arXiv subjects

Sanjib Sabhapandit

Publications and source records attributed to Sanjib Sabhapandit.

At least 55 records · Page 3Linked to original sources

arXiv2018

Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension

We investigate the motion of a run-and-tumble particle (RTP) in one dimension. We find the exact probability distribution of the particle with and without diffusion on the infinite line, as well as in a finite interval. In the infinite domain, this probability distribution approaches a Gaussian form in the long-time limit, as in the case of a regular Brownian particle. At intermediate times, this distribution exhibits unexpected multi-modal forms. In a finite domain, the probability distribution reaches a steady state form with peaks at the boundaries, in contrast to a Brownian particle. We also study the relaxation to the steady state analytically. Finally we compute the survival probability of the RTP in a semi-infinite domain. In the finite interval, we compute the exit probability and the associated exit times. We provide numerical verifications of our analytical results.

cond-mat.stat-mech↗

arXiv2018

Statistics of overtakes by a tagged agent

We consider a one-dimensional infinite lattice where at each site there sits an agent carrying a velocity, which is drawn initially for each agent independently from a common distribution. This system evolves as a Markov process where a pair of agents at adjacent sites exchange their positions with a specified rate, while retaining their respective velocities, only if the velocity of the agent on the left site is higher. We study the statistics of the net displacement of a tagged agent $m(t)$ on the lattice, in a given duration $t$, for two different kinds of rates: one in which a pair of agents at sites $i$ and $i+1$ exchange their sites with rate $1$, independent of the velocity difference between the neighbors, and another in which a pair exchange their sites with a rate equal to their relative speed. In both cases, we find $m(t)\sim t$ for large $t$. In the first case, for a randomly picked agent, $m/t$, in the limit $t\to \infty$, is distributed uniformly on $[-1,1]$ for all continuous distributions of velocities. In the second case, the distribution is given by the distribution of the velocities itself, with a Galilean shift by the mean velocity. We also find the large time approach to the limiting forms and compare the results with numerical simulations. In contrast, if the exchange of velocities occurs at unit rate, independent of their values, and irrespective of which is faster, $m(t)/t$ for large $t$ is has a gaussian distribution, whose width varies as $t^{-1/2}$.

cond-mat.stat-mech↗

arXiv2018

Extreme statistics and index distribution in the classical $1d$ Coulomb gas

We consider a one-dimensional gas of $N$ charged particles confined by an external harmonic potential and interacting via the one-dimensional Coulomb potential. For this system we show that in equilibrium the charges settle, on an average, uniformly and symmetrically on a finite region centred around the origin. We study the statistics of the position of the rightmost particle $x_{\max}$ and show that the limiting distribution describing its typical fluctuations is different from the Tracy-Widom distribution found in the one-dimensional log-gas. We also compute the large deviation functions which characterise the atypical fluctuations of $x_{\max}$ far away from its mean value. In addition, we study the gap between the two rightmost particles as well as the index $N_+$, i.e., the number of particles on the positive semi-axis. We compute the limiting distributions associated to the typical fluctuations of these observables as well as the corresponding large deviation functions. We provide numerical supports to our analytical predictions. Part of these results were announced in a recent Letter, Phys. Rev. Lett. 119, 060601 (2017).

cond-mat.stat-mech↗

arXiv2017

Partial entropy production in heat transport

We consider a system of two Brownian particles (say A and B), coupled to each other via harmonic potential of stiffness constant $k$. Particle-A is connected to two heat baths of constant temperatures $T_1$ and $T_2$, and particle-B is connected to a single heat bath of a constant temperature $T_3$. In the steady state, the total entropy production for both particles obeys the fluctuation theorem. We compute the total entropy production due to one of the particles called as partial or apparent entropy production, in the steady state for a time segment $τ$. When both particles are weakly interacting with each other, the fluctuation theorem for partial and apparent entropy production is studied. We find a significant deviation from the fluctuation theorem. The analytical results are also verified using numerical simulations.

cond-mat.stat-mech↗

arXiv2017

Gaps between avalanches in 1D Random Field Ising Models

We analyze the statistics of gaps ($ΔH$) between successive avalanches in one dimensional random field Ising models (RFIMs) in an external field $H$ at zero temperature. In the first part of the paper we study the nearest-neighbour ferromagnetic RFIM. We map the sequence of avalanches in this system to a non-homogeneous Poisson process with an $H$-dependent rate $ρ(H)$. We use this to analytically compute the distribution of gaps $P(ΔH)$ between avalanches as the field is increased monotonically from $-\infty$ to $+\infty$. We show that $P(ΔH)$ tends to a constant $\mathcal{C}(R)$ as $ΔH \to 0^+$, which displays a non-trivial behaviour with the strength of disorder $R$. We verify our predictions with numerical simulations. In the second part of the paper, motivated by avalanche gap distributions in driven disordered amorphous solids, we study a long-range antiferromagnetic RFIM. This model displays a gapped behaviour $P(ΔH) = 0$ up to a system size dependent offset value $ΔH_{\text{off}}$, and $P(ΔH) \sim (ΔH - ΔH_{\text{off}})^θ$ as $ΔH \to H_{\text{off}}^+$. We perform numerical simulations on this model and determine $θ\approx 0.95(5)$. We also discuss mechanisms which would lead to a non-zero exponent $θ$ for general spin models with quenched random fields.

cond-mat.dis-nn↗

arXiv2017

Unusual equilibration of a particle in a potential with a thermal wall

We consider a particle in a one-dimensional box of length $L$ with a Maxwell bath at one end and a reflecting wall at the other end. Using a renewal approach, as well as directly solving the master equation, we show that the system exhibits a slow power law relaxation with a logarithmic correction towards the final equilibrium state. We extend the renewal approach to a class of confining potentials of the form $U(x) \propto x^α$, $x>0$, where we find that the relaxation is $\sim t^{-(α+2)/(α-2)}$ for $α>2$, with a logarithmic correction when $(α+2)/(α-2)$ is an integer. For $α<2$ the relaxation is exponential. Interestingly for $α=2$ (harmonic potential) the localised bath can not equilibrate the particle.

cond-mat.stat-mech↗

arXiv2017

Exact extremal statistics in the classical $1d$ Coulomb gas

We consider a one-dimensional classical Coulomb gas of $N$ like-charges in a harmonic potential -- also known as the one-dimensional one-component plasma (1dOCP). We compute analytically the probability distribution of the position $x_{\max}$ of the rightmost charge in the limit of large $N$. We show that the typical fluctuations of $x_{\max}$ around its mean are described by a non-trivial scaling function, with asymmetric tails. This distribution is different from the Tracy-Widom distribution of $x_{\max}$ for the Dyson's log-gas. We also compute the large deviation functions of $x_{\max}$ explicitly and show that the system exhibits a third-order phase transition, as in the log-gas. Our theoretical predictions are verified numerically.

cond-mat.stat-mech↗

arXiv2017

Driven inelastic Maxwell gas in one dimension

A lattice version of the driven inelastic Maxwell gas is studied in one dimension with periodic boundary conditions. Each site $i$ of the lattice is assigned with a scalar `velocity', $v_i$. Nearest neighbors on the lattice interact, with a rate $τ_c^{-1}$, according to an inelastic collision rule. External driving, occurring with a rate $τ_w^{-1}$, sustains a steady state in the system. A set of closed coupled equations for the evolution of the variance and the two-point correlation is found. Steady state values of the variance, as well as spatial correlation functions, are calculated. It is shown exactly that the correlation function decays exponentially with distance, and the correlation length for a large system is determined. Furthermore, the spatio-temporal correlation $C(x,t)=\langle v_i(0) v_{i+x} (t)\rangle$ can also be obtained. We find that there is an interior region $-x^* < x < x^*$, where $C(x,t)$ has a time-dependent form, whereas in the exterior region $|x| > x^*$, the correlation function remains the same as the initial form. $C(x,t)$ exhibits second order discontinuity at the transition points $x=\pm x^*$ and these transition points move away from the $x=0$ with a constant speed.

cond-mat.stat-mech↗

arXiv2017

Velocity Distribution of Driven Inelastic One-component Maxwell gas

The nature of the velocity distribution of a driven granular gas, though well studied, is unknown as to whether it is universal or not, and if universal what it is. We determine the tails of the steady state velocity distribution of a driven inelastic Maxwell gas, which is a simple model of a granular gas where the rate of collision between particles is independent of the separation as well as the relative velocity. We show that the steady state velocity distribution is non-universal and depends strongly on the nature of driving. The asymptotic behavior of the velocity distribution are shown to be identical to that of a non-interacting model where the collisions between particles are ignored. For diffusive driving, where collisions with the wall are modelled by an additive noise, the tails of the velocity distribution is universal only if the noise distribution decays faster than exponential.

cond-mat.stat-mech↗

arXiv2016

Fluctuation theorem for entropy production of a partial system in the weak coupling limit

Small systems in contact with a heat bath evolve by stochastic dynamics. Here we show that, when one such small system is weakly coupled to another one, it is possible to infer the presence of such weak coupling by observing the violation of the steady state fluctuation theorem for the partial entropy production of the observed system. We give a general mechanism due to which the violation of the fluctuation theorem can be significant, even for weak coupling. We analytically demonstrate on a realistic model system that this mechanism can be realized by applying an external random force to the system. In other words, we find a new fluctuation theorem for the entropy production of a partial system, in the limit of weak coupling.

cond-mat.stat-mech↗

arXiv2016

Exact distributions of cover times for $N$ independent random walkers in one dimension

We study the probability density function (PDF) of the cover time $t_c$ of a finite interval of size $L$, by $N$ independent one-dimensional Brownian motions, each with diffusion constant $D$. The cover time $t_c$ is the minimum time needed such that each point of the entire interval is visited by at least one of the $N$ walkers. We derive exact results for the full PDF of $t_c$ for arbitrary $N \geq 1$, for both reflecting and periodic boundary conditions. The PDFs depend explicitly on $N$ and on the boundary conditions. In the limit of large $N$, we show that $t_c$ approaches its average value $\langle t_c \rangle \approx L^2/(16\, D \, \ln N)$, with fluctuations vanishing as $1/(\ln N)^2$. We also compute the centered and scaled limiting distributions for large $N$ for both boundary conditions and show that they are given by nontrivial $N$-independent scaling functions.

cond-mat.stat-mech↗

arXiv2015

Large deviations for Markov processes with resetting

Markov processes restarted or reset at random times to a fixed state or region in space have been actively studied recently in connection with random searches, foraging, and population dynamics. Here we study the large deviations of time-additive functions or observables of Markov processes with resetting. By deriving a renewal formula linking generating functions with and without resetting we are able to obtain the rate function of such observables, characterizing the likelihood of their fluctuations in the long-time limit. We consider as an illustration the large deviations of the area of the Ornstein-Uhlenbeck process with resetting. Other applications involving diffusions, random walks, and jump processes with resetting or catastrophes are discussed.

cond-mat.stat-mech↗

arXiv2015

Random walk with random resetting to the maximum

We study analytically a simple random walk model on a one-dimensional lattice, where at each time step the walker resets to the maximum of the already visited positions (to the rightmost visited site) with a probability $r$, and with probability $(1-r)$, it undergoes symmetric random walk, i.e., it hops to one of its neighboring sites, with equal probability $(1-r)/2$. For $r=0$, it reduces to a standard random walk whose typical distance grows as $\sqrt{n}$ for large $n$. In presence of a nonzero resetting rate $0<r\le 1$, we find that both the average maximum and the average position grow ballistically for large $n$, with a common speed $v(r)$. Moreover, the fluctuations around their respective averages grow diffusively, again with the same diffusion coefficient $D(r)$. We compute $v(r)$ and $D(r)$ explicitly. We also show that the probability distribution of the difference between the maximum and the location of the walker, becomes stationary as $n\to \infty$. However, the approach to this stationary distribution is accompanied by a dynamical phase transition, characterized by a weakly singular large deviation function. We also show that $r=0$ is a special `critical' point, for which the growth laws are different from the $r\to 0$ case and we calculate the exact crossover functions that interpolate between the critical $(r=0)$ and the off-critical $(r\to 0)$ behavior for finite but large $n$.

cond-mat.stat-mech↗

arXiv2015

Driven inelastic Maxwell gases

We consider the inelastic Maxwell model, which consists of a collection of particles that are characterized by only their velocities, and evolving through binary collisions and external driving. At any instant, a particle is equally likely to collide with any of the remaining particles. The system evolves in continuous time with mutual collisions and driving taken to be point processes with rates $τ_c^-{1}$ and $τ_w^{-1}$ respectively. The mutual collisions conserve momentum and are inelastic, with a coefficient of restitution $r$. The velocity change of a particle with velocity $v$, due to driving, is taken to be $Δv=-(1+r_w) v+η$, mimicking the collision with a vibrating wall, where $r_w$ the coefficient of restitution of the particle with the "wall" and $η$ is Gaussian white noise. The Ornstein-Uhlenbeck driving mechanism given by $\frac{dv}{dt}=-Γv+η$ is found to be a special case of the driving modeled as a point process. Using both the continuum and discrete versions we show that while the equations for the one-particle and the two-particle velocity distribution functions do not close, the joint evolution equations of the variance and the two-particle velocity correlation functions close. With the exact formula for the variance we find that, for $r_w\ne-1$, the system goes to a steady state. On the other hand, for $r_w=-1$, the system does not have a steady state. Similarly, the system goes to a steady state for the Ornstein-Uhlenbeck driving with $Γ\not=0$, whereas for the purely diffusive driving ($Γ=0$), the system does not have a steady state.

cond-mat.stat-mech↗

arXiv2015

Tagged particle diffusion in one-dimensional systems with Hamiltonian dynamics - II

We study various temporal correlation functions of a tagged particle in one-dimensional systems of interacting point particles evolving with Hamiltonian dynamics. Initial conditions of the particles are chosen from the canonical thermal distribution. The correlation functions are studied in finite systems, and their forms examined at short and long times. Various one-dimensional systems are studied. Results of numerical simulations for the Fermi-Pasta-Ulam chain are qualitatively similar to results for the harmonic chain, and agree unexpectedly well with a simple description in terms of linearized equations for damped fluctuating sound waves. Simulation results for the alternate mass hard particle gas reveal that - in contradiction to our earlier results [1] with smaller system sizes - the diffusion constant slowly converges to a constant value, in a manner consistent with mode coupling theories. Our simulations also show that the behaviour of the Lennard-Jones gas depends on its density. At low densities, it behaves like a hard-particle gas, and at high densities like an anharmonic chain. In all the systems studied, the tagged particle was found to show normal diffusion asymptotically, with convergence times depending on the system under study. Finite size effects show up at time scales larger than sound traversal times, their nature being system-specific.

cond-mat.stat-mech↗

arXiv2015

Exact probability distribution for the two-tag displacement in single-file motion

We consider a gas of point particles moving on the one-dimensional line with a hard-core inter-particle interaction that prevents particle crossings --- this is usually referred to as single-file motion. The individual particle dynamics can be arbitrary and they only interact when they meet. Starting from initial conditions such that particles are uniformly distributed, we observe the displacement of a tagged particle at time $t$, with respect to the initial position of another tagged particle, such that their tags differ by $r$. For $r=0$, this is the usual well studied problem of the tagged particle motion. Using a mapping to a non-interacting particle system we compute the exact probability distribution function for the two-tagged particle displacement, for general single particle dynamics. As by-products, we compute the large deviation function, various cumulants and, for the case of Hamiltonian dynamics, the two-particle velocity auto-correlation function.

cond-mat.stat-mech↗

arXiv2015

Dynamical transition in the temporal relaxation of stochastic processes under resetting

A stochastic process, when subject to resetting to its initial condition at a constant rate, generically reaches a non-equilibrium steady state. We study analytically how the steady state is approached in time and find an unusual relaxation mechanism in these systems. We show that as time progresses, an inner core region around the resetting point reaches the steady state, while the region outside the core is still transient. The boundaries of the core region grow with time as power laws at late times. Alternatively, at a fixed spatial point, the system undergoes a dynamical transition from the transient to the steady state at a characteristic space dependent timescale $t^*(x)$. We calculate analytically in several examples the large deviation function associated with this spatio-temporal fluctuation and show that generically it has a second order discontinuity at a pair of critical points characterizing the edges of the inner core. Our results are verified in the numerical simulations of several models, such as simple diffusion and fluctuating one-dimensional interfaces.

cond-mat.stat-mech↗

arXiv2014

Universal Large Deviations for the Tagged Particle in Single File Motion

We consider a gas of point particles moving in a one-dimensional channel with a hard-core inter-particle interaction that prevents particle crossings --- this is called single-file motion. Starting from equilibrium initial conditions we observe the motion of a tagged particle. It is well known that if the individual particle dynamics is diffusive, then the tagged particle motion is sub-diffusive, while for ballistic particle dynamics, the tagged particle motion is diffusive. Here we compute exactly the large deviation function for the tagged particle displacement and show that this is universal, independent of the individual dynamics.

cond-mat.stat-mech↗