non-Markovian Model Research Articles

Exact Closed Master Equation for Gaussian Non-Markovian Dynamics.

Non-Markovian master equations describe general open quantum systems when no approximation is made. We provide the exact closed master equation for the class of Gaussian, completely positive, trace preserving, non-Markovian dynamics. This very general result allows us to investigate a vast variety of physical systems. We show that the master equation for non-Markovian quantum Brownian motion is a particular case of our general result. Furthermore, we derive the master equation unraveled by a non-Markovian, dissipative stochastic Schrödinger equation, paving the way for the analysis of dissipative non-Markovian collapse models.

Metaphysics of Planning Domain Descriptions

STRIPS-like languages (SLLs) have fostered immense advances in automated planning. In practice, SLLs are used to express highly abstract versions of real-world planning problems, leading to more concise models and faster solution times. Unfortunately, as we show in the paper, simple ways of abstracting solvable real-world problems may lead to SLL models that are unsolvable, SLL models whose solutions are incorrect with respect to the real-world problem, or models that are inexpressible in SLLs. There is some evidence that such limitations have restricted the applicability of AI planning technology in the real world, as is apparent in the case of task and motion planning in robotics. We show that the situation can be ameliorated by a combination of increased expressive power — for example, allowing angelic nondeterminism in action effects — and new kinds of algorithmic approaches designed to produce correct solutions from initially incorrect or non-Markovian abstract models.

Loop-weighted walk

Loop-weighted walk with parameter \lambda \geq 0 is a non-Markovian model of random walks that is related to the loop O(N) model of statistical mechanics. A walk receives weight \lambda^{k} if it contains k loops; whether this is a reward or punishment for containing loops depends on the value of \lambda . A challenging feature of loop-weighted walk is that it is not purely repulsive, meaning the weight of the future of a walk may either increase or decrease if the past is forgotten. Repulsion is typically an essential property for lace expansion arguments. This article circumvents the lack of repulsion and proves, for any \lambda > 0 , that loop-weighted walk is diffusive in high dimensions by lace expansion methods.

PH FITTING ALGORITHM AND ITS APPLICATION TO RELIABILITY ENGINEERING

This article presents parameter estimation of phase-type (PH) distribution. PH distribution is widely used in model-based performance evaluation such as reliability and queueing system, and is defied by an absorbing continuous-time Markov chain (CTMC). Since non-Markovian model can be approximated by a CTMC by replacing general distributions with PH distributions, the parameter estimation is a challenging issue in stochastic modeling. We focus on the statistical inference algorithms to estimate not only the parameters of PH distribution with grouped, truncated and missing data but also the probability density function, and give two examples of PH fitting in reliability engineering.

Dynamics of social contagions with heterogeneous adoption thresholds: crossover phenomena in phase transition

Heterogeneous adoption thresholds exist widely in social contagions, such as behavior spreading, but were always neglected in previous studies. To this end, we introduce heterogeneous adoption threshold distribution into a non-Markovian spreading threshold model, in which an individual adopts a behavior only when the received cumulative pieces of behavioral information from neighbors exceeds his adoption threshold. In order to understand the effects of heterogeneous adoption thresholds quantitatively, an edge-based compartmental theory is developed. A two-state spreading threshold model is taken as an example, in which some individuals have a low adoption threshold (i.e., activists) while the remaining ones hold a relatively higher adoption threshold (i.e., bigots). We find a hierarchical characteristic in adopting behavior, i.e., activists first adopt the behavior and then stimulate bigots to adopt the behavior. Interestingly, two types of crossover phenomena in phase transition occur: for a relatively low adoption threshold of bigots, a change from first-order to second-order phase transition can be triggered by increasing the fraction of activists; for a relatively higher adoption threshold of bigots, a change from hybrid to second-order phase transition can be induced by varying the fraction of activists, decreasing mean degree or enhancing network heterogeneity. The theoretical predictions based on the suggested theory agree very well with the simulation results.

Self-organized anomalous aggregation of particles performing nonlinear and non-Markovian random walks

We present a nonlinear and non-Markovian random walks model for stochastic movement and the spatial aggregation of living organisms that have the ability to sense population density. We take into account social crowding effects for which the dispersal rate is a decreasing function of the population density and residence time. We perform stochastic simulations of random walks and discover the phenomenon of self-organized anomaly (SOA), which leads to a collapse of stationary aggregation pattern. This anomalous regime is self-organized and arises without the need for a heavy tailed waiting time distribution from the inception. Conditions have been found under which the nonlinear random walk evolves into anomalous state when all particles aggregate inside a tiny domain (anomalous aggregation). We obtain power-law stationary density-dependent survival function and define the critical condition for SOA as the divergence of mean residence time. The role of the initial conditions in different SOA scenarios is discussed. We observe phenomenon of transient anomalous bimodal aggregation.

Temporal Gillespie Algorithm: Fast Simulation of Contagion Processes on Time-Varying Networks.

Stochastic simulations are one of the cornerstones of the analysis of dynamical processes on complex networks, and are often the only accessible way to explore their behavior. The development of fast algorithms is paramount to allow large-scale simulations. The Gillespie algorithm can be used for fast simulation of stochastic processes, and variants of it have been applied to simulate dynamical processes on static networks. However, its adaptation to temporal networks remains non-trivial. We here present a temporal Gillespie algorithm that solves this problem. Our method is applicable to general Poisson (constant-rate) processes on temporal networks, stochastically exact, and up to multiple orders of magnitude faster than traditional simulation schemes based on rejection sampling. We also show how it can be extended to simulate non-Markovian processes. The algorithm is easily applicable in practice, and as an illustration we detail how to simulate both Poissonian and non-Markovian models of epidemic spreading. Namely, we provide pseudocode and its implementation in C++ for simulating the paradigmatic Susceptible-Infected-Susceptible and Susceptible-Infected-Recovered models and a Susceptible-Infected-Recovered model with non-constant recovery rates. For empirical networks, the temporal Gillespie algorithm is here typically from 10 to 100 times faster than rejection sampling.

A functional Itô’s calculus approach to convex risk measures with jump diffusion

Convex risk measures for European contingent claims are studied in a non-Markovian jump-diffusion modeling framework using functional Itô’s calculus. Two representations for a convex risk measure are considered, one based on a nonlinear g-expectation and another one based on a representation theorem. Functional Itô’s calculus for càdlàg processes, backward stochastic differential equations (BSDEs) with jumps and stochastic optimal control theory are used to discuss the evaluation of convex risk measures. FPDIEs and PDIEs for convex risk measures are derived in the Markovian and non-Markovian situations, respectively. An entropic risk measure, which is a particular case of a convex risk measure, is discussed.

Kinetic Contributions to Gating by Interactions Unique to N-methyl-d-aspartate (NMDA) Receptors

Among glutamate-gated channels, NMDA receptors produce currents that subside with unusually slow kinetics, and this feature is essential to the physiology of central excitatory synapses. Relative to the homologous AMPA and kainate receptors, NMDA receptors have additional intersubunit contacts in the ligand binding domain that occur at both conserved and non-conserved sites. We examined GluN1/GluN2A single-channel currents with kinetic analyses and modeling to probe these class-specific intersubunit interactions for their role in glutamate binding and receptor gating. We found that substitutions that eliminate such interactions at non-conserved sites reduced stationary gating, accelerated deactivation, and imparted sensitivity to aniracetam, an AMPA receptor-selective positive modulator. Abolishing unique contacts at conserved sites also reduced stationary gating and accelerated deactivation. These results show that contacts specific to NMDA receptors, which brace the heterodimer interface within the ligand binding domain, stabilize actively gating receptor conformations and result in longer bursts and slower deactivations. They support the view that the strength of the heterodimer interface modulates gating in both NMDA and non-NMDA receptors and that unique interactions at this interface are responsible in part for basic differences between the kinetics of NMDA and non-NMDA currents at glutamatergic synapses.

Dynamics of social contagions with limited contact capacity

Individuals are always limited by some inelastic resources, such as time and energy, which restrict them to dedicate to social interaction and limit their contact capacities. Contact capacity plays an important role in dynamics of social contagions, which so far has eluded theoretical analysis. In this paper, we first propose a non-Markovian model to understand the effects of contact capacity on social contagions, in which each adopted individual can only contact and transmit the information to a finite number of neighbors. We then develop a heterogeneous edge-based compartmental theory for this model, and a remarkable agreement with simulations is obtained. Through theory and simulations, we find that enlarging the contact capacity makes the network more fragile to behavior spreading. Interestingly, we find that both the continuous and discontinuous dependence of the final adoption size on the information transmission probability can arise. There is a crossover phenomenon between the two types of dependence. More specifically, the crossover phenomenon can be induced by enlarging the contact capacity only when the degree exponent is above a critical degree exponent, while the final behavior adoption size always grows continuously for any contact capacity when degree exponent is below the critical degree exponent.

Non-Markovian models of the growth of a polymer chain

Using simple exactly solvable models, we show that event-dependent time delays may lead to significant non-Poisson effects in the statistics of polymer chain growth. The results are confirmed by stochastic simulation of various growth scenarios. Our interest in mathematical aspects of non-Markovian growth arises from recent successful application of delayed probability density functions in stochastic modelling of controlled radical polymerization.

Zeno limit in frequency estimation with non-Markovian environments

A quantum system can be prepared in an entangled state that is very sensitive to changes introduced to its dynamics and thus potentially can be utilized as a precise sensor of dynamics parameters. $N$ entangled two-level atoms evolving unitarily can be used for frequency estimation with Heisenberg-limited precision which scales as ${N}^{\ensuremath{-}2}$, whereas uncorrelated states give only a standard-limited precision which scales as ${N}^{\ensuremath{-}1}$. As entangled states are also sensitive to disturbances introduced by interactions with external environments, quantum enhancement of the precision may be significantly limited for open quantum systems. This is the case when an interaction with an environment leads to local Markovian dephasing and the precision scales as ${N}^{\ensuremath{-}1}$ for all atom preparations. For local non-Markovian dephasing, however, Matsuzaki et al. [Phys. Rev. A 84, 012103 (2011)] and Chin et al. [Phys. Rev. Lett. 109, 233601 (2012)] showed that the atoms prepared in the maximally entangled state achieve Zeno scaling $\ensuremath{\propto}\phantom{\rule{0.16em}{0ex}}{N}^{\ensuremath{-}3/2}$ in precision and argued that this limit should hold for any entangled state. Here we prove that Zeno scaling is indeed the best possible scaling for all local non-Markovian dephasing models which feature initial Zeno dynamics. Moreover, Zeno scaling can hold only for experiments in the Zeno-dynamics regime, otherwise scaling of the precision is standard.

Entropy production in a non-Markovian environment.

Stochastic thermodynamics and the associated fluctuation relations provide the means to extend the fundamental laws of thermodynamics to small scales and systems out of equilibrium. The fluctuating thermodynamic variables are usually treated in the context of either isolated Hamiltonian evolution, or Markovian dynamics in open systems. However, there is no reason a priori why the Markovian approximation should be valid in driven systems under nonequilibrium conditions. In this work, we introduce an explicitly non-Markovian model of dynamics of an open system, where the correlations between the system and the environment drive a subset of the environment out of equilibrium. Such an environment gives rise to a new type of non-Markovian entropy production term. Such non-Markovian components must be taken into account in order to recover the fluctuation relations for entropy. As a concrete example, we explicitly derive such modified fluctuation relations for the case of an overheated single electron box.

Forward equations for option prices in semimartingale models

We derive a forward partial integro-differential equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a -possibly discontinuous- semimartingale. A uniqueness theorem is given for the solutions of this equation. This result generalizes Dupire's forward equation to a large class of non-Markovian models with jumps.

Non-Markovian model for the study of pitting corrosion in a water pipe system

The main studies on pitting consist in proposing Markovian stochastic models, based on the statistics of extreme values and focused on growing the depth of wells, especially the deepest one. We show that a non-Markovian model, described by a nonlinear Fokker–Planck (nFP) equation, properly depicts the time evolution of a distribution of depth values of pits that were experimentally obtained. The solution of this equation in a steady-state regime is a q-Gaussian distribution, i.e. a long-tail probability distribution that is the main characteristic of a nonextensive statistical mechanics. The proposed model, that is applied to data from four inspections conducted on a section of a line of regular water service in power water reactor (PWR) nuclear power plants, is in agreement with experimental results.

Time-average based on scaling law in anomalous diffusions

To solve the obscureness in measurement brought about from the weak ergodicity breaking appeared in anomalous diffusions, we have suggested the time-averaged mean squared displacement (MSD) [Formula: see text] with an integral interval depending linearly on the lag time τ. For the continuous time random walk describing a subdiffusive behavior, we have found that [Formula: see text] like that of the ensemble-averaged MSD, which makes it be possible to measure the proper exponent values through time-average in experiments like a single molecule tracking. Also, we have found that it has originated from the scaling nature of the MSD at an aging time in anomalous diffusion and confirmed them through numerical results of the other microscopic non-Markovian model showing subdiffusions and superdiffusions with the origin of memory enhancement.

Quaternary AlInGaN/InGaN quantum well on vicinal c-plane substrate for high emission intensity of green wavelengths

Electronic and optical properties of non-trivial semipolar AlInGaN/InGaN quantum well (QW) structures are investigated by using the multiband effective-mass theory and non-Markovian optical model. On vicinal c-plane GaN substrate miscut by a small angle (θ < 40°) from c-plane, the AlInGaN/InGaN system is shown to have ∼3 times larger spontaneous emission peak intensity than the conventional InGaN/GaN system at green wavelength. It is attributed to much larger optical matrix element of the quaternary AlInGaN/InGaN system, derived from the reduction of internal electric field induced by polarizations. This effect exceeds the performance-degrading factor of smaller quasi-Fermi-level separation for the quaternary AlInGaN/InGaN system than that for the conventional InGaN/GaN system. Results indicate that the use of quaternary III-nitride QWs on vicinal substrates may be beneficial in improving the performance of optical devices emitting green light.

Improving the quality of noisy spatial quantum channels

We show, for the non-Markovian or time-dependent Markovian model of noise, by breaking the noisy spatial quantum channel (SQC) into a series of periodically arranged sub-components, that the quality of information transmission described by the purity, fidelity and concurrence of the output states can be improved. The physical mechanism and possible implementation of the idea have been discussed.

Improving the quality of noisy spatial quantum channels

We show, for the non-Markovian or time-dependent Markovian model of noise, by breaking the noisy spatial quantum channel (SQC) into a series of periodically arranged sub-components, that the qualit...

Effects of AlGaN delta-layer insertion on light emission characteristics of ultraviolet AlGaN/AlN quantum well structures

The light emission characteristics of ultraviolet (UV) AlGaN/AlN quantum well (QW) structures with the AlGaN delta-layer in the active region were investigated using the multiband effective-mass theory and non-Markovian gain model. In a wavelength range of 220–280 nm, AlGaN/AlN QW structures with the ultrathin AlGaN layer are shown to have much larger light emission than conventional AlGaN/AlN QW structures. This can be explained by the fact that the internal field in the well is reduced and the optical matrix element is greatly enhanced with the inclusion of the low-bandgap AlGaN delta-layer. We expect that AlGaN/AlN QW structures with the ultrathin AlGaN layer can be used as a light source with a high efficiency in UV region.