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For almost a decade the consensus has held that the random walk propagator for the elephant random walk (ERW) model is a Gaussian. Here we present strong numerical evidence that the propagator is, in general, non-Gaussian and, in fact, non-Lévy. Motivated by this surprising finding, we seek a second, non-Gaussian solution to the associated Fokker-Planck equation. We prove mathematically, by calculating the skewness, that the ERW Fokker-Planck equation has a non-Gaussian propagator for the superdiffusive regime. Finally, we discuss some unusual aspects of the propagator in the context of higher order terms needed in the Fokker-Planck equation.

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We consider a three-state model comprising tumor cells, effector cells, and tumor-detecting cells under the influence of noises. It is demonstrated that inevitable stochastic forces existing in all three cell species are able to suppress tumor cell growth completely. Whereas the deterministic model does not reveal a stable tumor-free state, the auto-correlated noise combined with cross-correlation functions can either lead to tumor-dormant states, tumor progression, as well as to an elimination of tumor cells. The auto-correlation function exhibits a finite correlation time τ, while the cross-correlation functions shows a white-noise behavior. The evolution of each of the three kinds of cells leads to a multiplicative noise coupling. The model is investigated by means of a multivariate Fokker-Planck equation for small τ. The different behavior of the system is, above all, determined by the variation of the correlation time and the strength of the cross-correlation between tumor and tumor-detecting cells. The theoretical model is based on a biological background discussed in detail, and the results are tested using realistic parameters from experimental observations.

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The analytical solution of the Poisson-Nernst-Planck equations is found in the linear regime as response to a dc-voltage. In deriving the results a new approach is suggested, which allows to fulfill all initial and boundary conditions and guarantees the absence of Faradaic processes explicitly. We obtain the spatiotemporal distribution of the electric field and the concentration of the charge carriers valid in the whole time interval and for an arbitrary initial concentration of ions. A different behavior in the short- and the long-time regime is observed. The crossover between these regimes is estimated.

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We analyze a stochastic model for tumor cell growth with both multiplicative and additive colored noises as well as nonzero cross correlations in between. Whereas the death rate within the logistic model is altered by a deterministic term characterizing immunization, the birth rate is assumed to be stochastically changed due to biological motivated growth processes leading to a multiplicative internal noise. Moreover, the system is subjected to an external additive noise which mimics the influence of the environment of the tumor. The stationary probability distribution P_{s} is derived depending on the finite correlation time, the immunization rate, and the strength of the cross correlation. P_{s} offers a maximum which becomes more pronounced for increasing immunization rate. The mean-first-passage time is also calculated in order to find out under which conditions the tumor can suffer extinction. Its characteristics are again controlled by the degree of immunization and the strength of the cross correlation. The behavior observed can be interpreted in terms of a biological model of tumor evolution.

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The susceptible-infectious-recovered (SIR) model describes the evolution of three species of individuals which are subject to an infection and recovery mechanism. A susceptible S can become infectious with an infection rate beta by an infectious I type provided that both are in contact. The I type may recover with a rate gamma and from then on stay immune. Due to the coupling between the different individuals, the model is nonlinear and out of equilibrium. We adopt a stochastic individual-based description where individuals are represented by nodes of a graph and contact is defined by the links of the graph. Mapping the underlying master equation onto a quantum formulation in terms of spin operators, the hierarchy of evolution equations can be solved exactly for arbitrary initial conditions on a linear chain. In the case of uncorrelated random initial conditions, the exact time evolution for all three individuals of the SIR model is given analytically. Depending on the initial conditions and reaction rates beta and gamma , the I population may increase initially before decaying to zero. Due to fluctuations, isolated regions of susceptible individuals evolve, and unlike in the standard mean-field SIR model, one observes a finite stationary distribution of the S type even for large population size. The exact results for the ensemble-averaged population size are compared with simulations for single realizations of the process and also with standard mean-field theory, which is expected to be valid on large fully connected graphs.

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A random walk of N particles on a lattice with M sites is studied under the constraint that each lattice site is coupled to its own mesoscopic heat bath. Such a situation can be conveniently described by using the master equation in a quantized Hamiltonian formulation where the exclusion principle is included by using Pauli operators. If all reservoirs are mutually in contact, giving rise to a temperature gradient, an evolution equation for the particle density with two different currents already results in the mean-field approximation. One is the conventional diffusive current, driven by the density gradient, whereas the other includes a coupling between the local density and the temperature gradient. Due to the competitive currents, the system exhibits a stationary solution, where the local density is determined by the local temperature field and depends on the filling factor M/N. The stability of the solution is related to the eigenvalues of a Schrödinger-like equation. In the case of a fixed temperature gradient the stationary density distribution remains stable. The approach used is totally different from and an alternative to the conventional Onsager ansatz.

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The Glauber model is reconsidered based on a quantum formulation of the master equation. Unlike the conventional approach the temperature and the Ising energy are included from the beginning by introducing a Heisenberg-like picture of the second quantized operators. This method enables us to get an exact expression for the transition rate of a single flip process w(i)(sigma(i)) which is in accordance with the principle of detailed balance. The transition rate differs significantly from the conventional one due to Glauber in the low-temperature regime. Here the behavior is controlled by the Ising energy and not by the microscopic time scale.

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A simple spin-flip process is analyzed under the presence of two heat reservoirs. While one flip process is triggered by a bath at temperature T, the inverse process is activated by a bath at a different temperature T'. The situation can be described by using a master equation approach in a second quantized Hamiltonian formulation. The stationary solution leads to a generalized Fermi-Dirac distribution with an effective temperature Te. Likewise the relaxation time is given in terms of Te. Introducing a spin representation we perform a Landau expansion for the averaged spin as order parameter and consequently, a free energy functional can be derived. Owing to the two reservoirs the model is invariant with respect to a simultaneous change sigma<-->-sigma and T<-->T'. This symmetry generates a third order term in the free energy which gives rise a dynamically induced first order transition.

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We consider a discrete-time random walk where the random increment at time step t depends on the full history of the process. We calculate exactly the mean and variance of the position and discuss its dependence on the initial condition and on the memory parameter p . At a critical value p((1) )(c ) =1/2 where memory effects vanish there is a transition from a weakly localized regime [where the walker (elephant) returns to its starting point] to an escape regime. Inside the escape regime there is a second critical value where the random walk becomes superdiffusive. The probability distribution is shown to be governed by a non-Markovian Fokker-Planck equation with hopping rates that depend both on time and on the starting position of the walk. On large scales the memory organizes itself into an effective harmonic oscillator potential for the random walker with a time-dependent spring constant k=(2p-1)/t . The solution of this problem is a Gaussian distribution with time-dependent mean and variance which both depend on the initiation of the process.

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A microscopic model is studied numerically to describe wearless dry friction without thermal fluctuations between atomically flat contact interfaces. The analysis is based on a double-chain model with a Lennard-Jones interaction between the chains which are the respective upper flexible monolayers of the rigid bulk systems. Whereas below a critical interaction strength epsilon(c) the system exhibits a frictionless state, it offers static friction above epsilon(c) . Introducing an appropriate order parameter function we demonstrate the analogy of the critical behavior to a phase transition of second order. The order parameter is related to a hull function describing uniquely the incommensurate ground state of the model. The breakdown of analyticity of the hull function is identified with the phase transition. Critical exponents are calculated and the validity of finite-size scaling is displayed.

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Memory effects require for their incorporation into random-walk models an extension of the conventional equations. The linear Fokker-Planck equation for the probability density p(r,t) is generalized by including nonlinear and nonlocal spatial-temporal memory effects. The realization of the memory kernel is restricted due the conservation of the basic quantity p. A general criteria is given for the existence of stationary solutions. In case the memory kernel depends on p polynomially, transport may be prevented. Owing to the delay effects a finite amount of particles remains localized and the further transport is terminated. For diffusion with nonlinear memory effects we find an exact solution in the long-time limit. Although the mean square displacement exhibits diffusive behavior, higher order cumulants offer differences to diffusion and they depend on the memory strength.

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We propose a model of weighted scale-free networks incorporating a stochastic scheme for weight assignments to the links, taking into account both the popularity and fitness of a node. As the network grows, the weights of links are driven either by the connectivity with probability p or by the fitness with probability 1-p. Numerical results show that the total weight exhibits a power-law distribution with an exponent sigma that depends on the probability p. The exponent sigma decreases continuously as p increases. For p=0, the scaling behavior is the same as that of the connectivity distribution. An analytical expression for the total weight is derived so as to explain the features observed in the numerical results. Numerical results are also presented for a generalized model with a fitness-dependent link formation mechanism.

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We study numerically a random walk under the competitive processes of a self-organized feedback coupling, characterized by a strength lambda and an underlying fractal lattice. Whereas a fractal structure favors a subdiffusive behavior, a dynamical feedback leads either to localization in case of an attractive feedback, lambda>0, or to superdiffusion for a repulsive memory strength lambda<0. Under the influence of both processes the dynamical exponent z is changed. For a Sierpinski gasket or a Sierpinski carpet with repulsive feedback coupling we get 2/z=1.04 or 2/z=1.08, respectively. When an attractive feedback is dominant, the system offers localization as in the case of a random walk in regular lattices. The numerical results are strongly supported by analytical studies based on scaling arguments.

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The time evolution of a bistable Ginzburg-Landau model (GL) with a non-Markovian memory term of strength lambda is studied. Due to the nonlinear feedback coupling, the two branches of the stationary solution are not only controlled by the sign of the initial condition P(0), but also by the strength and the sign of lambda. Whereas in case of a positive lambda the stationary solution is ever reduced through the memory, it may be increasing for lambda<0. In that case the system is also able to switch over between both branches of the stationary solution. Such an ability is exclusively achieved for a negative lambda within an interval -u

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The paper is concerned with a toy model that generalizes the standard Lotka-Volterra equation for a certain population by introducing a competition between instantaneous and accumulative, history-dependent nonlinear feedback the origin of which could be a contribution from any kind of mismanagement in the past. The results depend on the sign of that additional cumulative loss or gain term of strength lambda. In case of a positive coupling the system offers a maximum gain achieved after a finite time but the population will die out in the long time limit. In this case the instantaneous loss term of strength u is irrelevant and the model exhibits an exact solution. In the opposite case lambda<0 the time evolution of the system is terminated in a crash after t(s) provided u=0. This singularity after a finite time can be avoided if u not equal to 0. The approach may well be of relevance for the qualitative understanding of more realistic descriptions.

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An autocatalytic reaction combined with spontaneous creation and annihilation processes of particles are studied in a quantum formalism of the master equation in a lattice gas representation with unrestricted occupancy. In case the system is activated by a linear coupling to a heat bath the problem can be solved exactly and the stationary particle density follows the Bose distribution. The relation to spin-flip processes with a restricted occupancy is discussed. Different from those processes the relaxation time and the density fluctuation increase in the high-temperature limit. On a small scale the mutual interaction between the particles is relevant. While in case of a repulsive interaction the stationary solution becomes unstable against short wavelength fluctuations, an attractive interaction leads to an instability for long wavelength fluctuations. The system decays in domains, the size of which can be estimated as a function of temperature and interaction strength. The model is also appropriate to study the growth of open bacterial colonies under the influence of a competitive interaction between the species.