Stochastic processes with time delay are invaluable for modeling in science and engineering when finite signal transmission and processing speeds can not be neglected. However, they can seldom be treated with sufficient precision analytically if the corresponding stochastic delay differential equations (SDDEs) are nonlinear. This work presents a numerical algorithm for calculating the probability densities of processes described by nonlinear SDDEs. The algorithm is based on Markovian embedding and solves the problem by basic matrix operations. We validate it for a broad class of parameters using exactly solvable linear SDDEs and a cubic SDDE. Besides, we show how to apply the algorithm to calculate transition rates and first passage times for a Brownian particle diffusing in a time-delayed cusp potential.
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The connection between absorbing boundary conditions and hard walls is well established in the mathematical literature for a variety of stochastic models, including for instance the Brownian motion. In this paper we explore this duality for a different type of process which is of particular interest in physics and biology, namely the run-tumble-particle, a toy model of active particle. For a one-dimensional run-and-tumble particle subjected to an arbitrary external force, we provide a duality relation between the exit probability, i.e. the probability that the particle exits an interval from a given boundary before a certain time $t$, and the cumulative distribution of its position in the presence of hard walls at the same time $t$. We show this relation for a run-and-tumble particle in the stationary state by explicitly computing both quantities. At finite time, we provide a derivation using the Fokker-Planck equation. All the results are confirmed by numerical simulations.
The one-loop quantum corrections to the internal energy of some lattices due to the quantum fluctuations of the scalar field of phonons are studied. The band spectrum of the lattice is characterised in terms of the scattering data, allowing to compute the quantum vacuum interaction energy between nodes at zero temperature, as well as the total Helmholtz free energy, the entropy, and the Casimir pressure between nodes at finite non-zero temperature. Some examples of periodic potentials built from the repetition in one of the three spatial dimensions of the same punctual or compact supported potential are addressed: a stack of parallel plates constructed by positioning $\delta\delta'$-functions at the lattice nodes, and an ``upside-down tiled roof" of parallel two-dimensional P"oschl-Teller wells centred at the nodes. They will be called \textit{generalised Dirac comb} and \textit{P"oschl-Teller comb}, respectively. Positive one-loop quantum corrections to the entropy appear for both combs at non-zero temperatures. Moreover, the Casimir force between the lattice nodes is always repulsive for both chains when non-trivial temperatures are considered, implying that the primitive cell increases its size due to the quantum interaction of the phonon field.
Fractional Brownian motion (fBm) is an experimentally-relevant, non-Markovian Gaussian stochastic process with long-ranged correlations between the increments, parametrised by the so-called Hurst exponent H; depending on its value the process can be sub-diffusive (0 < H < 1⁄2), diffusive (H = 1⁄2) or super-diffusive (1⁄2 < H < 1). There exist three alternative equally often used definitions of fBm ‒ due to Lévy and due to Mandelbrot and van Ness (MvN), which differ by the interval on which the time variable is formally defined. Respectively, the covariance functions of these fBms have different functional forms. Moreover, the MvN fBms have stationary increments, while for the Lévy fBm this is not the case. One may therefore be tempted to conclude that these are, in fact, different processes which only accidentally bear the same name. Recently determined explicit path integral representations also appear to have very different functional forms, which only reinforces the latter conclusion. Here we develope a unifying equivalent path integral representation of all three fBms in terms of Riemann-Liouville fractional integrals, which links the fBms and proves that they indeed belong to the same family. We show that the action in such a representation involves the fractional integral of the same form and order (dependent on whether H < 1⁄2 or H > 1⁄2) for all three cases, and differs only by the integration limits.
In this work, we study a model in nonlinear electrodynamics in the presence of a CPT-even term that violates Lorentz symmetry. The Lorentz-breaking vector, in addition to the usual background magnetic field, produces interesting effects in the dispersion relations. The consequences on the vacuum refractive index and the group velocity are studied. Vacuum birefringence is discussed in the case the nonlinear electrodynamics is a Euler-Heisenberg model.
We study general "normally" (Gaussian) distributed random unitary
transformations. These distributions can be defined in terms of a diffusive random
walk in the respective group manifold. On the one hand, a Gaussian distribution
induces a unital quantum channel, which we will call "normal". On the other
hand, the diffusive random walk defines a unital quantum process, generated by a
Lindblad master equation. In the single qubit case, we show different distributions
may induce the same quantum channel.
In the case of two qubits, we study normal quantum channels, induced by
Gaussian distributions in SU(2)⊗SU(2). They provide an appropriate framework
for modeling quantum errors with classical correlations. In contrast to correlated
Pauli errors, for instance, they conserve their Markovianity, and they lead to very
different results in error correcting codes. This is illustrated with an application
to entanglement distillation.
We characterize the spectrum of the transition matrix for simple random walk on graphs consisting of a finite graph with a finite number of infinite Cayley trees attached. We show that there is a continuous spectrum identical to that for a Cayley tree and, in general, a non-empty pure point spectrum. We apply our results to studying continuous time quantum walk on these graphs. If the pure point spectrum is nonempty the walk is in general confined with a nonzero probability.
We introduce a Q-operator $\mathcal{Q}_z$ for the hyperbolic Calogero–Moser system as a one-parameter family of explicit integral operators. We establish the standard properties of a Q-operator, i.e. invariance of Hamiltonians, commutativity for different parameter values and that its eigenvalues satisfy an explicitly given first order ordinary difference equation in the parameter z.
A broad class of blocked or jammed configurations of particles on the one-dimensional lattice can be characterized in terms of local rules involving only the lengths of clusters of particles (occupied sites) and of holes (empty sites). Examples of physical relevance include the metastable states reached by kinetically constrained spin chains, the attractors of totally irreversible processes such as random sequential adsorption, and arrays of Rydberg atoms in the blockade regime. The configurational entropy of ensembles of such blocked configurations has been investigated recently by means of an approach inspired from the theory of stochastic renewal processes. This approach provides a valuable alternative to the more traditional transfer-matrix formalism. We show that the renewal approach is also an efficient tool to investigate a range of observables in uniform ensembles of blocked configurations, besides their configurational entropy.
The main emphasis is on their structure factor and correlation function.
Brownian Motion (BM) is the paradigmatic model of diffusion. Transcending from diffusion to anomalous diffusion, the prominent Gaussian generalizations of BM are Scaled BM (SBM) and Fractional BM (FBM). In the sub/super diffusivity regimes: SBM is characterized by aging/anti-aging, and FBM is characterized by anti-persistence/persistence. BM is neither aging/anti-aging, nor persistent/anti-persistent. Within the realm of diffusion, a recent Gaussian generalization of BM, Weird BM (WBM), was shown to display aging/anti-aging and persistence /anti-persistence. This paper introduces and explores the anomalous-diffusion counterpart of WBM, Beta BM (BBM), and shows that: the weird behaviors of WBM become even weirder when elevating to BBM. Indeed, BBM displays a rich assortment of anomalous behaviors, and an even richer assortment of combinations of anomalous behaviors. In particular, the BBM anomalous behaviors include aging/anti-aging and persistence/anti-persistence -- which BBM displays in both the sub and super diffusivity regimes. So, anomalous behaviors that are unattainable by the prominent models of SBM and FBM are well attainable by the BBM model.