Nonlinear Master Equation Research Articles

A Theoretical Solution for a Nonlinear Master Equation

Based on the master equation describing the interaction of a single-mode bosonic state with a heat bath at finite temperature in the Born-Markov approximation, we constructed a new nonlinear master equation and derived the infinite operator sum representation of quasi-Kraus operators for the density operator

Non-Markovian models for migration-proliferation dichotomy of cancer cells: Anomalous switching and spreading rate

Proliferation and migration dichotomy of the tumor cell invasion is examined within two non-Markovian models. We consider the tumor spheroid, which consists of the tumor core with a high density of cells and the outer invasive zone. We distinguish two different regions of the outer invasive zone and develop models for both zones. In model I we analyze the near-core-outer region, where biased migration away from the tumor spheroid core takes place. We suggest non-Markovian switching between the migrating and proliferating phenotypes of tumor cells. Nonlinear master equations for mean densities of cancer cells of both phenotypes are derived. In anomalous switching case we estimate the average size of the near-core-outer region that corresponds to sublinear growth (r(t)) ~ t(μ) for 0 < μ < 1. In model II we consider the outer zone, where the density of cancer cells is very low. We suggest an integrodifferential equation for the total density of cancer cells. For proliferation rate we use the classical logistic growth, while the migration of cells is subdiffusive. The exact formulas for the overall spreading rate of cancer cells are obtained by a hyperbolic scaling and Hamilton-Jacobi techniques.

Nucleated polymerization with secondary pathways. II. Determination of self-consistent solutions to growth processes described by non-linear master equations

Nucleated polymerisation processes are involved in many growth phenomena in nature, including the formation of cytoskeletal filaments and the assembly of sickle hemoglobin and amyloid fibrils. Closed form rate equations have, however, been challenging to derive for these growth phenomena in cases where secondary nucleation processes are active, a difficulty exemplified by the highly non-linear nature of the equation systems that describe monomer dependent secondary nucleation pathways. We explore here the use of fixed point analysis to provide self-consistent solutions to such growth problems. We present iterative solutions and discuss their convergence behaviour. We establish a range of closed form results for linear growth processes, including the scaling behaviours of the maximum growth rate and of the reaction end-point. We further show that a self-consistent approach applied to the master equation of filamentous growth allows the determination of the evolution of the shape of the length distribution including the mean, the standard deviation, and the mode. Our results highlight the power of fixed-point approaches in finding closed form self-consistent solutions to growth problems characterised by the highly non-linear master equations.

Stochastic Processes and Mean Field Systems Defined by Nonlinear Markov Chains: An Illustration for a Model of Evolutionary Population Dynamics

In physics, there is a growing interest in studying stochastic processes described by evolution equations such as nonlinear master equations and nonlinear Fokker–Planck equations that define the so-called nonlinear Markov processes and are nonlinear with respect to probability densities. In this context, however, relatively little is known about nonlinear Markov processes defined by nonlinear Markov chains. In the present work, we demonstrate explicitly how the nonlinear Markov chain approach can be carried out by addressing a model for evolutionary population dynamics. In line with the nonlinear Markov chain approach, we derive a measure that tells us how attractive it is for a biological entity to evolve towards a particular biological type. Likewise, a measure for the noise level of the evolutionary process is obtained. Both measures are found to be implicitly time dependent. Finally, a simulation scheme for the many-body system corresponding to the Markov chain model is discussed.

Nonlinear thermodynamic quantum master equation: Properties and examples

The quantum master equation obtained from two different thermodynamic arguments is seriously nonlinear. We argue that, for quantum systems, nonlinearity occurs naturally in the step from reversible to irreversible equations and we analyze the nature and consequences of the nonlinear contribution. The thermodynamic nonlinearity naturally leads to canonical equilibrium solutions and extends the range of validity to lower temperatures. We discuss the Markovian character of the thermodynamic quantum master equation and introduce a solution strategy based on coupled evolution equations for the eigenstates and eigenvalues of the density matrix. The general ideas are illustrated for the two-level system and for the damped harmonic oscillator. Several conceptual implications of the nonlinearity of the thermodynamic quantum master equation are pointed out, including the absence of a Heisenberg picture and the resulting difficulties with defining multitime correlations.

Solving nonlinear master equation describing quantum damping by virtue of the entangled state representation

This paper solves the newly constructed nonlinear master equation dρ/dt = κ[2f (N) aρ (1/f (N − 1))a† − a†aρ−ρa†a], where f(N) is an operator-valued function of N = a†a, for describing amplitude damping channel, and derives the infinite operator sum representation of quasi-Kraus operators for the density operator. It also shows that in this nonlinear process the initial pure number state density operator will evolve into the binomial field (a mixed state) when

Anomalous transport and nonlinear reactions in spiny dendrites

We present a mesoscopic description of the anomalous transport and reactions of particles in spiny dendrites. As a starting point we use two-state Markovian model with the transition probabilities depending on residence time variable. The main assumption is that the longer a particle survives inside spine, the smaller becomes the transition probability from spine to dendrite. We extend a linear model presented in Fedotov [Phys. Rev. Lett. 101, 218102 (2008)] and derive the nonlinear Master equations for the average densities of particles inside spines and parent dendrite by eliminating residence time variable. We show that the flux of particles between spines and parent dendrite is not local in time and space. In particular the average flux of particles from a population of spines through spines necks into parent dendrite depends on chemical reactions in spines. This memory effect means that one cannot separate the exchange flux of particles and the chemical reactions inside spines. This phenomenon does not exist in the Markovian case. The flux of particles from dendrite to spines is found to depend on the transport process inside dendrite. We show that if the particles inside a dendrite have constant velocity, the mean particle's position <x(t)> increases as t(μ) with μ<1 (anomalous advection). We derive a fractional advection-diffusion equation for the total density of particles.

NUMERIC AND EXACT SOLUTIONS OF THE NONLINEAR CHAPMAN–KOLMOGOROV EQUATION: A CASE STUDY FOR A NONLINEAR SEMI-GROUP MARKOV MODEL

Nonlinear Markov processes are known to arise from diffusion approximations of nonlinear master equations and generalized Boltzmann equations and have been studied in their own merit in the context of nonlinear Fokker–Planck equations. Nonlinear Markov processes can account for collective phenomena, collective oscillations, and collective chaotic behavior. Despite the importance of nonlinear Markov processes for the physics of stochastic processes, relatively little is known about the Chapman–Kolmogorov equation of nonlinear Markov processes. The manuscript derives the exact solution of the Chapman–Kolmogorov equation for a recently proposed deterministic Markov model. Numerical solutions of the Chapman–Kolmogorov equation are used to demonstrate the nonlinear semi-group property of conditional probabilities of nonlinear Markov processes.

Effects of the environment on the electric conductivity of double-stranded DNAmolecules

We present a theoretical analysis of the effects of the environment on chargetransport in double-stranded synthetic poly(G)–poly(C) DNA molecules attached totwo ideal leads. Coupling of the DNA to the environment results in two effects:(i) localization of carrier functions due to static disorder and (ii) phonon-inducedscattering of the carriers between the localized states, resulting in hoppingconductivity. A nonlinear Pauli master equation for populations of localized states isused to describe the hopping transport and calculate the electric current as afunction of the applied bias. We demonstrate that, although the electronic gap inthe density of states shrinks as the disorder increases, the voltage gap in theI–V characteristics becomes wider. A simple physical explanation of this effect is provided.

Fluctuation-induced phase transition in a spatially extended model for catalyticCOoxidation

A reaction-diffusion master equation has been introduced in order to model the bistable CO oxidation on single crystal metal surfaces at high pressure where the diffusion length becomes small and local fluctuations are important. Analytical solutions can be found in a reduced one-component nonlinear master equation after applying the Weiss mean-field approximation together with the adiabatic elimination of oxygen. It is shown that the Weiss mean-field approximation predicts a symmetry-breaking bifurcation associated with a phase transition. The corresponding stationary solutions of the nonlinear master equation are supported by Gillespie-type Monte Carlo simulations.

Coupled Ito equations of continuous quantum state measurement and estimation

We discuss a non-linear stochastic master equation that governs the time-evolution of the estimated quantum state. Its differential evolution corresponds to the infinitesimal updates that depend on the time-continuous measurement of the true quantum state. The new stochastic master equation couples to the two standard stochastic differential equations of time-continuous quantum measurement. For the first time, we can prove that the calculated estimate almost always converges to the true state, also at low-efficiency measurements. We show that our single-state theory can be adapted to weak continuous ensemble measurements as well.

Theory of Nonlinear Dispersive Waves and Selection of the Ground State

A theory of time-dependent nonlinear dispersive equations of the Schrödinger or Gross-Pitaevskii and Hartree type is developed. The short, intermediate and large time behavior is found, by deriving nonlinear master equations (NLME), governing the evolution of the mode powers, and by a novel multitime scale analysis of these equations. The scattering theory is developed and coherent resonance phenomena and associated lifetimes are derived. Applications include Bose-Einstein condensate large time dynamics and nonlinear optical systems. The theory reveals a nonlinear transition phenomenon, "selection of the ground state," and NLME predicts the decay of excited state, with half its energy transferred to the ground state and half to radiation modes. Our results predict the recent experimental observations of Mandelik et al. in nonlinear optical waveguides.

Stochastic modeling of nonlinear epidemiology

The objectives of this paper to analyse, model and simulate the spread of an infectious disease by resorting to modern stochastic algorithms. The approach renders it possible to circumvent the simplifying assumption of linearity imposed in the majority of the past works on stochastic analysis of epidemic processes. Infectious diseases are often transmitted through contacts of those infected with those susceptible; hence the processes are inherently nonlinear. According to the classical model of Kermack and McKendrick, or the SIR model, three classes of populations are involved in two types of processes: conversion of susceptibles (S) to infectives (I) and conversion of infectives to removed (R). The master equations of the SIR process have been formulated through the probabilistic population balance around a particular state by considering the mutually exclusive events. The efficacy of the present methodology is mainly attributable to its ability to derive the governing equations for the means, variances and covariance of the random variables by the method of system-size expansion of the nonlinear master equations. Solving these equations simultaneously along with rates associated influenza epidemic data yields information concerning not only the means of the three populations but also the minimal uncertainties of these populations inherent in the epidemic. The stochastic pathways of the three different classes of populations during an epidemic, i.e. their means and the fluctuations around these means, have also been numerically simulated independently by the algorithm derived from the master equations, as well as by an event-driven Monte Carlo algorithm. The master equation and Monte Carlo algorithms have given rise to the identical results.

SELECTION OF THE GROUND STATE FOR NONLINEAR SCHRÖDINGER EQUATIONS

We prove for a class of nonlinear Schrödinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as ground state selection. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree–Fock type.

On a Nonlinear Master Equation and the Haken-Kelso-Bunz Model

A nonlinear master equation (NLME) is proposed basedon general information measures.Classical and cut-off solutions of the NLME are considered.In the former case, the NLME exhibits uniquely defined stationary distributions. In the latter case, there are multiple stationary distributions.In particular, for classical solutions, it is shown that transient solutions converge to stationary distributions that maximize information measures (H-theorem). Cut-off distributions arestudied numerically for the Haken-Kelso-Bunz model. The Haken-Kelso-Bunz modelis known to describe multistable human motor control systems. It is shownthat a stochastic Haken-Kelso-Bunz model based on a NLME can exhibit multiplestationary cut-off distributions.In doing so, we illustrate that multistability in stochastic biological systems can beestablished by means of cut-off distributions.

Folding stabilizes the evolution of catalysts.

Sequence folding is known to determine the spatial structure and catalytic function of proteins and nucleic acids. We show here that folding also plays a key role in enhancing the evolutionary stability of the intermolecular recognition necessary for the prevalent mode of catalytic action in replication, namely, in trans, one molecule catalyzing the replication of another copy, rather than itself. This points to a novel aspect of why molecular life is structured as it is, in the context of life as it could be: folding allows limited, structurally localized recognition to be strongly sensitive to global sequence changes, facilitating the evolution of cooperative interactions. RNA secondary structure folding, for example is shown to be able to stabilize the evolution of prolonged functional sequences, using only a part of this length extension for intermolecular recognition, beyond the limits of the (cooperative) error threshold. Such folding could facilitate the evolution of polymerases in spatially heterogeneous systems. This facilitation is, in fact, vital because physical limitations prevent complete sequence-dependent discrimination for any significant-size biopolymer substrate. The influence of partial sequence recognition between biopolymer catalysts and complex substrates is investigated within a stochastic, spatially resolved evolutionary model of trans catalysis. We use an analytically tractable nonlinear master equation formulation called PRESS (McCaskill et al., Biol. Chem. 382: 1343-1363), which makes use of an extrapolation of the spatial dynamics down from infinite dimensional space, and compare the results with Monte Carlo simulations.

Modeling Growth of a Heterogeneous Tumor

It has long been recognized that the growth of tumor population depends on the initial age distribution of the cells in the tumor and the age-dependent cellular birth rate. Deterministic dual-cell models have been available for sometime; these models take into account the effects of the resultant cell heterogeneity. Nevertheless, these models ignore various variables significantly affecting the growth, such as those characterizing the cells' inherent properties and environmental factors. Uncertainties, or fluctuations, arise when the growth is simulated with the models. Stochastic analysis of these fluctuations is the focus of the current work. Two types of cells are visualized to proliferate separately and to transform mutually during the process. The master equations of the system have been formulated through probabilistic population balance around a particular state by considering all mutually exclusive events. The governing equations for the means, variances, and covariance of the random variables have been derived through the system-size expansion of these nonlinear master equations. The stochastic pathways of the two different types of cells have been numerically simulated by the algorithm derived from the master equation for two different physical situations, one without and, the other, with the chemotherapeutic treatment. The results of the current study illuminate the significance of stochastically modeling the responses of the tumor to a variety of medicinal treatments: The coefficient of variation of the malignant cells' population magnifies with time under chemotherapeutic regimens. Consequently, the impact of the uncertainties in the exact number of malignant cells as expressed by this coefficient of variation is highly unpredictable. For example, it becomes increasingly uncertain if or how fast these cells will reactivate to become a full-blown carcinogenic tumor after treatment.

Dynamics of a nonlinear master equation: Low-dimensional manifolds and the nature of vibrational relaxation

The dynamics of nonlinear master equations describing vibrational relaxation in shock-heated molecules are studied. The nonlinearity results principally from inclusion of self-collisions. The master equations were derived in a previous paper by fitting experimental data and besides being nonlinear they vary according to changes in the bath temperature. It is demonstrated that, except for brief transients, the dynamics lie on one-dimensional, nonlinear manifolds, including the full time of experimental observation. The one-dimensional nature of the dynamics allows for an in depth study of vibrational relaxation. It is shown that vibrational distributions cannot be characterized accurately by a vibrational temperature until they are close to equilibrium and that one-dimensional rate laws accurately describe the dynamics on the one-dimensional manifold. The latter characteristic is important, because it allows results generated from master equations which include self-collisions to be easily incorporated into kinetic modeling.

Modeling of nonlinear vibrational relaxation of large molecules in shock waves with a nonlinear, temperature-varying master equation

We model recent experiments on the vibrational relaxation of oxirane in a shock tube. A master equation is developed which includes self-collisions of the oxirane, leading to a nonlinear master equation. This master equation is also applied to a more limited study of vibrational relaxation for cyclopropane in a shock tube. The time variation of the temperature dependence of the bath is also included in the calculations. Good agreement between the modeling and experiments are obtained through a fit to the energy transfer parameters. These fits demonstrate that self-collisions are dominant in promoting the relaxation even for mixtures of Kr and oxirane where the oxirane is 2% and 4% dilute. This dominance comes from two sources: (1) much larger energy transfer per collision for oxirane–oxirane collisions and (2) resonant energy transfer effects. For cyclopropane, some of the good fits show smaller energy transfer characteristics for self-collisions than buffer gas collisions. Even in these cases self-collisions are an important part of the energy transfer process through resonant energy transfer effects.

Geometric Approach to Multiple-Time-Scale Kinetics: A Nonlinear Master Equation Describing Vibration-to-Vibration Relaxation

A geometric approach to the study of multiple-time-scale kinetics is taken here. The approach to equilibrium for kinetic systems is studied via low-dimensional manifolds, with an application to a nonlinear master equation for vibrational relaxation. One of our main concerns is the asymptotic (in time) behavior of the system and whether there is a well-defined rate of approach to equilibrium. One-dimensional slow manifolds provide a good means for studying such behavior in nonlinear systems, because they are the analogue of the eigenvector with least negative eigenvalue for linear kinetics.