arXiv subjects

Anupam Kundu

Publications and source records attributed to Anupam Kundu.

89 records · Page 5Linked to original sources

arXiv2017

Classification of Population Using Voronoi Area Based Density Estimation

Given two sets of training samples, general method is to estimate the density function and classify the test sample according to higher values of estimated densities. Natural way to estimate the density should be histogram tending to frequency curve. But nature of different amount of data accumulation at different points, obstruct us to take common interval length in drawing histogram. Here we followed a method similar to drawing histogram with an intention to density estimation and classification, but without doing arbitrarily we considered diagram of the scattered plot and has obtained estimate of density based on areas of the Voronoi cells. But area of arbitrary Voronoi cell may be difficult to find out particularly in two and higher dimensions, and also in manifolds. For this reason we estimate the area of the Voronoi cell by drawing many independent uniform variates covering the support of the training samples. Please note that though we have in mind density estimation and classification problem in higher dimension and in typical manifolds, but our simulations and algebraic calculations have been mostly shown for densities on real line. We expect in higher dimension and in manifolds the work will be in similar lines and plan to do in near future. Also we expect discrete cases can be handled in slightly different but similar lines which are also left for future works.

stat.ME↗

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

Tagged particle in single-file diffusion with arbitrary initial conditions

We compute the full probability distribution of the positions of a tagged particle exactly for given arbitrary initial positions of the particles and for general single-particle propagators. We consider the thermodynamic limit of our exact expressions in quenched and annealed settings. For a particular class of single-particle propagators, the exact formula is expressed in a simple integral form in the quenched case whereas in the annealed case, it is expressed as a simple combination of Bessel functions. In particular, we focus on the step and the power-law initial configurations. In the former case, a drift is induced even when the one-particle propagators are symmetric. On the other hand, in the later case the scaling of the cumulants of the position of the tracer becomes different than the uniform case. We provide numerical verifications of our results.

cond-mat.stat-mech↗

arXiv2017

Center of mass perturbation as a normalizable estimate of dynamic balance in gait: an application comparing typically developing children with spastic cerebral palsy

It is well recognized that the relative position of the center of mass (pCOM) with respect to the base of support (BOS) is a determining factor in the maintenance of balance. However, during gait the dynamic nature of the BOS is not well defined and in most studies is completely ignored. Prior work tends to focus on the variability in the position of the center of mass (COM) with respect to the laboratory reference frame to attempt to quantify dynamic balance. We propose a modified method whereby the position of the COM in an end-effector based coordinate system may be used as an improved estimate of threat to balance during gait. The distance (DN) of the projection of the COM (pCOM) from the reference axis, normalized by half the foot length for inter-subject comparison, is an estimate of the gravitational moment arm. It can be shown that in concordance with theory, balance impaired subjects with spastic cerebral palsy (CP) have a larger DN than their typically developing peers. Furthermore, when compared with variational methods, DN shows better discriminative ability in characterizing groups.

physics.med-ph↗

arXiv2016

Diffusion under time-dependent resetting

We study a Brownian particle diffusing under a time-modulated stochastic resetting mechanism to a fixed position. The rate of resetting r(t) is a function of the time t since the last reset event. We derive a sufficient condition on r(t) for a steady-state probability distribution of the position of the particle to exist. We derive the form of the steady-state distributions under some particular choices of r(t) and also consider the late time relaxation behavior of the probability distribution. Finally we consider first passage time properties for the Brownian particle to reach the origin and derive a formula for the mean first passage time. We study optimal properties of the mean first passage time and show that a threshold function is at least locally optimal for the problem of minimizing the mean first passage time.

cond-mat.stat-mech↗

arXiv2015

Long range correlations in stochastic transport with energy and momentum conservation

We consider a simple one dimensional stochastic model of heat transport which locally conserves both energy and momentum and which is coupled to heat reservoirs with different temperatures at its two ends. The steady state is analyzed and the model is found to obey the Fourier law with finite heat conductivity. In the infinite length limit, the steady state is described locally by an equilibrium Gibbs state. However finite size corrections to this local equilibrium state are present. We analyze these finite size corrections by calculating the on-site fluctuations of the momentum and the two point correlation of the momentum and energy. These correlations are long ranged and have scaling forms which are computed explicitly. We also introduce a multi-lane variant of the model in which correlations vanish in the steady state. The deviation from local equilibrium in this model as expressed in terms of the on-site momentum fluctuations is calculated in the large length limit.

cond-mat.stat-mech↗

arXiv2015

Exact gap statistics for the random average process on a ring with a tracer

We study statistics of the gaps in Random Average Process (RAP) on a ring with particles hopping symmetrically, except one tracer particle which could be driven. These particles hop either to the left or to the right by a random fraction $η$ of the space available till next particle in the respective directions. The random fraction $η\in [0,~1)$ is chosen from a distribution $R(η)$. For non-driven tracer, when $R(η)$ satisfies a necessary and sufficient condition, the stationary joint distribution of the gaps between successive particles takes an universal form that is factorized except for a global constraint. Some interesting explicit forms of $R(η)$ are found which satisfy this condition. In case of driven tracer, the system reaches a current-carrying steady state where such factorization does not hold. Analytical progress has been made in the thermodynamic limit, where we computed the single site mass distribution inside the bulk. We have also computed the two point gap-gap correlation exactly in that limit. Numerical simulations support our analytical results.

cond-mat.stat-mech↗

arXiv2014

Winding statistics of a Brownian particle on a ring

We consider a Brownian particle moving on a ring. We study the probability distributions of the total number of turns and the net number of counter-clockwise turns the particle makes till time t. Using a method based on the renewal properties of Brownian walker, we find exact analytical expressions of these distributions. This method serves as an alternative to the standard path integral techniques which are not always easily adaptable for certain observables. For large t, we show that these distributions have Gaussian scaling forms. We also compute large deviation functions associated to these distributions characterizing atypically large fluctuations. We provide numerical simulations in support of our analytical results.

cond-mat.stat-mech↗

arXiv2014

Time auto-correlation function and Green-Kubo formula: A study on disordered harmonic chain

We have considered heat conduction in a one-dimensional mass disordered harmonic chain of $N$ particles connected to two Langevin type reservoirs at different temperatures. An exact expression for the boundary heat current-current auto-correlation function in the non-equilibrium steady state (NESS) is obtained in terms of non-equilibrium phonon Green's functions. The time integral of the correlation function gives expected result, both in non-equilibrium as well as equilibrium cases. Using the form of this correlation function we show that asymptotic system size dependence of current fluctuation in NESS for a mass disordered harmonic chain is $N^{-α}$ for different boundary conditions. For free and fixed boundary conditions we get $α=1/2$ and 3/2 respectively, while for pinned case the fluctuation decays exponentially with system size.

cond-mat.stat-mech↗

arXiv2014

Maximal distance travelled by N vicious walkers till their survival

We consider $N$ Brownian particles moving on a line starting from initial positions ${\bf{u}}\equiv \{u_1,u_2,\dots u_N\}$ such that $0<u_1 < u_2 < \cdots < u_N$. Their motion gets stopped at time $t_s$ when either two of them collide or when the particle closest to the origin hits the origin for the first time. For $N=2$, we study the probability distribution function $p_1(m|{\bf{u}})$ and $p_2(m|{\bf{u}})$ of the maximal distance travelled by the $1^{\text{st}}$ and $2^{\text{nd}}$ walker till $t_s$. For general $N$ particles with identical diffusion constants $D$, we show that the probability distribution $p_N(m|{\bf u})$ of the global maximum $m_N$, has a power law tail $p_i(m|{\bf{u}}) \sim {N^2B_N\mathcal{F}_{N}({\bf u})}/{m^{ν_N}}$ with exponent $ν_N =N^2+1$. We obtain explicit expressions of the function $\mathcal{F}_{N}({\bf u})$ and of the $N$ dependent amplitude $B_N$ which we also analyze for large $N$ using techniques from random matrix theory. We verify our analytical results through direct numerical simulations.

cond-mat.stat-mech↗

arXiv2013

Exact distributions of the number of distinct and common sites visited by N independent random walkers

We study the number of distinct sites S_N(t) and common sites W_N(t) visited by N independent one dimensional random walkers, all starting at the origin, after t time steps. We show that these two random variables can be mapped onto extreme value quantities associated to N independent random walkers. Using this mapping, we compute exactly their probability distributions P_N^d(S,t) and P_N^d(W,t) for any value of N in the limit of large time t, where the random walkers can be described by Brownian motions. In the large N limit one finds that S_N(t)/\sqrt{t} \propto 2 \sqrt{\log N} + \widetilde{s}/(2 \sqrt{\log N}) and W_N(t)/\sqrt{t} \propto \widetilde{w}/N where \widetilde{s} and \widetilde{w} are random variables whose probability density functions (pdfs) are computed exactly and are found to be non trivial. We verify our results through direct numerical simulations.

cond-mat.stat-mech↗

arXiv2011

Large deviations of heat flow in harmonic chains

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 $τ$ has a distribution $P(Q,τ)$. We study the large time form of the corresponding moment generating function $ \sim g(λ) e^{τμ(λ)}$. Exact formal expressions, in terms of phonon Green's functions, are obtained for both $μ(λ)$ and also the lowest order correction $g(λ)$. We point out that, in general a knowledge of both $μ(λ)$ and $g(λ)$ is required for finding the large deviation function associated with $P(Q,τ)$. The function $μ(λ)$ is known to be the largest eigenvector of an appropriate Fokker-Planck type operator and our method also gives the corresponding eigenvector exactly.

cond-mat.stat-mech↗

arXiv2011

Application of importance sampling to the computation of large deviations in non-equilibrium processes

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.

cond-mat.stat-mech↗

arXiv2010

Linear response formula for finite frequency thermal conductance of open systems

An exact linear response expression is obtained for the heat current in a classical Hamiltonian system coupled to heat baths with time-dependent temperatures. The expression is equally valid at zero and finite frequencies. We present numerical results on the frequency dependence of the response function for three different one-dimensional models of coupled oscillators connected to Langevin baths with oscillating temperatures. For momentum conserving systems, a low frequency peak is seen that, is higher than the zero frequency response for large systems. For momentum non-conserving systems, there is no low frequency peak. The momentum non-conserving system is expected to satisfy Fourier's law, however, at the single bond level, we do not see any clear agreement with the predictions of the diffusion equation even at low frequencies. We also derive an exact analytical expression for the response of a chain of harmonic oscillators to a (not necessarily small) temperature difference; the agreement with the linear response simulation results for the same system is excellent.

cond-mat.stat-mech↗

arXiv2010

Heat conduction and phonon localization in disordered harmonic crystals

We investigate the steady state heat current in two and three dimensional isotopically disordered harmonic lattices. Using localization theory as well as kinetic theory we estimate the system size dependence of the current. These estimates are compared with numerical results obtained using an exact formula for the current given in terms of a phonon transmission function, as well as by direct nonequilibrium simulations. We find that heat conduction by high-frequency modes is suppressed by localization while low-frequency modes are strongly affected by boundary conditions. Our {\color{black}heuristic} arguments show that Fourier's law is valid in a three dimensional disordered solid except for special boundary conditions. We also study the pinned case relevant to localization in quantum systems and often used as a model system to study the validity of Fourier's law. Here we provide the first numerical verification of Fourier's law in three dimensions. In the two dimensional pinned case we find that localization of phonon modes leads to a heat insulator.

cond-mat.dis-nn↗

arXiv2009

Heat transport and phonon localization in mass-disordered harmonic crystals

We investigate the steady state heat current in two and three dimensional disordered harmonic crystals in a slab geometry, connected at the boundaries to stochastic white noise heat baths at different temperatures.The disorder causes short wavelength phonon modes to be localized so the heat current in this system is carried by the extended phonon modes which can be either diffusive or ballistic. Using ideas both from localization theory and from kinetic theory we estimate the contribution of various modes to the heat current and from this we obtain the asymptotic system size dependence of the current. These estimates are compared with results obtained from a numerical evaluation of an exact formula for the current, given in terms of a frequency transmission function, as well as from direct nonequilibrium simulations. These yield a strong dependence of the heat flux on boundary conditions. Our analytical arguments show that for realistic boundary conditions the conductivity is finite in three dimensions but we are not able to verify this numerically, except in the case where the system is subjected to an external pinning potential. This case is closely related to the problem of localization of electrons in a random potential and here we numerically verify that the pinned three dimensional system satisfies Fourier's law while the two dimensional system is a heat insulator. We also investigate the inverse participation ratio of different normal modes.

cond-mat.dis-nn↗

arXiv2009

Green-Kubo formula for heat conduction in open systems

We obtain an exact Green-Kubo type linear response result for the heat current in an open system. The result is derived for classical Hamiltonian systems coupled to heat baths. Both lattice models and fluid systems are studied and several commonly used implementations of heat baths, stochastic as well as deterministic, are considered. The results are valid in arbitrary dimensions and for any system sizes. Our results are useful for obtaining the linear response transport properties of mesoscopic systems. Also we point out that for systems with anomalous heat transport, as is the case in low-dimensional systems, the use of the standard Green-Kubo formula is problematic and the open system formula should be used.

cond-mat.stat-mech↗