- Filling an empty lattice by local injection of quantum particles. Akash Trivedi, Sparsh Gupta, Bijay Kumar Agarwalla, Abhishek Dhar, Manas Kulkarni, Anupam Kundu, and Sanjib Sabhapandit, Phys. Rev. A 108, 052204 (2023).
We study the quantum dynamics of filling an empty lattice of size $L$ by connecting it locally with an equilibrium thermal bath that injects noninteracting bosons or fermions. We adopt four different approaches, namely, (i) direct exact numerics, (ii) Redfield equation, (iii) Lindblad equation, and (iv) quantum Langevin equation, which are unique in their ways for solving the time dynamics and the steady state. In this simple setup we investigate open quantum dynamics and subsequent approach to thermalization. The quantities of interest that we consider are the spatial density profile and the total number of bosons and fermions in the lattice. The spatial spread is ballistic in nature and the local occupation eventually settles down owing to equilibration. The ballistic spread of local density admits a universal scaling form. We show that this universality is only seen when the condition of detailed balance is satisfied by the baths. The difference between bosons and fermions shows up in the early time growth rate and the saturation values of the profile. The techniques developed here are applicable to systems in arbitrary dimensions and for arbitrary geometries.
DOI: 10.1103/PhysRevA.108.052204. arXiv e-print: 2209.08014.
- Long time behavior of run-and-tumble particles in two dimensions. Ion Santra, Urna Basu, and Sanjib Sabhapandit, J. Stat. Mech. 2023, 033203 (2023).
We study the long-time asymptotic behavior of the position distribution of a run-and-tumble particle (RTP) in two dimensions in the presence of translational diffusion and show that the distribution at a time $t$ can be expressed as a perturbative series in $(\gamma t)^{-1}$, where $\gamma^{-1}$ is the persistence time of the RTP. We show that the higher order corrections to the leading order Gaussian distribution generically satisfy an inhomogeneous diffusion equation where the source term depends on the previous order solutions. The explicit solution of the inhomogeneous equation requires the position moments, and we develop a recursive formalism to compute the same. We find that the subleading corrections undergo shape transitions as the translational diffusion is increased.
DOI: 10.1088/1742-5468/acbc22. arXiv e-print: 2211.07337.
- Effect of stochastic resetting on Brownian motion with stochastic diffusion coefficient. Ion Santra, Urna Basu, and Sanjib Sabhapandit, J. Phys. A: Math. Theor. 55, 414002 (2022).
We study the dynamics of a Brownian motion with a diffusion coefficient which evolves stochastically. We first study this process in arbitrary dimensions and find the scaling form and the corresponding scaling function of the position distribution. We find that the tails of the distribution have exponential tails with a ballistic scaling. We then introduce the resetting dynamics where, at a constant rate, both the position and the diffusion coefficient are reset to zero. This eventually leads to a nonequilibrium stationary state, which we study in arbitrary dimensions. In stark contrast to ordinary Brownian motion under resetting, the stationary position distribution in one dimension has a logarithmic divergence at the origin. For higher dimensions, however, the divergence disappears and the distribution attains a dimension-dependent constant value at the origin, which we compute exactly. The distribution has a generic stretched exponential tail in all dimensions. We also study the approach to the stationary state and find that, as time increases, an inner core region around the origin attains the stationary state, while the outside region still has a transient distribution—this inner stationary region grows $\sim t^2$, i.e., with a constant acceleration, much faster than ordinary Brownian motion.
DOI: 10.1088/1751-8121/ac8dcc. arXiv e-print: 2205.00549.
- Universal framework for the long-time position distribution of free active particles. Ion Santra, Urna Basu, and Sanjib Sabhapandit, J. Phys. A: Math. Theor. 55, 385002 (2022).
Active particles self-propel themselves with a stochastically evolving velocity, generating a persistent motion leading to a non-diffusive behavior of the position distribution. Nevertheless, an effective diffusive behavior emerges at times much larger than the persistence time. Here we develop a general framework for studying the long-time behavior for a class of active particle dynamics and illustrate it using the examples of run-and-tumble particle, active Ornstein–Uhlenbeck particle, active Brownian particle, and direction reversing active Brownian particle. Treating the ratio of the persistence-time to the observation time as the small parameter, we show that the position distribution generically satisfies the diffusion equation at the leading order. We further show that the sub-leading contributions, at each order, satisfies an inhomogeneous diffusion equation, where the source term depends on the previous order solutions. We explicitly obtain a few sub-leading contributions to the Gaussian position distribution. As a part of our framework, we also prescribe a way to find the position moments recursively and compute the first few explicitly for each model.
DOI: 10.1088/1751-8121/ac864c. arXiv e-print: 2202.12117.
- Record statistics for random walks and Lévy flights with resetting. Satya N Majumdar, Philippe Mounaix, Sanjib Sabhapandit, and Grégory Schehr, J. Phys. A: Math. Theor. 55, 034002 (2022).
We compute exactly the mean number of records $\langle R_N\rangle$ for a time-series of size $N$ whose entries represent the positions of a discrete time random walker on the line with resetting. At each time step, the walker jumps by a length $\eta$ drawn independently from a symmetric and continuous distribution $f(\eta)$ with probability $1 – r$ (with $0 \le r < 1$) and with the complementary probability $r$ it resets to its starting point $x = 0$. This is an exactly solvable example of a weakly correlated time-series that interpolates between a strongly correlated random walk series (for $r = 0$) and an uncorrelated time-series (for $(1 – r) \ll 1$). Remarkably, we found that for every fixed and any $N$, the mean number of records $\langle R_N\rangle$ is completely universal, i.e. independent of the jump distribution $f(\eta)$. In particular, for large $N$, we show that $\langle R_N\rangle$ grows very slowly with increasing $N$ as for $0 < r < 1$. We also computed the exact universal crossover scaling functions for $\langle R_N\rangle$ in the two limits $r\to 0$ and $r \to 1$. Our analytical predictions are in excellent agreement with numerical simulations.
DOI: 10.1088/1751-8121/ac3fc1. arXiv e-print: 2110.01539.
- Direction reversing active Brownian particle in a harmonic potential. Ion Santra, Urna Basu, and Sanjib Sabhapandit, Soft Matter 17, 10108 (2021).
We study the two-dimensional motion of an active Brownian particle of speed $v_0$, with intermittent directional reversals in the presence of a harmonic trap of strength $\mu$. The presence of the trap ensures that the position of the particle eventually reaches a steady state where it is bounded within a circular region of radius $v_0/\mu$, centered at the minimum of the trap. Due to the interplay between the rotational diffusion constant $D_R$, reversal rate $\gamma$, and the trap strength $\mu$, the steady state distribution shows four different types of shapes, which we refer to as active-I & II, and passive-I & II phases. In the active-I phase, the weight of the distribution is concentrated along an annular region close to the circular boundary, whereas in active-II, an additional central diverging peak appears giving rise to a Mexican hat-like shape of the distribution. The passive-I is marked by a single Boltzmann-like centrally peaked distribution in the large $D_R$ limit. On the other hand, while the passive-II phase also shows a single central peak, it is distinguished from passive-I by a non-Boltzmann like divergence near the origin. We characterize these phases by calculating the exact analytical forms of the distributions in various limiting cases. In particular, we show that for $D_R \ll \gamma$, the shape transition of the two-dimensional position distribution from active-II to passive-II occurs at $\mu=\gamma$. We compliment these analytical results with numerical simulations beyond the limiting cases and obtain a qualitative phase diagram in the ($D_R$, $\gamma$, $\mu-1$) space.
DOI: 10.1039/D1SM01118A. arXiv e-print: 2107.12640.
- Active Brownian motion with directional reversals. Ion Santra, Urna Basu, and Sanjib Sabhapandit, Phys. Rev. E 104, L012601 (2021). [Letter]
Active Brownian motion with intermittent direction reversals are common in a class of bacteria like {\it Myxococcus xanthus} and {\it Pseudomonas putida}. We show that, for such a motion in two dimensions, the presence of the two time scales set by the rotational diffusion constant $D_R$ and the reversal rate $\gamma$ gives rise to four distinct dynamical regimes: (I) $t\ll \min (\gamma^{-1}, D_R^{-1}),$ (II) $\gamma^{-1}\ll t\ll D_R^{-1}$, (III) $D_R^{-1} \ll t \ll \gamma^{-1}$, and (IV) $t\gg \max (\gamma^{-1}$, $D_R^{-1})$, showing distinct behaviors. We characterize these behaviors by analytically computing the position distribution and persistence exponents. The position distribution shows a crossover from a strongly non-diffusive and anisotropic behavior at short-times to a diffusive isotropic behavior via an intermediate regime (II) or (III). In regime (II), we show that, the position distribution along the direction orthogonal to the initial orientation is a function of the scaled variable $z\propto x_{\perp}/t$ with a non-trivial scaling function, $f(z)=(2\pi^3)^{-1/2}\Gamma(1/4+iz)\Gamma(1/4-iz)$. Furthermore, by computing the exact first-passage time distribution, we show that a novel persistence exponent $\alpha=1$ emerges due to the direction reversal in this regime.
DOI: 10.1103/PhysRevE.104.L012601. arXiv e-print: 2101.11327.
- Run-and-tumble particles in two dimensions under stochastic resetting conditions. Ion Santra, Urna Basu, and Sanjib Sabhapandit, J. Stat. Mech. 2020, 113206 (2020).
We study the effect of stochastic resetting on a run-and-tumble particle (RTP) in two spatial dimensions. We consider a resetting protocol which affects both the position and orientation of the RTP: the particle undergoes constant-rate positional resetting to a fixed point in space and a random orientation. We compute the radial and $x$-marginal stationary-state distributions and show that while the former approaches a constant value as $r \to 0$, the latter diverges logarithmically as $x \to 0$. On the other hand, both the marginal distributions decay exponentially with the same exponent when they are far from the origin. We also study the temporal relaxation of the RTP and show that the positional distribution undergoes a dynamic transition to a stationary state. We also study the first-passage properties of the RTP in the presence of resetting and show that the optimization of the resetting rate can minimize the mean first-passage time. We also provide a brief discussion of the stationary states for resetting a particle to an initial position with a fixed orientation.
DOI: 10.1088/1742-5468/abc7b7. arXiv e-print: 2009.09891.
- Freezing Transition in the Barrier Crossing Rate of a Diffusing Particle. Sanjib Sabhapandit and Satya N. Majumdar, Phys. Rev. Lett. 125, 200601 (2020).
We study the decay rate $\theta(a)$ that characterizes the late time exponential decay of the first-passage probability density $F_a(t|0) \sim e^{-\theta(a)\, t}$ of a diffusing particle in a one dimensional confining potential $U(x)$, starting from the origin, to a position located at $a>0$. For general confining potential $U(x)$ we show that $\theta(a)$, a measure of the barrier (located at $a$) crossing rate, has three distinct behaviors as a function of $a$, depending on the tail of $U(x)$ as $x\to -\infty$. In particular, for potentials behaving as $U(x)\sim |x|$ when $x\to -\infty$, we show that a novel freezing transition occurs at a critical value $a=a_c$, i.e, $\theta(a)$ increases monotonically as $a$ decreases till $a_c$, and for $a \le a_c$ it freezes to $\theta (a)=\theta(a_c)$. Our results are established using a general mapping to a quantum problem and by exact solution in three representative cases, supported by numerical simulations. We show that the freezing transition occurs when in the associated quantum problem, the gap between the ground state (bound) and the continuum of scattering states vanishes.
DOI: 10.1103/PhysRevLett.125.200601. arXiv e-print: 2005.00024.
- Run-and-tumble particle in inhomogeneous media in one dimension. Prashant Singh, Sanjib Sabhapandit, and Anupam Kundu, J. Stat. Mech. 2020, 083207 (2020).
We investigate the run and tumble particle (RTP), also known as persistent Brownian motion, in one dimension. A telegraphic noise $\sigma(t)$ drives the particle which changes between $\pm 1$ values with some rates. Denoting the rate of flip from $1$ to $-1$ as $R_1$ and the converse rate as $R_2$, we consider the position and direction dependent rates of the form $R_1(x)=\left(\frac{\mid x \mid}{l}\right) ^{\alpha}\left[\gamma_1~\theta(x)+\gamma_2 ~\theta (-x)\right]$ and $R_2(x)=\left(\frac{\mid x \mid}{l}\right) ^{\alpha}\left[\gamma_2~\theta(x)+\gamma_1 ~\theta (-x)\right]$ with $\alpha \geq 0$. For $\gamma_1 >\gamma_2$, we find that the particle exhibits a steady-state probability distriution even in an infinite line whose exact form depends on $\alpha$. For $\alpha =0$ and $1$, we solve the master equations exactly for arbitrary $\gamma_1$ and $\gamma_2$ at large $t$. From our explicit expression for time-dependent probability distribution $P(x,t)$ we find that it exponentially relaxes to the steady-state distribution for $\gamma_1 > \gamma_2$. On the other hand, for $\gamma_1<\gamma_2$, the large $t$ behaviour of $P(x,t)$ is drastically different than $\gamma_1=\gamma_2$ case where the distribution decays as $t^{-\frac{1}{2}}$. Contrary to the latter, detailed balance is not obeyed by the particle even at large $t$ in the former case. For general $\alpha$, we argue that the approach to the steady state in $\gamma_1>\gamma_2$ case is exponential which we numerically demonstrate. On the other hand for $\gamma_1\leq \gamma_2$, the distribution $P(x,t)$ does not reach a steady state, however posseses certain scaling behaviour. For $\gamma_1=\gamma_2$ we derive this scaling behaviour as well as the scaling function rigorously whereas for $\gamma_1< \gamma_2$ we provide heuristic arguments for the scaling behaviour and the corresponding scaling functions. We also study the dynamics in semi-infinite line with an absorbing barrier at the origin. We analytically compute the survival probabilities and first-passage time distributions for $\alpha =0$ and $1$. For general $\alpha \geq 0$, once again we compute the value of survival probability at large $t$ and approach to it. Finally, we consider RTP in an finite interval $[0,M]$ and compute the associated exit probability from that interval for all $\alpha$. All our analytic results match with the numerical simulation of the same.
DOI: 10.1088/1742-5468/aba7b1. arXiv e-print: 2004.11041.
- Run-and-tumble particles in two dimensions: Marginal position distributions. Ion Santra, Urna Basu, and Sanjib Sabhapandit, Phys. Rev. E 101, 062120 (2020).
We study a set of Run-and-tumble particle (RTP) dynamics in two spatial dimensions. In the first case of the orientation $\theta$ of the particle can assume a set of $n$ possible discrete values while in the second case $\theta$ is a continuous variable. We calculate exactly the marginal position distributions for $n=3,4$ and the continuous case and show that in all the cases the RTP shows a cross-over from a ballistic to diffusive regime. The ballistic regime is a typical signature of the active nature of the systems and is characterized by non-trivial position distributions which depends on the specific model. We also show that, the signature of activity at long-times can be found in the atypical fluctuations which we also characterize by computing the large deviation functions explicitly.
DOI: 10.1103/PhysRevE.101.062120. arXiv e-print: 2004.07562.
- Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap. Urna Basu, Satya N Majumdar, Alberto Rosso, Sanjib Sabhapandit, and Grégory Schehr, J. Phys. A: Math. Theor. 53, 09LT01 (2020).
We study the motion of a one-dimensional run-and-tumble particle with three discrete internal states in the presence of a harmonic trap of stiffness $\mu.$ The three internal states, corresponding to positive, negative and zero velocities respectively, evolve following a jump process with rate $\gamma$. We compute the stationary position distribution exactly for arbitrary values of $\mu$ and $\gamma$ which turns out to have a finite support on the real line. We show that the distribution undergoes a shape-transition as $\beta=\gamma/\mu$ is changed. For $\beta<1,$ the distribution has a double-concave shape and shows algebraic divergences with an exponent $(\beta-1)$ both at the origin and at the boundaries. For $\beta>1,$ the position distribution becomes convex, vanishing at the boundaries and with a single, finite, peak at the origin. We also show that for the special case $\beta=1,$ the distribution shows a logarithmic divergence near the origin while saturating to a constant value at the boundaries.
DOI: 10.1088/1751-8121/ab6af0. arXiv e-print: 1910.10083.
- Dynamical correlations of conserved quantities in the one-dimensional equal mass hard particle gas. Aritra Kundu, Abhishek Dhar, and Sanjib Sabhapandit, J. Stat. Mech. 2020, 023205 (2020).
We study a gas of point particles with hard-core repulsion in one dimension where the particles move freely in-between elastic collisions. We prepare the system with a uniform density on the infinite line. The velocities $\{v_i; i \in \mathbb{Z} \}$ of the particles are chosen independently from a thermal distribution. Using a mapping to the non-interacting gas, we analytically compute the equilibrium spatio-temporal correlations $\langle v_i^m(t) v_j^n(0)\rangle$ for arbitrary integers $m,n$. The analytical results are verified with microscopic simulations of the Hamiltonian dynamics. The correlation functions have ballistic scaling, as expected in an integrable model.
DOI: 10.1088/1742-5468/ab5d0c. arXiv e-print: 1909.01698.
- Entropy production for partially observed harmonic systems. Deepak Gupta and Sanjib Sabhapandit, J. Stat. Mech. 2020, 013204 (2020).
The probability distribution of the total entropy production in the non-equilibrium steady state follows a symmetry relation called the fluctuation theorem. When a certain part of the system is masked or hidden, it is difficult to infer the exact estimate of the total entropy production. Entropy produced from the observed part of the system shows significant deviation from the steady state fluctuation theorem. This deviation occurs due to the interaction between the observed and the masked part of the system. A naive guess would be that the deviation from the steady state fluctuation theorem may disappear in the limit of small interaction between both parts of the system. In contrast, we investigate the entropy production of a particle in a harmonically coupled Brownian particle system (say, particle A and B) in a heat reservoir at a constant temperature. The system is maintained in the non-equilibrium steady state using stochastic driving. When the coupling between particle A and B is infinitesimally weak, the deviation from the steady state fluctuation theorem for the entropy production of a partial system of a coupled system is studied. Furthermore, we consider a harmonically confined system (i.e. a harmonically coupled system of particle A and B in harmonic confinement). In the weak coupling limit, the entropy produced by the partial system (e.g. particle A) of the coupled system in a harmonic trap satisfies the steady state fluctuation theorem. Numerical simulations are performed to support the analytical results. Part of these results were announced in a recent letter, see Gupta and Sabhapandit, Europhys. Lett. 115, 60003 (2016).
DOI: 10.1088/1742-5468/ab54b6. arXiv e-print: 1710.11339.
- Non-equilibrium dynamics of the piston in the Szilard engine. Deepak Bhat, Abhishek Dhar, Anupam Kundu, and Sanjib Sabhapandit, Europhy. Lett. 127, 10004 (2019).
We consider a Szilard engine in one dimension, consisting of a single particle of mass $m$, moving between a piston of mass $M$, and a heat reservoir at temperature $T$. In addition to an external force, the piston experiences repeated elastic collisions with the particle. We find that the motion of a heavy piston ($M \gg m$), can be described effectively by a Langevin equation. Various numerical evidences suggest that the frictional coefficient in the Langevin equation is given by $\gamma = (1/X)\sqrt{8 \pi m k_BT}$, where $X$ is the position of the piston measured from the thermal wall. Starting from the exact master equation for the full system and using a perturbation expansion in $\epsilon= \sqrt{m/M}$, we integrate out the degrees of freedom of the particle to obtain the effective Fokker-Planck equation for the piston albeit with a different frictional coefficient. Our microscopic study shows that the piston is never in equilibrium during the expansion step, contrary to the assumption made in the usual Szilard engine analysis — nevertheless the conclusions of Szilard remain valid.
DOI: 10.1209/0295-5075/127/10004. arXiv e-print: 1810.07890.
- Velocity distribution of driven granular gases. V V Prasad, Dibyendu Das, Sanjib Sabhapandit, and R Rajesh, J. Stat. Mech. 2019, 063201 (2019).
The granular gas is a paradigm for understanding the effects of inelastic interactions in granular materials. Kinetic theory provides a general theoretical framework for describing the granular gas. Its central result is that the tail of the velocity distribution of a driven granular gas is a stretched exponential that, counterintuitively, decays slower than that of the corresponding elastic gas in equilibrium. However, a derivation of this result starting from a microscopic model is lacking. Here, we obtain analytical results for a microscopic model for a granular gas where particles with two-dimensional velocities are driven homogeneously and isotropically by reducing the velocities by a factor and adding a stochastic noise. We find two universal regimes. For generic physically relevant driving, we find that the tail of the velocity distribution is a Gaussian with additional logarithmic corrections. Thus, the velocity distribution decays faster than the corresponding equilibrium gas. The second universal regime is less generic and corresponds to the scenario described by kinetic theory. Here, the velocity distribution is shown to decay as an exponential with additional logarithmic corrections, in contradiction to the predictions of the phenomenological kinetic theory, necessitating a re-examination of its basic assumptions.
DOI: 10.1088/1742-5468/ab11da. arXiv e-print: 1804.02558.
- Run-and-tumble particle in one-dimensional confining potentials: Steady-state, relaxation, and first-passage properties. Abhishek Dhar, Anupam Kundu, Satya N. Majumdar, Sanjib Sabhapandit, and Grégory Schehr, Phys. Rev. E 99, 032132 (2019).
We study the dynamics of a one-dimensional run and tumble particle subjected to confining potentials of the type $V(x) = \alpha \, |x|^p$, 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 $(p, \alpha)$-plane. For $p>1$, the distribution has a finite support in $[x_-,x_+]$ and there is a critical line $\alpha_c(p)$ that separates an active-like phase for $\alpha > \alpha_c(p)$ where $P(x)$ diverges at $x_\pm$, from a passive-like phase for $\alpha < \alpha_c(p)$ where $P(x)$ vanishes at $x_\pm$. For $p<1$, the stationary density $P(x)$ collapses to a delta function at the origin, $P(x) = \delta(x)$. In the marginal case $p=1$, we show that, for $\alpha < \alpha_c$, the stationary density $P(x)$ is a symmetric exponential, while for $\alpha > \alpha_c$, it again is a delta function $P(x) = \delta(x)$. 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.
DOI: 10.1103/PhysRevE.99.032132. arXiv e-print: 1811.03808.
- Statistics of overtake events by a tagged agent. Santanu Das, Deepak Dhar, and Sanjib Sabhapandit, Phys. Rev. E 98, 052122 (2018).
We consider a minimalist model of overtaking dynamics in one dimension. On each site of a one-dimensional infinite lattice sits an agent carrying a random number specifying the agent’s preferred velocity, which is drawn initially for each agent independently from a common distribution. The time evolution is Markovian, where a pair of agents at adjacent sites exchange their positions with a specified rate, while retaining their respective preferred velocities, only if the preferred velocity of the agent on the “left” site is higher. We discuss two different cases: one in which a pair of agents at sites $i$ and $i+1$ exchange their positions with rate 1, independent of their velocity difference, and another in which a pair exchange their positions with a rate equal to the modulus of the velocity difference. In both cases, we find that the net number of overtake events by a tagged agent in a given duration $t$, denoted by $m(t)$, increases linearly with time $t$, for large $t$. In the first case, for a randomly picked agent, $ m/t$, in the limit $t\to\infty$, is distributed uniformly on $[-1,1]$, independent of the distributions of preferred velocities. In the second case, the distribution is given by the distribution of the preferred 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.
DOI: 10.1103/PhysRevE.98.052122. arXiv e-print: 1804.07280.
- Extreme statistics and index distribution in the classical 1d Coulomb gas. Abhishek Dhar, Anupam Kundu, Satya N Majumdar, Sanjib Sabhapandit, and Grégory Schehr, J. Phys. A: Math. Theor. 51, 295001 (2018).
We consider a 1D gas of $N$ charged particles confined by an external harmonic potential and interacting via the 1D 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 1D 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).
DOI: 10.1088/1751-8121/aac75f. arXiv e-print: 1802.10374.
- Partial entropy production in heat transport. Deepak Gupta and Sanjib Sabhapandit, J. Stat. Mech. 2018, 063203 (2018).
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 $\tau$. 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.
DOI: 10.1088/1742-5468/aabfca. arXiv e-print: 1712.02102.
- Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension. Kanaya Malakar, V Jemseena, Anupam Kundu, K Vijay Kumar, Sanjib Sabhapandit, Satya N Majumdar, S Redner, and Abhishek Dhar, J. Stat. Mech. 2018, 043215 (2018).
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.
DOI: 10.1088/1742-5468/aab84f. arXiv e-print: 1711.08474.
- Unusual equilibration of a particle in a potential with a thermal wall. Deepak Bhat, Sanjib Sabhapandit, Anupam Kundu, and Abhishek Dhar, J. Stat. Mech. 2017, 113210 (2017).
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 $U(x) \propto x^\alpha$, $x>0$, where we find that the relaxation is $\sim t^{-(\alpha+2)/(\alpha-2)}$ for $\alpha >2$, with a logarithmic correction when $(\alpha+2)/(\alpha-2)$ is an integer. For $\alpha <2$ the relaxation is exponential. Interestingly for $\alpha=2$ (harmonic potential) the localised bath can not equilibrate the particle.
DOI: 10.1088/1742-5468/aa9683. arXiv e-print: 1707.00814.
- Stochastic efficiency of an isothermal work-to-work converter engine. Deepak Gupta and Sanjib Sabhapandit, Phys. Rev. E 96, 042130 (2017).
We investigate the efficiency of an isothermal Brownian work-to-work converter engine, composed of a Brownian particle coupled to a heat bath at a constant temperature. The system is maintained out of equilibrium by using two external time-dependent stochastic Gaussian forces, where one is called load force and the other is called drive force. Work done by these two forces are stochastic quantities. The efficiency of this small engine is defined as the ratio of stochastic work done against load force to stochastic work done by the drive force. The probability density function as well as large deviation function of the stochastic efficiency are studied analytically and verified by numerical simulations.
DOI: 10.1103/PhysRevE.96.042130. arXiv e-print: 1707.06843.
- Gaps between avalanches in one-dimensional random-field Ising models. Jishnu N. Nampoothiri, Kabir Ramola, Sanjib Sabhapandit, and Bulbul Chakraborty, Phys. Rev. E 96, 032107 (2017).
We analyze the statistics of gaps ($\Delta H$) 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-neighbor ferromagnetic RFIM. We map the sequence of avalanches in this system to a non-homogeneous Poisson process with an $H$-dependent rate $\rho(H)$. We use this to analytically compute the distribution of gaps $P(\Delta H)$ between avalanches as the field is increased monotonically from $-\infty$ to $+\infty$. We show that $P(\Delta H)$ tends to a constant $\mathcal{C}(R)$ as $\Delta H \to 0^+$, which displays a non-trivial behavior 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 behavior $P(\Delta H) = 0$ up to a system size dependent offset value $\Delta H_{\text{off}}$, and $P(\Delta H) \sim (\Delta H – \Delta H_{\text{off}})^{\theta}$ as $\Delta H \to H_{\text{off}}^+$. We perform numerical simulations on this model and determine $\theta \approx 0.95(5)$. We also discuss mechanisms which would lead to a non-zero exponent $\theta$ for general spin models with quenched random fields.
DOI: 10.1103/PhysRevE.96.032107. arXiv e-print: 1705.09069.
- Exact Extremal Statistics in the Classical 1D Coulomb Gas. Abhishek Dhar, Anupam Kundu, Satya N. Majumdar, Sanjib Sabhapandit, and Grégory Schehr, Phys. Rev. Lett. 119, 060601 (2017).
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.
DOI: 10.1103/PhysRevLett.119.060601. arXiv e-print: 1704.08973.
- Velocity distribution of a driven inelastic one-component Maxwell gas. V. V. Prasad, Dibyendu Das, Sanjib Sabhapandit, and R. Rajesh, Phys. Rev. E 95, 032909 (2017).
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 nonuniversal and depends strongly on the nature of driving. The asymptotic behavior of the velocity distribution is shown to be identical to that of a noninteracting model where the collisions between particles are ignored. For diffusive driving, where collisions with the wall are modeled by an additive noise, the tails of the velocity distribution is universal only if the noise distribution decays faster than exponential.
DOI: 10.1103/PhysRevE.95.032909. arXiv e-print: 1701.03600.
- Driven inelastic Maxwell gas in one dimension. V. V. Prasad, Sanjib Sabhapandit, Abhishek Dhar, and Onuttom Narayan, Phys. Rev. E 95, 022115 (2017).
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 $\tau_c^{-1}$, according to an inelastic collision rule. External driving, occurring with a rate $\tau_w^{-1}$, 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.
DOI: 10.1103/PhysRevE.95.022115. arXiv e-print: 1606.09561.
- Exact distributions of cover times for $N$ independent random walkers in one dimension. Satya N. Majumdar, Sanjib Sabhapandit, and Grégory Schehr, Phys. Rev. E 94, 062131 (2016).
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.
DOI: 10.1103/PhysRevE.94.062131. arXiv e-print: 1609.06325.
- Fluctuation theorem for entropy production of a partial system in the weak-coupling limit. Deepak Gupta and Sanjib Sabhapandit, Europhy. Lett. 115, 60003 (2016).
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.
DOI: 10.1209/0295-5075/115/60003. arXiv e-print: 1603.03506.
- Large deviations for Markov processes with resetting. Janusz M. Meylahn, Sanjib Sabhapandit, and Hugo Touchette, Phys. Rev. E 92, 062148 (2015).
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.
DOI: 10.1103/PhysRevE.92.062148. arXiv e-print: 1510.02431.
- Random walk with random resetting to the maximum position. Satya N. Majumdar, Sanjib Sabhapandit, and Grégory Schehr, Phys. Rev. E 92, 052126 (2015). [Editors’ Suggestion]
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 $0 < r \leq 1$, 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 $n$.
DOI: 10.1103/PhysRevE.92.052126. arXiv e-print: 1509.04516.
- Exact probability distribution for the two-tag displacement in single-file motion. Sanjib Sabhapandit and Abhishek Dhar, J. Stat. Mech. 2015, P07024 (2015).
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.
DOI: 10.1088/1742-5468/2015/07/p07024. arXiv e-print: 1506.01824.
- Dynamical transition in the temporal relaxation of stochastic processes under resetting. Satya N. Majumdar, Sanjib Sabhapandit, and Grégory Schehr, Phys. Rev. E 91, 052131 (2015).
A stochastic process, when subject to resetting to its initial condition at a constant rate, generically reaches a nonequilibrium 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 with new exponents. 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 spatiotemporal fluctuation and show that, generically, it has a second-order discontinuity at a pair of critical points characterizing the edges of the inner core. These singularities act as separatrices between typical and atypical trajectories. Our results are verified in the numerical simulations of several models, such as simple diffusion and fluctuating one-dimensional interfaces.
DOI: 10.1103/PhysRevE.91.052131. arXiv e-print: 1502.07124.
- Tagged Particle Diffusion in One-Dimensional Systems with Hamiltonian Dynamics-II. Anjan Roy, Abhishek Dhar, Onuttom Narayan, and Sanjib Sabhapandit, J. Stat. Phys. 160, 73 (2015).
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 [Roy et al. in J Stat Phys 150(5):851-866, (2013)] 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.
DOI: 10.1007/s10955-015-1232-y. arXiv e-print: 1405.5718.
- Driven inelastic Maxwell gases. V. V. Prasad, Sanjib Sabhapandit, and Abhishek Dhar, Phys. Rev. E 90, 062130 (2014).
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 $\tau_c^{-1}$ and $\tau_w^{-1}$, 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 $\Delta v=-(1+r_w) v+\eta$, where $r_w\in [-1,1]$ and $\eta$ is Gaussian white noise. For $r_w\in(0,1]$, this driving mechanism mimics the collision with a randomly moving wall, where $r_w$ is the coefficient of restitution. Another special limit of this driving is the so-called Ornstein-Uhlenbeck process given by $\frac{dv}{dt}=-\Gamma v+\eta$. We show that while the equations for the $n$-particle velocity distribution functions ($n=1,2,\dotsc$) 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. Also we obtain the exact tail of the velocity distribution in the 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 $\Gamma\not=0$, whereas for the purely diffusive driving ($\Gamma=0$), the system does not have a steady state.
DOI: 10.1103/PhysRevE.90.062130. arXiv e-print: 1408.3964.
- First Order Transition for the Optimal Search Time of Lévy Flights with Resetting. Lukasz Kusmierz, Satya N. Majumdar, Sanjib Sabhapandit, and Grégory Schehr, Phys. Rev. Lett. 113, 220602 (2014).
We study analytically an intermittent search process in one dimension. There is an immobile target at the origin and a searcher undergoes a discrete time jump process starting at $x_0\geq0$, where successive jumps are drawn independently from an arbitrary jump distribution $f(\eta)$. In addition, with a probability $0\leq r \leq1$ the position of the searcher is reset to its initial position $x_0$. The efficiency of the search strategy is characterized by the mean time to find the target, i.e., the mean first passage time (MFPT) to the origin. For arbitrary jump distribution $f(\eta)$, initial position $x_0$ and resetting probability $r$, we compute analytically the MFPT. For the heavy-tailed Lévy stable jump distribution characterized by the Lévy index $0<\mu < 2$, we show that, for any given $x_0$, the MFPT has a global minimum in the $(\mu,r)$ plane at $(\mu^*(x_0),r^*(x_0))$. We find a remarkable first-order phase transition as $x_0$ crosses a critical value $x_0^*$ at which the optimal parameters change discontinuously. Our analytical results are in good agreement with numerical simulations.
DOI: 10.1103/PhysRevLett.113.220602. arXiv e-print: 1409.1733.
- Work fluctuations for a Brownian particle driven by a correlated external random force. Arnab Pal and Sanjib Sabhapandit, Phys. Rev. E 90, 052116 (2014).
We have considered the underdamped motion of a Brownian particle in the presence of a correlated external random force. The force is modeled by an Ornstein-Uhlenbeck process. We investigate the fluctuations of the work done by the external force on the Brownian particle in a given time interval in the steady state. We calculate the large deviation functions as well as the complete asymptotic form of the probability density function of the performed work. We also discuss the symmetry properties of the large deviation functions for this system. Finally we perform numerical simulations and they are in a very good agreement with the analytic results.
DOI: 10.1103/PhysRevE.90.052116. arXiv e-print: 1407.6191.
- Universal Large Deviations for the Tagged Particle in Single-File Motion. Chaitra Hegde, Sanjib Sabhapandit, and Abhishek Dhar, Phys. Rev. Lett. 113, 120601 (2014).
We consider a gas of point particles moving in a one-dimensional channel with a hard-core interparticle 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 subdiffusive, while for ballistic particle dynamics, the tagged particle motion is diffusive. Here we compute the exact large deviation function for the tagged particle displacement and show that this is universal, independent of the individual dynamics.
DOI: 10.1103/PhysRevLett.113.120601. arXiv e-print: 1406.6191.
- High-energy tail of the velocity distribution of driven inelastic Maxwell gases. V. V. Prasad, Sanjib Sabhapandit, and Abhishek Dhar, Europhy. Lett. 104, 54003 (2013).
A model of homogeneously driven dissipative system, consisting of a collection of $N$ particles that are characterized by only their velocities, is considered. Adopting a discrete time dynamics, at each time step, a pair of velocities is randomly selected. They undergo inelastic collision with probability $p$. With probability $(1-p)$, energy of the system is changed by changing the velocities of both the particles independently according to $v\rightarrow -r_w v +\eta$, where $\eta$ is a Gaussian noise drawn independently for each particle as well as at each time steps. For the case $r_w=- 1$, although the energy of the system seems to saturate (indicating a steady state) after time steps of $O(N)$, it grows linearly with time after time steps of $O(N^2)$, indicating the absence of a eventual steady state. For $-1 < r_w \leq 1$, the system reaches a steady state, where the average energy per particle and the correlation of velocities are obtained exactly. In the thermodynamic limit of large $N$, an exact equation is obtained for the moment generating function. In the limit of nearly elastic collisions and weak energy injection, the velocity distribution is shown to be a Gaussian. Otherwise, for $|r_w| < 1$, the high-energy tail of the velocity distribution is Gaussian, with a different variance, while for $r_w=+1$ the velocity distribution has an exponential tail.
DOI: 10.1209/0295-5075/104/54003. arXiv e-print: 1307.3564.
- Work fluctuations for a Brownian particle in a harmonic trap with fluctuating locations. Arnab Pal and Sanjib Sabhapandit, Phys. Rev. E 87, 022138 (2013).
We consider a Brownian particle in a harmonic trap. The location of the trap is modulated according to an Ornstein-Uhlenbeck process. We investigate the fluctuation of the work done by the modulated trap on the Brownian particle in a given time interval in the steady state. We compute the large deviation as well as the complete asymptotic form of the probability density function of the work done. The theoretical asymptotic forms of the probability density function are in very good agreement with the numerics. We also discuss the validity of the fluctuation theorem for this system.
DOI: 10.1103/PhysRevE.87.022138. arXiv e-print: 1212.0704.
- Tagged Particle Diffusion in One-Dimensional Gas with Hamiltonian Dynamics. Anjan Roy, Onuttom Narayan, Abhishek Dhar, and Sanjib Sabhapandit, J. Stat. Phys. 150, 851 (2013).
We consider a one-dimensional gas of hard point particles in a finite box that are in thermal equilibrium and evolving under Hamiltonian dynamics. Tagged particle correlation functions of the middle particle are studied. For the special case where all particles have the same mass, we obtain analytic results for the velocity auto-correlation function in the short time diffusive regime and the long time approach to the saturation value when finite-size effects become relevant. In the case where the masses are unequal, numerical simulations indicate sub-diffusive behaviour with mean square displacement of the tagged particle growing as $t/\ln(t)$ with time $t$. Also various correlation functions, involving the velocity and position of the tagged particle, show damped oscillations at long times that are absent for the equal mass case.
DOI: 10.1007/s10955-012-0673-9. arXiv e-print: 1209.1572.
- Scaling behavior in the convection-driven Brazil nut effect. Prakhyat Hejmady, Ranjini Bandyopadhyay, Sanjib Sabhapandit, and Abhishek Dhar, Phys. Rev. E 86, 050301 (2012). [Rapid Communication]
The Brazil nut effect is the phenomenon in which a large intruder particle immersed in a vertically shaken bed of smaller particles rises to the top, even when it is much denser. The usual practice while describing these experiments has been to use the dimensionless acceleration $\Gamma=a\omega^2/g$, where $a$ and $\omega$ are, respectively, the amplitude and the angular frequency of vibration and $g$ is the acceleration due to gravity. Considering a vibrated quasi-two-dimensional bed of mustard seeds, we show here that the peak-to-peak velocity of shaking $v=a\omega$, rather than $\Gamma$, is the relevant parameter in the regime where boundary-driven granular convection is the main driving mechanism. We find that the rise time $\tau$ of an intruder is described by the scaling law $\tau \sim (v-v_c)^{-\alpha}$, where $v_c$ is identified as the critical vibration velocity for the onset of convective motion of the mustard seeds. This scaling form holds over a wide range of $(a,\omega)$, diameter, and density of the intruder.
DOI: 10.1103/PhysRevE.86.050301. arXiv e-print: 1111.4484.
- Heat and work fluctuations for a harmonic oscillator. Sanjib Sabhapandit, Phys. Rev. E 85, 021108 (2012).
The formalism of Kundu et al. [J. Stat. Mech. P03007 (2011)], for computing the large deviations of heat flow in harmonic systems, is applied to the case of single Brownian particle in a harmonic trap and coupled to two heat baths at different temperatures. The large-$\tau$ form of the moment generating function $\langle e^{-\lambda }\rangle \approx g(\lambda) \exp[\tau \mu(\lambda)]$, of the total heat flow $Q$ from one of the baths to the particle in a given time interval $\tau$, is studied and exact explicit expressions are obtained for both $\mu(\lambda)$ and $g(\lambda)$. For a special case of the single particle problem that corresponds to the work done by an external stochastic force on a harmonic oscillator coupled to a thermal bath, the large-$\tau$ form of the moment generating function is analyzed to obtain the exact large deviation function as well as the complete asymptotic forms of the probability density function of the work.
DOI: 10.1103/PhysRevE.85.021108. arXiv e-print: 1202.4257.
- Work fluctuations for a harmonic oscillator driven by an external random force. Sanjib Sabhapandit, Europhy. Lett. 96, 20005 (2011).
The fluctuations of the work done by an external Gaussian random force on a harmonic oscillator that is also in contact with a thermal bath are studied. We have obtained the exact large deviation function as well as the complete asymptotic forms of the probability density function. The distribution of the work done is found to be non-Gaussian. The steady-state fluctuation theorem holds only if the ratio of the variances, of the external random forcing and the thermal noise, respectively, is less than 1/3. On the other hand, the transient fluctuation theorem holds (asymptotically) for all the values of that ratio. The theoretical asymptotic forms of the probability density function are in very good agreement with the numerics as well as with an experiment.
DOI: 10.1209/0295-5075/96/20005. arXiv e-print: 1104.0895.
- Dynamics of a flexible polymer in planar mixed flow. Dipjyoti Das, Sanjib Sabhapandit, and Dibyendu Das, J. Phys.: Conf. Ser. 297, 012007 (2011).
We present exact spatio-temporal correlation functions of a Rouse polymer chain submerged in a fluid having planar mixed flow, in the steady state. Using these correlators, determination of the time scale distribution functions associated with the first-passage tumbling events is difficult in general; it was done recently in Phys. Rev. Lett. 101, 188301 (2008), for the special case of “simple shear” flow. We show here that the method used in latter paper fails for the general mixed flow problem. We also give many new estimates of the exponent $\theta$ associated with the exponential tail of the angular tumbling time distribution in the case of simple shear.
DOI: 10.1088/1742-6596/297/1/012007. arXiv e-print: 1109.0804.
- Record statistics of continuous time random walk. Sanjib Sabhapandit, Europhy. Lett. 94, 20003 (2011).
The statistics of records for a time series generated by a continuous time random walk is studied, and found to be independent of the details of the jump length distribution, as long as the latter is continuous and symmetric. However, the statistics depend crucially on the nature of the waiting-time distribution. The probability of finding $M$ records within a given time duration $t$, for large $t$, has a scaling form, and the exact scaling function is obtained in terms of the one-sided Lévy stable distribution. The mean of the ages of the records, defined as $\langle t/M \rangle$, differs from $t/\langle M \rangle$. The asymptotic behaviour of the shortest and the longest ages of the records are also studied.
DOI: 10.1209/0295-5075/94/20003. arXiv e-print: 1008.1762.
- Large deviations of heat flow in harmonic chains. Anupam Kundu, Sanjib Sabhapandit, and Abhishek Dhar, J. Stat. Mech. 2011, P03007 (2011).
We consider heat transport across a harmonic chain connected at its two ends to white-noise Langevin reservoirs at different temperatures. In the steady state of this system the heat $Q$ flowing from one reservoir into the system in a finite time $\tau$ has a distribution $P(Q, \tau)$. We study the large time form of the corresponding moment generating function $\langle e^{-\lambda Q} \rangle \sim g(\lambda) e^{\tau \mu(\lambda)}$. Exact formal expressions, in terms of phonon Green’s functions, are obtained for both $\mu(\lambda)$ and also the lowest order correction $g(\lambda)$. We point out that, in general, a knowledge of both $\mu(\lambda)$ and $g(\lambda)$ is required for finding the large deviation function associated with $P(Q, \tau)$. The function $\mu(\lambda)$ is known to be the largest eigenvector of an appropriate Fokker-Planck type operator and our method also gives the corresponding eigenvector exactly.
DOI: 10.1088/1742-5468/2011/03/p03007. arXiv e-print: 1101.3669.
- Application of importance sampling to the computation of large deviations in nonequilibrium processes. Anupam Kundu, Sanjib Sabhapandit, and Abhishek Dhar, Phys. Rev. E 83, 031119 (2011).
We present an algorithm for finding the probabilities of rare events in nonequilibrium processes. The algorithm consists of evolving the system with a modified dynamics for which the required event occurs more frequently. By keeping track of the relative weight of phase-space trajectories generated by the modified and the original dynamics one can obtain the required probabilities. The algorithm is tested on two model systems of steady-state particle and heat transport where we find a huge improvement from direct simulation methods.
DOI: 10.1103/PhysRevE.83.031119. arXiv e-print: 1003.2106.
- Statistical properties of the final state in one-dimensional ballistic aggregation. Satya N. Majumdar, Kirone Mallick, and Sanjib Sabhapandit, Phys. Rev. E 79, 021109 (2009).
We investigate the long time behavior of the one-dimensional ballistic aggregation model that represents a sticky gas of $N$ particles with random initial positions and velocities, moving deterministically, and forming aggregates when they collide. We obtain a closed formula for the stationary measure of the system which allows us to analyze some remarkable features of the final “fan” state. In particular, we identify universal properties which are independent of the initial position and velocity distributions of the particles. We study cluster distributions and derive exact results for extreme value statistics (because of correlations these distributions do not belong to the Gumbel-Fréchet-Weibull universality classes). We also derive the energy distribution in the final state. This model generates dynamically many different scales and can be viewed as one of the simplest exactly solvable model of $N$-body dissipative dynamics.
DOI: 10.1103/PhysRevE.79.021109. arXiv e-print: 0811.0908.
- Accurate Statistics of a Flexible Polymer Chain in Shear Flow. Dibyendu Das and Sanjib Sabhapandit, Phys. Rev. Lett. 101, 188301 (2008).
We present exact and analytically accurate results for the problem of a flexible polymer chain in shear flow. Under such a flow the polymer tumbles, and the probability distribution of the tumbling time $\tau$ of the polymer decays e enti ally as $\sim \exp(-\alpha\tau/\tau_0)$ (where $\tau_0$ is the longest relaxation time). We show that for a Rouse chain this nontrivial constant $\alpha$ can be calculated in the limit of a large Weissenberg number (high shear rate) and is in excellent agreement with our simulation result of $\alpha \simeq 0.324$. We also derive exactly the distribution functions for the length and the orientational angles of the end-to-end vector $\boldsymbol{R}$ of the polymer.
DOI: 10.1103/PhysRevLett.101.188301. arXiv e-print: 0809.2131.
- Statistics of the total number of collisions and the ordering time in a freely expanding hard-point gas. Sanjib Sabhapandit, Ioana Bena, and Satya N Majumdar, J. Stat. Mech. 2008, P05012 (2008).
We consider a Jepsen gas of $N$ hard-point particles undergoing free expansion on a line, starting from random initial positions of the particles having random initial velocities. The particles undergo binary elastic collisions upon contact and move freely in-between collisions. After a certain ordering time $T_\mathrm{o}$, the system reaches a ‘fan’ state where all the velocities are completely ordered from left to right in an increasing fashion and there is no further collision. We compute analytically the distributions of (i) the total number of collisions and (ii) the ordering time $T_\mathrm{o}$. We show that several features of these distributions are universal.
DOI: 10.1088/1742-5468/2008/05/p05012. arXiv e-print: 0804.1866.
- Crowding at the front of marathon packs. Sanjib Sabhapandit, Satya N Majumdar, and S Redner, J. Stat. Mech. 2008, L03001 (2008).
We study the crowding of near-extreme events in the time gaps between successive finishers in major international marathons. Naively, one might expect these gaps to become progressively larger for better-placing finishers. While such an increase does indeed occur from the middle of the finishing pack down to approximately 20th place, the gaps saturate for the first 10-20 finishers. We give a probabilistic account of this feature. However, the data suggest that the gaps have a weak maximum around the 10th place, a feature that seems to have a sociological origin.
DOI: 10.1088/1742-5468/2008/03/l03001. arXiv e-print: 0802.1702.
- A note on limit shapes of minimal difference partitions. Alain Comtet, Satya N Majumdar, and Sanjib Sabhapandit, Journal of Mathematical Physics, Analysis, Geometry 4, 24 (2008).
We provide a variational derivation of the limit shape of minimal difference partitions and discuss the link with exclusion statistics.
arXiv e-print: 0801.4300.
- Integer partitions and exclusion statistics: limit shapes and the largest parts of Young diagrams. Alain Comtet, Satya N Majumdar, Stéphane Ouvry, and Sanjib Sabhapandit, J. Stat. Mech. 2007, P10001 (2007).
We compute the limit shapes of the Young diagrams of the minimal difference $p$ partitions and provide a simple physical interpretation for the limit shapes. We also calculate the asymptotic distribution of the largest part of the Young diagram and show that the scaled distribution has a Gumbel form for all $p$. This Gumbel statistics for the largest part remains unchanged even for general partitions of the form $E=\sum_i n_i i^{1/\nu}$ with $\nu>0$ where $n_i$ is the number of times the part $i$ appears.
DOI: 10.1088/1742-5468/2007/10/p10001. arXiv e-print: 0707.2312.
- Statistical properties of a single-file diffusion front. Sanjib Sabhapandit, J. Stat. Mech. 2007, L05002 (2007).
Statistical properties of the front of a semi-infinite system of single-file diffusion (a one-dimensional system where particles cannot pass each other, but in between collisions each one independently exhibits diffusive motion) are investigated. Exact as well as asymptotic results are provided for the probability density function of (a) the front position, (b) the maximum of the front positions, and (c) the first-passage time to a given position. The asymptotic laws for the front position and the maximum front position are found to be governed by Fisher-Tippett-Gumbel extreme value statistics. The asymptotic properties of the first-passage time is dominated by a stretched-exponential tail in the distribution. The farness of the front with the rest of the system is investigated by considering (i) the gap from the front to the closest particle, and (ii) the density profile with respect to the front position, and analytical results are provided for late-time behaviours.
DOI: 10.1088/1742-5468/2007/05/l05002. arXiv e-print: cond-mat/0703231.
- Density of Near-Extreme Events. Sanjib Sabhapandit and Satya N. Majumdar, Phys. Rev. Lett. 98, 140201 (2007).
We provide a quantitative analysis of the phenomenon of crowding of near-extreme events by computing exactly the density of states (DOS) near the maximum of a set of independent and identically distributed random variables. We show that the mean DOS converges to three different limiting forms depending on whether the tail of the distribution of the random variables decays slower than pure exponential, faster than pure exponential, or as a pure exponential function. We argue that some of these results would remain valid even for certain correlated cases and verify it for power-law correlated stationary Gaussian sequences. Satisfactory agreement is found between the near-maximum crowding in the summer temperature reconstruction data of western Siberia and the theoretical prediction.
DOI: 10.1103/PhysRevLett.98.140201. arXiv e-print: cond-mat/0701375.
- Statistical properties of functionals of the paths of a particle diffusing in a one-dimensional random potential. Sanjib Sabhapandit, Satya N. Majumdar, and Alain Comtet, Phys. Rev. E 73, 051102 (2006).
We present a formalism for obtaining the statistical properties of functionals and inverse functionals of the paths of a particle diffusing in a one-dimensional quenched random potential. We demonstrate the implementation of the formalism in two specific examples: (1) where the functional corresponds to the local time spent by the particle around the origin and (2) where the functional corresponds to the occupation time spent by the particle on the positive side of the origin, within an observation time window of size $t$. We compute the disorder average distributions of the local time, the inverse local time, the occupation time, and the inverse occupation time and show that in many cases disorder modifies the behavior drastically.
DOI: 10.1103/PhysRevE.73.051102. arXiv e-print: cond-mat/0601455.
- Absence of jump discontinuity in the magnetization in quasi-one-dimensional random-field Ising models. Sanjib Sabhapandit, Phys. Rev. B 70, 224401 (2004).
We consider the zero-temperature random-field Ising model in the presence of an external field, on ladders and in one dimension with finite range interactions, for unbounded continuous distributions of random fields, and show that there is no jump discontinuity in the magnetizations for any quasi-one-dimensional model. We show that the evolution of the system at an external field can be described by a stochastic matrix and the magnetization can be obtained using the eigenvector of the matrix corresponding to the eigenvalue one, which is continuous and differentiable function of the external field.
DOI: 10.1103/PhysRevB.70.224401. arXiv e-print: cond-mat/0405376.
- Performance Limitations of Flat-Histogram Methods. P. Dayal, S. Trebst, S. Wessel, D. Würtz, M. Troyer, S. Sabhapandit, and S. N. Coppersmith, Phys. Rev. Lett. 92, 097201 (2004).
We determine the optimal scaling of local-update flat-histogram methods with system size by using a perfect flat-histogram scheme based upon the exact density of states of 2D Ising models. The typical tunneling time needed to sample the entire bandwidth does not scale with the number of spins $N$ as the minimal $N^2$ of an unbiased random walk in energy space. While the scaling is power law for the ferromagnetic and fully frustrated Ising model, for the $\pm J$ nearest-neighbor spin glass the distribution of tunneling times is governed by a fat-tailed Fréchet extremal value distribution that obeys exponential scaling. Furthermore, the shape parameters of these distributions indicate that statistical sample means become ill defined already for moderate system sizes within these complex energy landscapes.
DOI: 10.1103/PhysRevLett.92.097201. arXiv e-print: cond-mat/0306108.
- Hysteresis in the Random-Field Ising Model and Bootstrap Percolation. Sanjib Sabhapandit, Deepak Dhar, and Prabodh Shukla, Phys. Rev. Lett. 88, 197202 (2002).
We study hysteresis in the random-field Ising model with an asymmetric distribution of quenched fields, in the limit of low disorder in two and three dimensions. We relate the spin flip process to bootstrap percolation, and show that the characteristic length for self-averaging $L^\star$ increases as $\exp[\exp(J/\Delta)]$ in 2D, and as $\exp{\exp[\exp(J/\Delta)]}$ in 3D, for disorder strength $\Delta$ much less than the exchange coupling $J$. For system size $1 \ll L \ll L^\star$, the coercive field $h_\mathrm{coer}$ varies as $2 J – \Delta \ln\ln L$ for the square lattice, and as $2J – \Delta \ln\ln\ln L$ on the cubic lattice. Its limiting value is 0 for $L\to\infty$ for both square and cubic lattices. For lattices with coordination number 3, the limiting magnetization shows no jump, and $h_\mathrm{coer}$ tends to $J$.
DOI: 10.1103/PhysRevLett.88.197202. arXiv e-print: cond-mat/0204618.
- Distribution of Avalanche Sizes in the Hysteretic Response of the Random-Field Ising Model on a Bethe Lattice at Zero Temperature. Sanjib Sabhapandit, Prabodh Shukla, and Deepak Dhar, J. Stat. Phys. 98, 103 (2000).
We consider the zero-temperature single-spin-flip dynamics of the random-field Ising model on a Bethe lattice in the presence of an external field h. We derive the exact self-consistent equations to determine the distribution $\mathrm{Prob}(s)$ of avalanche sizes $s$, as the external field increases from large negative to positive values. We solve these equations explicitly for a rectangular distribution of the random fields for a linear chain and the Bethe lattice of coordination number $z=3$, and show that in these cases, $\mathrm{Prob}(s)$ decreases exponentially with $s$ for large $s$ for all $h$ on the hysteresis loop. We found that for $z >3$ and for small disorder, the magnetization shows a first order discontinuity for several continuous and unimodel distributions of random fields. The avalanche distribution $\mathrm{Prob}(s)$ varies as $s^{-3/2}$ for large $s$ near the discontinuity.
DOI: 10.1023/A:1018622805347. arXiv e-print: cond-mat/9905236.
- Out-of-plane phase segregation and in-plane clustering in a binary mixture of amphiphiles at the air-water interface. Debashish Chowdhury, Prabal K. Maiti, S. Sabhapandit, and P. Taneja, Phys. Rev. E 56, 667 (1997).
We have carried out extensive Monte Carlo simulations of a microscopic lattice model of a binary mixture of amphiphilic molecules, of two different lengths, in a system where water is separated from the air above it by a sharp well-defined interface. We have demonstrated an entropy-driven phase segregation in a direction perpendicular to the air-water interface when the initial total surface density of the amphiphiles is sufficiently high. We have also investigated (a) the conformations of the amphiphiles, (b) the distribution of the sizes of the clusters of monomers belonging to the long amphiphiles as well as of those belonging to the short amphiphiles in planes parallel to the interface, (c) the effects of varying the lengths, total concentration, and the ratio of the numbers of the two types of the amphiphilic molecules as well as those of varying the strength of the intermonomer interactions and temperature. We have critically examined the interpretations of some recent experimental results in the light of our observations.
DOI: 10.1103/PhysRevE.56.667.
