajectories. These illustrative examples include Brownian motion with a death rate or in the presence of an absorbing wall, for which we consider a set of empirical observables such as the net displacement, local time, residence time, and area under the trajectory. Using a backward Fokker-Planck approach, we derive the large deviation functions of these observables, and demonstrate how singularities emerge from a competition between survival and diffusion. Furthermore, we analyse this scenario using an alternative approach with tilted operators, showing that at the singular point, the effective dynamics undergoes an abrupt transition. Extending this approach, we show that similar transitions may generically arise in Markov chains with transient states. This scenario is robust and generalizable for non-Markovian dynamics and for many-body systems, potentially leading to multiple dynamical phase transitions. We have confirmed most of our findings on the singular large-deviation function using rare-event simulation techniques.

cond-mat.stat-mech↗

arXiv2025

Dynamically emergent correlations in a Brownian gas with diffusing diffusivity

We study a gas of $N$ Brownian particles in the presence of a common stochastic diffusivity $D(t)=B^2(t)$, where $B(t)$ represents a one-dimensional Brownian motion at time $t$. Starting from all the particles localized at the origin, the gas expands with a ballistic scaling $x\sim t$. We show that because of the common stochastic diffusivity, the expanding gas gets dynamically correlated, and the joint probability density function of the position of the particles has a CIID structure that was recently found in several other systems. The special structure allows us to compute the average density profile of the gas, extreme and order statistics, gap distribution between successive particles, and the full counting statistics (FCS) that describes the probability density function (PDF) $H(κ, t)$ of the fraction of particles $κ$ in a given region $[-L,L]$. Interestingly, the position fluctuation of the central particles and the average density profiles are described by the same scaling function. The PDF describing the FCS has an essential singularity near $κ=0$, indicating the presence of particles inside the box $[-L,L]$ at all times. Near the upper limit $κ=1$, the scaling function $H(κ,t)$ has a rather unusual behavior: $H(κ,t)\sim (1-κ)^{β(t)}$ where the exponent $β(t)$ changes continuously with time. At early times $β(t)$ is negative, indicating a divergence of $H(κ,t)$ as $κ\to 1$, whereas $β(t)$ becomes positive for $t>t_c$ where $t_c$ is computed exactly. Thus, as a function of $t$, the FCS exhibits an interesting shape transition. We also obtain the PDFs of the first-passage time to a given position $x$ and first-exit time from a box $[-L,L]$, by any one of the particles, and find that both PDFs are described by the same scaling function.

cond-mat.stat-mech↗

arXiv2025

First Passage Resetting Gas

We study a one-dimensional gas of $N$ Brownian particles that diffuse independently but are simultaneously reset whenever any of them reaches a fixed threshold located at $L > 0$. For any $N > 2$, the system reaches a non-equilibrium stationary state (NESS) at long-times with strong long-range correlations. These correlations emerge purely from the dynamics, and not from built-in interactions. Despite being strongly correlated, the NESS has a solvable structure that allows for an exact computation of several physical observables, both global and local. These include the average density profile, the distribution of the position of the $k$-th ordered particles, the distribution of the gap between two consecutive particles and the full counting statistics, i.e., the distribution of the number of particles in a finite interval around the origin.

cond-mat.stat-mech↗

arXiv2025

Extreme statistics and spacing distribution in a Brownian gas correlated by resetting

We study a one-dimensional gas of $N$ Brownian particles that diffuse independently, but are {\it simultaneously} reset to the origin at a constant rate $r$. The system approaches a non-equilibrium stationary state (NESS) with long-range interactions induced by the simultaneous resetting. Despite the presence of strong correlations, we show that several observables can be computed exactly, which include the global average density, the distribution of the position of the $k$-th rightmost particle and the spacing distribution between two successive particles. Our analytical results are confirmed by numerical simulations. We also discuss a possible experimental realisation of this resetting gas using optical traps.

cond-mat.stat-mech↗

arXiv2025

Two dimensional Coulomb gas in a non-conservative trap

We study the nonequilibrium steady state of a two dimensional Coulomb gas under the action of an anisotropic harmonic trapping potential along with a non-conservative rotational force. In the case without rotation, the equilibrium (zero temperature) steady state has a uniform density supported over a a static elliptical droplet. The addition of a rotational force drives the system into a nonequilibrium steady state where the density is still uniform inside an ellipse, but the ellipse gets tilted by a a fixed angle compared to the non-rotational case. In addition, a nonzero current is generated inside the droplet which run concentrically to the droplet boundary. For large rotational force, the droplet develops a purely circular form. Our results are predicted by a simple hydrodynamic calculation and are confirmed by numerical simulations and provide a full understanding of a novel driven non-equilibrium state in a strongly interacting system.

cond-mat.stat-mech↗

arXiv2025

Optimal detection of quantum states via projective measurements

We consider the quantum dynamical evolution of a fully-connected quantum system subjected to random projective measurements and study the first detection time of an extended subspace of the Hilbert space. Exact analytical expressions are obtained for the mean first detection time and the full first detection probability distribution as functionals of the initial state $\ket{ψ_0}$ and the measurement rate $r$. Exact enumeration agrees perfectly with our analytical predictions.

quant-ph↗

arXiv2025

Dynamically generated correlations in a trapped bosonic gas via frequency quenches

We study a system of $N$ noninteracting bosons in a harmonic trap subjected to repeated quantum quenches, where the trap frequency is switched from one value to another after a random time duration drawn from an exponential distribution. Each cycle contains two steps: (i) changing the trap frequency to enable unitary evolution under a Hamiltonian, and (ii) reapplying the original trap at stochastic times to cool the gas back to its initial state. This protocol effectively makes it an open quantum system and drives it into a unique nonequilibrium steady state (NESS). We analytically and numerically characterize the NESS, uncovering a conditionally independent and identically distributed (CIID) structure in the joint probability density function (JPDF) of the positions. The JPDF in the CIID structure is a product of Gaussians with a common random variance, which is then averaged with respect to its distribution, making the JPDF non-factorizable, giving rise to long-range emergent dynamical correlations. The average density profile of the gas shows significant deviations from the initial Gaussian shape. We further compute the order and the gap statistics, revealing universal scaling in both bulk and edge regimes. We also analyze the full counting statistics, exposing rich parameter-dependent structure. Our results demonstrate how stochastic quenches can generate nontrivial correlations in quantum many-body systems.

cond-mat.quant-gas↗

arXiv2025

First-passage properties of the jump process with a drift. Two exactly solvable cases

We investigate the first-passage properties of a jump process with a constant drift, focusing on two key observables: the first-passage time $τ$ and the number of jumps $n$ before the first-passage event. By mapping the problem onto an effective discrete-time random walk, we derive an exact expression for the Laplace transform of the joint distribution of $τ$ and $n$ using the generalized Pollaczek-Spitzer formula. This result is then used to analyze the first-passage properties for two exactly solvable cases: (i) both the inter-jump intervals and jump amplitudes are exponentially distributed, and (ii) the inter-jump intervals are exponentially distributed while all jumps have the same fixed amplitude. We show the existence of two distinct regimes governed by the strength of the drift: (i) a survival regime, where the process remains positive indefinitely with finite probability; (ii) an absorption regime, where the first-passage eventually occurs; and (iii) a critical point at the boundary between these two phases. We characterize the asymptotic behavior of survival probabilities in each regime: they decay exponentially to a constant in the survival regime, vanish exponentially fast in the absorption regime, and exhibit power-law decay at the critical point. Furthermore, in the absorption regime, we derive large deviation forms for the marginal distributions of $τ$ and n. The analytical predictions are validated through extensive numerical simulations.

cond-mat.stat-mech↗

arXiv2025

Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions

We study the diffusion of a particle with a time-dependent diffusion constant $D(t)$ that switches between random values drawn from a distribution $W(D)$ at a fixed rate $r$. Using a renewal approach, we compute exactly the moments of the position of the particle $\langle x^{2n}(t) \rangle$ at any finite time $t$, and for any $W(D)$ with finite moments $\langle D^n \rangle$. For $t \gg 1$, we demonstrate that the cumulants $\langle x^{2n}(t) \rangle_c$ grow linearly with $t$ and are proportional to the free cumulants of a random variable distributed according to $W(D)$. For specific forms of $W(D)$, we compute the large deviations of the position of the particle, uncovering rich behaviors and dynamical transitions of the rate function $I(y=x/t)$. Our analytical predictions are validated numerically with high precision, achieving accuracy up to $10^{-2000}$.

cond-mat.stat-mech↗

arXiv2025

Power-law relaxation of a confined diffusing particle subject to resetting with memory

We study the relaxation of a Brownian particle with long range memory under confinement in one dimension. The particle diffuses in an arbitrary confining potential and resets at random times to previously visited positions, chosen with a probability proportional to the local time spent there by the particle since the initial time. This model mimics an animal which moves erratically in its home range and returns preferentially to familiar places from time to time, as observed in nature. The steady state density of the position is given by the equilibrium Gibbs-Boltzmann distribution, as in standard diffusion, while the transient part of the density can be obtained through a mapping of the Fokker-Planck equation of the process to a Schrödinger eigenvalue problem. Due to memory, the approach at late times toward the steady state is critically self-organised, in the sense that it always follows a sluggish power-law form, in contrast to the exponential decay that characterises Markov processes. The exponent of this power-law depends in a simple way on the resetting rate and on the leading relaxation rate of the Brownian particle in the absence of resetting. We apply these findings to several exactly solvable examples, such as the harmonic, V-shaped and box potentials.

cond-mat.stat-mech↗

arXiv2025

The cost of resetting discrete-time random walks

We consider a discrete-time continuous-space random walk, with a symmetric jump distribution, under stochastic resetting. Associated with the random walker are cost functions for jumps and resets, and we calculate the distribution of the total cost for the random walker up to the first passage to the target. By using the backward master equation approach we demonstrate that the distribution of the total cost up to the first passage to the target can be reduced to a Wiener-Hopf integral equation. The resulting integral equation can be exactly solved (in Laplace space) for arbitrary cost functions for the jump and selected functions for the reset cost. We show that the large cost behaviour is dominated by resetting or the jump distribution according to the choice of the jump distribution. In the important case of a Laplace jump distribution, which corresponds to run-and-tumble particle dynamics, and linear costs for jumps and resetting, the Wiener-Hopf equation simplifies to a differential equation which can easily be solved.

cond-mat.stat-mech↗

arXiv2025

Exact height distribution in one-dimensional Edwards-Wilkinson interface with diffusing diffusivity

We study the height distribution of a one-dimensional Edwards-Wilkinson interface in the presence of a stochastic diffusivity $D(t)=B^2(t)$, where $B(t)$ represents a one-dimensional Brownian motion at time $t$. The height distribution at a fixed point is space is computed analytically. The typical height $h(x,t)$ at a given point in space is found to scale as $t^{3/4}$ and the distribution $G(H)$ of the scaled height $H=h/t^{3/4}$ is symmetric but with a nontrivial shape: while it approaches a nonzero constant quadratically as $H\to 0$, it has a non-Gaussian tail that decays exponentially for large $H$. We show that this exponential tail is rather robust and holds for a whole family of linear interface models parametrized by a dynamical exponent $z>1$, with $z=2$ corresponding to the Edwards-Wilkinson model.

cond-mat.stat-mech↗

arXiv2025

Extreme value statistics in a continuous time branching process: a pedagogical primer

We study a continuous time branching process where an individual splits into two daughters with rate b and dies with rate a, starting from a single individual at t=0. We show that the model can be mapped exactly to a random walk problem where the population size N(t) performs a random walk on a positive semi-infinite lattice. The hopping rate of this random walker out of a site labelled n is proportional to n, i.e., the walker gets more and more `agitated' as it moves further and further away from the origin--we call this an `agitated random walk' (ARW). We demonstrate that this random walk problem is particularly suitable to obtain exact explicit results on the extreme value statistics, namely, on the distribution of the maximal population size M(t)= \max_{0τ\le t}[N(τ)] up to time t. This extreme value distribution displays markedly different behaviors in the three phases: (i) subcritical (b a). In the subcritical and critical phases , Q(L,t) becomes independent of time t for large t and the stationary distribution Q(L, \infty) decays to zero with increasing L, respectively exponentially (subcritical) and algebraically (critical). For finite but large t, the distribution at the critical point exhibits a scaling form Q(L,t)\sim f_c(L/{at})/L^2 where the scaling function f_c(z) has a nontrivial shape that we compute analytically. In the supercritical phase, the distribution Q(L,t) has a `fluid' part that becomes independent of t for large t and a `condensate' part (a delta peak centered at e^{(b-a)t}) which gets disconnected from the `fluid' part and moves rapidly to \infty as time increases. We also verify our analytical predictions via numerical simulations finding excellent agreement.

cond-mat.stat-mech↗

arXiv2025

Exact joint distributions of three global characteristic times for Brownian motion

We consider three global characteristic times for a one-dimensional Brownian motion $x(τ)$ in the interval $τ\in [0,t]$: the occupation time $t_{\rm o}$ denoting the cumulative time where $x(τ)>0$, the time $t_{\rm m}$ at which the process achieves its global maximum in $[0,t]$ and the last-passage time $t_l$ through the origin before $t$. All three random variables have the same marginal distribution given by Lévy's arcsine law. We compute exactly the pairwise joint distributions of these three times and show that they are quite different from each other. The joint distributions display rather rich and nontrivial correlations between these times. Our analytical results are verified by numerical simulations.

cond-mat.stat-mech↗