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A colloidal particle is a prominent example of a stochastic system, and, if suspended in a simple viscous liquid, very closely resembles the case of an ideal random walker. A variety of new phenomena have been observed when such colloid is suspended in a viscoelastic fluid instead, for example pronounced nonlinear responses when the viscoelastic bath is driven out of equilibrium. Here, using a micron-sized particle in a micellar solution, we investigate in detail, how these nonlinear bath properties leave their fingerprints already in equilibrium measurements, for the cases where the particle is unconfined or trapped in a harmonic potential. We find that the coefficients in an effective linear (generalized) Langevin equation show intriguing inter-dependencies, which can be shown to arise only in nonlinear baths: for example, the friction memory can depend on the external potential that acts only on the colloidal particle (as recently noted in simulations of molecular tracers in water in (2017 Phys. Rev. X 7 041065)), it can depend on the mass of the colloid, or, in an overdamped setting, on its bare diffusivity. These inter-dependencies, caused by so-called fluctuation renormalizations, are seen in an exact small time expansion of the friction memory based on microscopic starting points. Using linear response theory, they can be interpreted in terms of microrheological modes of force-controlled or velocity-controlled driving. The mentioned nonlinear markers are observed in our experiments, which are astonishingly well reproduced by a stochastic Prandtl-Tomlinson model mimicking the nonlinear viscoelastic bath. The pronounced nonlinearities seen in our experiments together with the good understanding in a simple theoretical model make this system a promising candidate for exploration of colloidal motion in nonlinear stochastic environments.

The El Niño-Southern Oscillation (ENSO) is a complex and influential climate phenomenon critical to understanding global climate systems and enhancing climate predictions. Despite extensive research utilizing both statistical methods and numerical models for accurate ENSO forecasting, significant gaps remain in practical applications. Therefore, we proposed a novel memory kernel function-based approach to solve the inverse problem of ENSO time-varying systems. This method involves constructing differential equations through memory vectors composed of multiple initial values, effectively capturing the system’s evolutionary and trends. Unlike traditional inverse problem-solving methods, our research delved into the inherent properties exhibited by ENSO, such as memory and periodicity, and embedded these properties as specific targets in differential equations. By leveraging the flexibility of evolutionary algorithms to solve mathematical problems, we achieved a model targeted at ENSO and predicted at lead times up to 26 months. The results demonstrate that this scheme overcomes the limitations of traditional differential equations with a single initial value and extends these equations to memory vector equations based on multiple initial values. This not only enhances our ability to describe the evolutionary laws of complex systems but also improves the timeliness and reliability of ENSO predictions, achieving encouraging results.

We consider the Moore–Gibson–Thompson equation with memory of type II ∂ ttt u ( t ) + α ∂ tt u ( t ) + β A ∂ t u ( t ) + γ A u ( t ) - ∫ 0 t g ( t - s ) A ∂ t u ( s ) d s = 0 where A is a strictly positive selfadjoint linear operator (bounded or unbounded) and α , β , γ > 0 satisfy the relation γ ≤ α β . First, we prove well-posedness of finite energy solutions, without requiring any restriction on the total mass ϱ of g . This extends previous results in the literature, where such a restriction was imposed. Second, we address an open question within the context of longtime behavior of solutions. We show that an “overdamping” in the memory term can destabilize the originally stable dynamics. In fact, it is always possible to find memory kernels g , complying with the usual mass restriction ϱ < β , such that the equation admits solutions with energy growing exponentially fast, even in the regime γ < α β where the corresponding model without memory is exponentially stable. In particular, this provides an answer to a question recently raised in the literature.

In this work, analytical solution of hybrid Maxwell nanofluid of the vertical channel due to pressure gradient is discussed. By introducing dimensionless variables the governing equations with all levied initial and boundary conditions is converted into dimensionless form. Fractional model for Maxwell fluid is developed by Caputo time fractional differential operator by using the constitutive relation. The dimensionless expression for temperature and velocity are found using Laplace transform. Draw graphs of temperature and velocity by Mathcad software and discuss the behavior of flow parameters and the effect of fractional parameters. As a result, we have found by increasing the volumetric fraction of copper and alumina temperature increases and velocity decreases. Also, fluid flow properties showed dual behavior for small and large time, respectively, by increasing fractional parameters values.

Master equations are a useful tool to describe the evolution of open quantum systems. In order to characterize the mathematical features and the physical origin of the dynamics, it is often useful to consider different kinds of master equations for the same system. Here, we derive an exact connection between the time-local and the integro-differential descriptions, focusing on the class of commutative dynamics. The use of the damping-basis formalism allows us to devise a general procedure to go from one master equation to the other and vice versa, by working with functions of time and their Laplace transforms only. We further analyze the Lindbladian form of the time-local and the integro-differential master equations, where we account for the appearance of different sets of Lindbladian operators. In addition, we investigate a Redfield-like approximation, that transforms the exact integro-differential equation into a time-local one by means of a coarse graining in time. Besides relating the structure of the resulting master equation to those associated with the exact dynamics, we study the effects of the approximation on Markovianity. In particular, we show that, against expectation, the coarse graining in time can possibly introduce memory effects, leading to a violation of a divisibility property of the dynamics.

Since there is no available vaccination for a contagious disease like HIV/AIDS, utilizing fractional order modeling in awareness, diagnosis, treatment, and control theory can be an effective method to prevent the spread of the disease. Consequently, we developed a novel mathematical model within a Caputo fractional framework to analyze the patterns of HIV/AIDS disease transmission. To make the unaware susceptibles aware, media-based global awareness and word-of-mouth communication based local awareness, together with psychological anxiety about infection among these conscious susceptibles, are incorporated here. Diagnosis process is taken into account to diagnosis the asymptomatic individuals to identify the AIDS and HIV symptomatic population. The local awareness and diagnosis rates are assumed dynamic, which are proportional to the aware, infectious and population under treatment. Taking advantage of the generalized mean value theorem, we first ensure the positivity, then applying the Mittag-Leffler function’s property, we show the biologically feasible region, and finally the solution’s existence and uniqueness are given by utilizing the Banach fixed point theorem. We present only local stability around the disease-free equilibrium point. Next local and global stabilities around the disease persistent equilibrium point-based are exhibited using the basic reproduction number ( R 0 ). The direction of transcritical bifurcation is illustrated analytically using the unit epidemic edge. In order to identify the most important parameter(s) for our defined epidemic model, which are embedded in R 0 , a quantitative assessment is conducted on R 0 . To achieve the population dynamics’s trends, a numerical experiment is conducted here. During and after transition phases, we impose a strive for understanding the behavior of the order and kernel of the fractional derivative on the system’s dynamics and in the prevention of disease transmission. Based on the fractional optimal control theory, it is demonstrated that HIV/AIDS can be mathematically eradicated with the intervention strategies, namely, enhanced awareness, diagnosis and treatment. Fractional derivatives serve a crucial role in preventing the propagation of the disease and have a significant impact on the system’s dynamics during the transition period.

Quantum collision models have proved to be useful for a clear and concise description of many physical phenomena in the field of open quantum systems: thermalization, decoherence, homogenization, nonequilibrium steady state, entanglement generation, simulation of many-body dynamics, and quantum thermometry. A challenge in the standard collision model, where the system and many ancillas are all initially uncorrelated, is how to describe quantum correlations among ancillas induced by successive system-ancilla interactions. Another challenge is how to deal with initially correlated ancillas. Here we develop a tensor network formalism to address both challenges. We show that the induced correlations in the standard collision model are well captured by a matrix product state (a matrix product density operator) if the colliding particles are in pure (mixed) states. In the case of the initially correlated ancillas, we construct a general tensor diagram for the system dynamics and derive a memory-kernel master equation. Analyzing the perturbation series for the memory kernel, we go beyond the recent results concerning the leading role of two-point correlations and consider multipoint correlations (Waldenfelds cumulants) that become relevant in the higher-order stroboscopic limits. These results open an avenue for the further analysis of memory effects in collisional quantum dynamics.

This article presents a new prediction model, the ordinary differential equations–memory kernel function (ODE–MKF), constructed from multiple backtracking initial values (MBIV). The model is similar to a simplified numerical model after spatial dimension reduction and has both nonlinear characteristics and the low-cost advantage of a time series model. The ODE–MKF focuses on utilizing more temporal information and includes machine learning to solve complex mathematical inverse problems to establish a predictive model. This study first validates the feasibility of the ODE–MKF via experiments using the Lorenz system. The results demonstrate that the ODE–MKF prediction model could describe the nonlinear characteristics of complex systems and exhibited ideal predictive robustness. The prediction of the El Niño-Southern Oscillation (ENSO) index further demonstrates its effectiveness, as it achieved 24-month lead predictions and effectively improved nonlinear problems. Furthermore, the reliability of the model was also tested, and approximately 18 months of prediction were achieved, which was verified with the Clouds and the Earth’s Radiant Energy System (CERES) Energy Balanced and Filled (EBAF) radiation fluxes. The short-term memory index Southern Oscillation (SO) was further used to examine the applicability of ODE–MKF. A six-month lead prediction of the SO trend was achieved, indicating that the predictability of complex systems is related to their inherent memory scales.

We study asymptotic properties of the Generalized Langevin Equation (GLE) in the presence of a wide class of external potential wells with a power-law decay memory kernel. When the memory can be expressed as a sum of exponentials, a class of Markovian systems in infinite-dimensional spaces is used to represent the GLE. The solutions are shown to converge in probability in the small-mass limit and the white-noise limit to appropriate systems under minimal assumptions, of which no global Lipschitz condition is required on the potentials. With further assumptions about space regularity and potentials, we obtain L1 convergence in the white-noise limit.

In this manuscript, a nonlinear time-fractional diffusion equation with a generalized memory kernel is studied. Initially, the original model problem is linearized by implementing the Newton’s quasilinearization technique. In the time-fractional term, a generalized Caputo derivative is considered and approximated using the non-uniform L 1-scheme as the solution has a singularity at t = 0 . The main contribution of this work is to develop a generalized discrete fractional Grönwall inequality. Thereafter, permitting its use to establish the stability and analyze the error estimate, under a proper regularity condition in the L 2 -norm, and an optimal convergence order O N - ( 2 - ζ ) is obtained for the L 1-scheme with respect to the graded mesh. Numerical results are inserted to corroborate the effectiveness of the theoretical analysis.