Nonlinear Master Research Articles

A New Information Soliton to Resist Decaying of the Excitation Based on the External Field Interaction

In this work, we reveal possibility using information soliton to resist senescence of certain important bio-excitations (such as Davydov solitons) which play a fundamental role in information processing of life. For this goal, a type of external field interaction with original system is introduced. This field enables the total system to be described by a nonlinear Master equation. Then we found that the nonlinear term in the equation drives the initial excitation to evolve as a kind of information soliton asymptotically. It is this information soliton to resist decaying of the excitations. This provides a constructive way to prolong age of biological excitations by exerting an external field, which forms a basis used in medical devices or treatments.

Nonlinear Liouville Equation and Information Soliton

In this work, some types of nonlinear Liouville equation (NLE) and nonlinear Master equations (NME) are studied. We found that the nonlinear terms in the equation can resist state of system damping so that an information solitonic structure appears. Furthermore, the power in the non-linear term is independent of limitation of the solution. This characteristic offers a possibility to construct complicated information solitons from some simple solutions, which allow one to solve complicated NLE or NME. The results obtained in this work may provide an innovated channel for the quantum information transmission over far distance against dissipation and decoherence, and also open a constructive way to resist age decaying of system by designing adjusted field interaction with the system nonlinearly.

Many-particle Sudarshan-Lindblad equation: mean-field approximation, nonlinearity and dissipation in a spin system

A system of N spin-1/2 particles interacting with a thermal reservoir is used as a pedagogical example for advanced undergraduate and graduate students. We introduce and illustrate some methods, approximations, and phenomena related to dissipation and nonlinearity in many-particle physics. We start our analysis from the dynamical Sudarshan-Lindblad quantum master equation for the density operator of a system S interacting with a thermal reservoir R. We derive the quantum version of the so-called Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) equations such that the master equation can be decomposed in a hierarchical set of N - 1 equations (N > 1). The hierarchy is broken by introducing the mean-field approximation and reducing the problem to a nonlinear single particle system. In this scenario, the Hamiltonian is nonlinear (i.e., it depends on the state of <S), although the superoperator responsible for the dissipation and decoherence of S remains unaffected. To provide a useful tool to students: (1) we discuss the physical approximations involved, (2) we derive the analytical solution to the mean values equations of motion resulting from the Hamiltonian, (3) we solve analytically the master equation in the stationary regime, (4) we obtain and discuss the solution of the nonlinear master equation, numerically, and finally, (5) we discuss the master equation beyond the mean-field approximation and show how to introduce higher order quantum correlations that have been previously neglected.

Time-driven quantum master equations and their compatibility with the fluctuation dissipation theorem

We contribute to a long-standing debate on the supposed failure of the fluctuation dissipation theorem (FDT) for the Davies master equation (DME), an important class of Lindblad quantum master equations, describing time-driven quantum systems weakly coupled to a heat bath. First we propose two simple and natural criteria on the driving which guarantee compatibility with the FDT. We show through our setting that, contrary to what is often stated in the literature, the DME is fully compatible with the FDT. We thus argue that the cause of the dispute lies in the adopted perturbation scheme, rather than in the Lindblad character of the master equation itself. We confirm our statement by proving that the Grabert master equation, first proposed by Grabert [Projection Operator Techniques in Nonequilibrium Statistical Mechanics (Springer, Berlin, 1982)] as an alternative linear dynamics fulfilling the FDT, is nothing else than the incriminated DME. Our criteria for the FDT can also be used in the analysis of the nonlinear thermodynamical master equation, first obtained in the Brownian motion limit [H. Grabert, Z. Phys. B 49, 161 (1982)] and later independently rediscovered and generalized on purely thermodynamic grounds [H. C. \"Ottinger, Europhys. Lett. 94, 10006 (2011)].

Kraus-Form Operator Solution to Nonlinear Master Equation by Virtue of the Nonlinear Thermo-Entangled State Representation

Our work solves the newly constructed nonlinear master equation d ρ d t = 2 κ ( 1 f ( N - 1 ) a † ρ f ( N ) a + f ( N ) a ρ 1 f ( N - 1 ) a † a ρ - ρ a a † ) , where f ( N ) is an operator-valued function of N = a † a , for describing the mechanism of evolution in the laser channel at extremely high temperature and derives the infinite operator sum representation of quasi-Kraus operators for the reduced density operator. It also shows that in this nonlinear process the initial nonlinear coherent state (NCS) will evolve into the nonlinear coherent excitation of thermo field, indicating that at extremly high temperature, after passing through the laser channel the NCS (a pure state) evolves into a nonlinear coherent excitation of thermo field (a mixed state, also called displaced chaotic state) by continually absorbing photons from the thermal reservoir. Additionally, universal evolving rules of the mean number of photons, variance in the photon number, relative fluctuation of the number of photons, quantum mechanical second-order degrees of coherence are derived.

4‐Channel Sliding Mode Control of Teleoperation Systems with Disturbances

AbstractSliding mode control can effectively account for the disturbances of a system. Among the different teleoperation architectures, 4‐channel architecture is the most successful for fulfilling transparency. In this paper, two sliding mode controllers are designed for nonlinear master and slave with external disturbances and are incorporated into a 4‐channel structure to achieve transparency. To this end, each of the controllers consists of a sliding mode position feedback law, a force feedback law, and two supplementary terms regarding gravity and contact force compensation. Stability and transparency of the overall system is studied via a Lyapunov function analysis. Simulations compared with the conventional adaptive control on teleoperation systems demonstrate the effectiveness of the proposed scheme.

Multiple re-encounter approach to radical pair reactions and the role of nonlinear master equations.

We formulate a multiple-encounter model of the radical pair mechanism that is based on a random coupling of the radical pair to a minimal model environment. These occasional pulse-like couplings correspond to the radical encounters and give rise to both dephasing and recombination. While this is in agreement with the original model of Haberkorn and its extensions that assume additional dephasing, we show how a nonlinear master equation may be constructed to describe the conditional evolution of the radical pairs prior to the detection of their recombination. We propose a nonlinear master equation for the evolution of an ensemble of independently evolving radical pairs whose nonlinearity depends on the record of the fluorescence signal. We also reformulate Haberkorn's original argument on the physicality of reaction operators using the terminology of quantum optics/open quantum systems. Our model allows one to describe multiple encounters within the exponential model and connects this with the master equation approach. We include hitherto neglected effects of the encounters, such as a separate dephasing in the triplet subspace, and predict potential new effects, such as Grover reflections of radical spins, that may be observed if the strength and time of the encounters can be experimentally controlled.

Asymptotic solutions of the Oosawa model for the length distribution of biofilaments.

Nucleated polymerisation phenomena are general linear growth processes that underlie the formation of a range of biofilaments in nature, including actin and tubulin that are key components of the cellular cytoskeleton. The conventional theoretical framework for describing this process is the Oosawa model that takes into account homogeneous nucleation coupled to linear growth. In his original work, Oosawa provided an analytical solution to the total mass concentration of filaments; the time evolution of the full length distribution has, however, been challenging to access, in large part due to the nonlinear nature of the rate equations inherent in the description of such phenomena and to date analytical solutions for the filament distribution are known only in certain special cases. Here, by exploiting a technique based on the method of matched asymptotics, we present an analytical treatment of the Oosawa model that describes the shape of the length distribution of biofilaments reversibly growing through primary nucleation and filament elongation. Our work highlights the power of matched asymptotics for obtaining closed-form analytical solutions to nonlinear master equations in biophysics and allows us to identify the key time scales that characterize biological polymerization processes.

Non-linear kinetic model for oscillatory relaxation of the photovoltage effect on a Si(111)7 × 7 surface

Oscillatory relaxation of surface photovoltage has been observed by time-resolved photoemission spectroscopy on a Si(111)7×7 surface with a temporal period of several tens of nanoseconds. The oscillation was reproduced by solving non-linear Master equations that describe surface recombination of electron–hole pairs associated with the formation of exciton-like states at the surface. Appearance of non-linear phenomena is naturally expected by the model when high density photo-excited carriers are generated by the high power laser pulse.

Stochasticity and noise-induced transition of genetic toggle switch

The ability to predict and analyze the function of genetic circuits will enhance the design of autonomous, programmable, complex regulatory genetic structures. An abundance of modeling techniques has recently been developed to delineate simple genetic structures in terms of their constituents. Simple systems with characteristics of feedback inhibition, multi-stability, switching, and oscillatory expression have often been the focus. The present work is an attempt to improve existing deterministic models that fail to oblige to the crucial aspect of noise in genetic modeling. The objective of this work is to analyze, model, and simulate the protein populations in gene expression mechanisms by resorting to stochastic algorithms. The system involves two types of genes; the protein produced from the expression of one gene is capable of turning off the expression of the other gene. Rates of degradation of these proteins are assumed to be proportional to their concentrations. The master equation of this ‘genetic toggle switch’ is formulated using the probabilistic population balance around a particular state and by considering five mutually exclusive events. The efficacy of the present methodology is mainly attributable to the 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 Equation. A less laborious approach based on Kurtz’s limit theorems for the derivation of the stochastic characteristics is also presented for comparison. Solving the resultant ordinary differential equations governing the means, variances, and covariance of the master equations simultaneously using the published data yield information concerning not only the means of the two populations of proteins but also the minimal uncertainties of the populations inherent in the expressions. It is demonstrated that systems with small populations are susceptible to large internal fluctuations (or uncertainties) in their population evolution. Large uncertainties are observed after the populations enter the proximity of the saddle node, which is likely to cause transition of system’s steady state from one to another. Independent Monte-Carlo simulation runs clearly demonstrates that the occurrence of such internal noise-induced transition.

Nonlinear subdiffusive fractional equations and the aggregation phenomenon

In this article we address the problem of the nonlinear interaction of subdiffusive particles. We introduce the random walk model in which statistical characteristics of a random walker such as escape rate and jump distribution depend on the mean density of particles. We derive a set of nonlinear subdiffusive fractional master equations and consider their diffusion approximations. We show that these equations describe the transition from an intermediate subdiffusive regime to asymptotically normal advection-diffusion transport regime. This transition is governed by nonlinear tempering parameter that generalizes the standard linear tempering. We illustrate the general results through the use of the examples from cell and population biology. We find that a nonuniform anomalous exponent has a strong influence on the aggregation phenomenon.

Open Quantum Systems Coupled to Time-Dependent Classical Environments

We investigate the behavior of open quantum systems interacting with classical time-dependent environments. As a simple example, we employ a two-level quantum system, and a thermodynamic oscillator serves as an environment. We analyze how the relationship between parameters of the classical environment and the quantum subsystem changes the evolution. Using the nonlinear thermodynamic master equation, we demonstrate how the energy of the quantum system evolves in time and how feedback effects from a quantum to a classical system influences the dynamics.

Synthesis and Linear and Nonlinear Melt Rheology of Well-Defined Comb Architectures of PS and PpMS with a Low and Controlled Degree of Long-Chain Branching

Well-defined, monodisperse homopolymer comb architectures with varied number and length of the branches under linear and nonlinear deformation were synthesized and examined to determine the effect of branching on different rheological properties. The correlation of the rheological properties with the comb topology is of special interest for the determination of the degree of branching. Therefore, well-defined polystyrene-based comb polymers with systematically varied number and molecular weight of the branches, narrow polydispersities, and a controlled, but low, number of branches (typically 0.1–1 mol % branches per backbone) were synthesized and compared with data from polystyrene combs of the Roovers series that have a higher number of branches (>1 mol % branches per backbone). To investigate the rheological properties in detail, various linear and nonlinear techniques were applied. Within the linear regime, the reduced van Gurp–Palmen plot (δ vs |G*|/GN0) was used to identify critical points that illustrated the influence of the branch molecular weight and number of branches on the resulting rheological properties. In the nonlinear regime large amplitude oscillatory shear (LAOS) measurements were performed to obtain the nonlinear parameter Q0(ω) via a quadratic scaling law from FT-rheology. An intrinsic nonlinear master curve based on the Q0(ω) parameter reflected the relaxation hierarchy and was shown to be a sensitive method to extract information on the different relaxation time scales. The nonlinear shear measurements were complemented by uniaxial extensional measurements to quantify the strain hardening effect and how the strain hardening was affected by branch relaxation. The results obtained from the uniaxial extensional measurements could be correlated to relaxation times obtained from the intrinsic nonlinear master curve Q0(ω). Pom-pom constitutive model predictions were performed for the comparison with experimental data for extensional rheology with focus on the strain hardening behavior and for LAOS with focus on the nonlinear parameter Q0(ω) as a function of increasing number and molecular weight of the branches in the pom-pom molecule. A comparison of the applied rheological methods—(1) small amplitude oscillatory shear (SAOS) in the linear regime, (2) LAOS in combination with FT-rheology, and (3) extensional rheology in the nonlinear regime—illustrated the detection limits as well as the advantages and disadvantages of each technique toward the investigation of rare, but entangled branched comb polymer topologies.

Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

Strongly nonlinear stochastic processes can be found in many applications in physics and the life sciences. In particular, in physics, strongly nonlinear stochastic processes play an important role in understanding nonlinear Markov diffusion processes and have frequently been used to describe order-disorder phase transitions of equilibrium and nonequilibrium systems. However, diffusion processes represent only one class of strongly nonlinear stochastic processes out of four fundamental classes of time-discrete and time-continuous processes evolving on discrete and continuous state spaces. Moreover, strongly nonlinear stochastic processes appear both as Markov and non-Markovian processes. In this paper the full spectrum of strongly nonlinear stochastic processes is presented. Not only are processes presented that are defined by nonlinear diffusion and nonlinear Fokker-Planck equations but also processes are discussed that are defined by nonlinear Markov chains, nonlinear master equations, and strongly nonlinear stochastic iterative maps. Markovian as well as non-Markovian processes are considered. Applications range from classical fields of physics such as astrophysics, accelerator physics, order-disorder phase transitions of liquids, material physics of porous media, quantum mechanical descriptions, and synchronization phenomena in equilibrium and nonequilibrium systems to problems in mathematics, engineering sciences, biology, psychology, social sciences, finance, and economics.

Gauge Invariance of Nonlinear Klein-Gordon

We have discussed the gauge invariance of nonlinear Klein-Gordon equation which describes the interaction of electromagnetic initially proposed by Hermann Weyl. The construction of nonlinear Klein-Gordon itself is formulated by two classical conservation laws, Hamilton-Jacobi of relativistic motion and continuity equations. Generally, this equation takes the similar concept to the derivation of the nonlinear master Schrödinger ignoring two fundamental concepts in ordinary quantum mechanics, Einstein and de Broglie’s postulates. In this paper, the proposing for general form of nonlinear Klein-Gordon including the external potential and its gauge invariance have been established. However, we have to ignore the external potential in order to satisfy gauge invariance. In addition, we also prove the derivation of nonlinear Dirac equation for including half-integer spin concept can not hold by similar derivation.

Information Soliton

In this work, for understanding bio-information transmission through long distance a type of nonlinear master equation is studied. We found that the nonlinear power term can introduce a novel solution of the equation, in which a possible invariant structure as an information soliton can exist when time elapses long enough. This provides a sort of constructive channel for bio-information transmission for long distance.

Quantum quasi-Markov processes in eventum mechanics dynamics, observation, filtering and control

Quantum mechanical systems exhibit an inherently probabilistic behavior upon measurement which excludes in principle the singular case of direct observability. The theory of quantum stochastic time continuous measurements and quantum filtering was earlier developed by the author on the basis of non-Markov conditionally-independent increment models for quantum noise and quantum nondemolition observability. Here this theory is generalized to the case of demolition indirect measurements of quantum unstable systems satisfying the microcausality principle. The exposition of the theory is given in the most general algebraic setting unifying quantum and classical theories as particular cases. The reduced quantum feedback-controlled dynamics is described equivalently by linear quasi-Markov and nonlinear conditionally-Markov stochastic master equations. Using this scheme for diffusive and counting measurements to describe the stochastic evolution of the open quantum system under the continuous indirect observation and working in parallel with classical indeterministic control theory, we derive the Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these measurements. The resulting Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian problem which emphasizes many similarities with the corresponding classical control problem.

Stochastic process behind nonlinear thermodynamic quantum master equation. II. Simulation

We simulate a piecewise deterministic Markovian jump process that represents an unraveling of a nonlinear quantum master equation describing the evolution of a quantum subsystem in contact with a heat bath. This process relies on running ensemble averages and state vectors normalized only on average. The latter causes the normalization of a small subset of trajectories to grow exponentially, resulting in numerical instability. We propose an efficient solution to this problem and illustrate the general ideas for a harmonic oscillator and a two-level system, each of them being weakly coupled to a heat reservoir. Our findings lay the foundation for a powerful simulation technique for dissipative quantum systems surrounded by general classical nonequilibrium environments.

Stochastic process behind nonlinear thermodynamic quantum master equation. I. Mean-field construction

We propose a piecewise deterministic Markovian jump process in Hilbert space such that the covariance matrix of this stochastic process solves the thermodynamic quantum master equation. The proposed stochastic process is particularly simple because the normalization of the vectors in Hilbert space is preserved only on average. As a consequence of the nonlinearity of the thermodynamic master equation, the construction of stochastic trajectories involves the density matrix as a running ensemble average. We identify a principle of detailed balance and a fluctuation-dissipation relation for our Markovian jump process.

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