The delayed uncoupled continuous-time random walks do not provide a model for the telegraph equation

It has been alleged in several papers that the so called delayed continuous-time random walks (DCTRWs) provide a model for the one-dimensional telegraph equation at microscopic level. This conclusion, being widespread now, is strange, since the telegraph equation describes phenomena with finite propagation speed, while the velocity of the motion of particles in the DCTRWs is infinite. In this paper we investigate how accurate are the approximations to the DCTRWs provided by the telegraph equation. We show that the diffusion equation, being the correct limit of the DCTRWs, gives better approximations in $L_2$ norm to the DCTRWs than the telegraph equation. We conclude therefore that, first, the DCTRWs do not provide any correct microscopic interpretation of the one-dimensional telegraph equation, and second, the kinetic (exact) model of the telegraph equation is different from the model based on the DCTRWs.Comment: 12 pages, 9 figure

Limit laws for distorted return time processes for infinite measure preserving transformations

We consider conservative ergodic measure preserving transformations on infinite measure spaces and investigate the asymptotic behaviour of distorted return time processes with respect to sets satisfying a type of Darling-Kac condition. We identify two critical cases for which we prove uniform distribution laws. For this we introduce the notion of uniformly returning sets and discuss some of their properties.Comment: 18 pages, 2 figure

Smoluchowski-Kramers approximation in the case of variable friction

We consider the small mass asymptotics (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The limit of the solution in the classical sense does not exist in this case. We study a modification of the Smoluchowski-Kramers approximation. Some applications of the Smoluchowski-Kramers approximation to problems with fast oscillating or discontinuous coefficients are considered.Comment: already publishe

Brownian motion of a charged particle driven internally by correlated noise

We give an exact solution to the generalized Langevin equation of motion of a charged Brownian particle in a uniform magnetic field that is driven internally by an exponentially-correlated stochastic force. A strong dissipation regime is described in which the ensemble-averaged fluctuations of the velocity exhibit transient oscillations that arise from memory effects. Also, we calculate generalized diffusion coefficients describing the transport of these particles and briefly discuss how they are affected by the magnetic field strength and correlation time. Our asymptotic results are extended to the general case of internal driving by correlated Gaussian stochastic forces with finite autocorrelation times.Comment: 10 pages, 4 figures with subfigures, RevTeX, v2: revise

Antipersistent binary time series

Completely antipersistent binary time series are sequences in which every time that an $N$-bit string $\mu$ appears, the sequence is continued with a different bit than at the last occurrence of $\mu$. This dynamics is phrased in terms of a walk on a DeBruijn graph, and properties of transients and cycles are studied. The predictability of the generated time series for an observer who sees a longer or shorter time window is investigated also for sequences that are not completely antipersistent.Comment: 6 pages, 6 figure

Statistical properties of spectral fluctuations for a quantum system with infinitely many components

Extending the idea formulated in Makino {\it{et al}}[Phys.Rev.E {\bf{67}},066205], that is based on the Berry--Robnik approach [M.V. Berry and M. Robnik, J. Phys. A {\bf{17}}, 2413], we investigate the statistical properties of a two-point spectral correlation for a classically integrable quantum system. The eigenenergy sequence of this system is regarded as a superposition of infinitely many independent components in the semiclassical limit. We derive the level number variance (LNV) in the limit of infinitely many components and discuss its deviations from Poisson statistics. The slope of the limiting LNV is found to be larger than that of Poisson statistics when the individual components have a certain accumulation. This property agrees with the result from the semiclassical periodic-orbit theory that is applied to a system with degenerate torus actions[D. Biswas, M.Azam,and S.V.Lawande, Phys. Rev. A {\bf 43}, 5694].Comment: 6 figures, 10 page

Detecting brute-force attacks on cryptocurrency wallets

Blockchain is a distributed ledger, which is protected against malicious modifications by means of cryptographic tools, e.g. digital signatures and hash functions. One of the most prominent applications of blockchains is cryptocurrencies, such as Bitcoin. In this work, we consider a particular attack on wallets for collecting assets in a cryptocurrency network based on brute-force search attacks. Using Bitcoin as an example, we demonstrate that if the attack is implemented successfully, a legitimate user is able to prove that fact of this attack with a high probability. We also consider two options for modification of existing cryptocurrency protocols for dealing with this type of attacks. First, we discuss a modification that requires introducing changes in the Bitcoin protocol and allows diminishing the motivation to attack wallets. Second, an alternative option is the construction of special smart-contracts, which reward the users for providing evidence of the brute-force attack. The execution of this smart-contract can work as an automatic alarm that the employed cryptographic mechanisms, and (particularly) hash functions, have an evident vulnerability.Comment: 10 pages, 2 figures; published versio

The optimal sink and the best source in a Markov chain

It is well known that the distributions of hitting times in Markov chains are quite irregular, unless the limit as time tends to infinity is considered. We show that nevertheless for a typical finite irreducible Markov chain and for nondegenerate initial distributions the tails of the distributions of the hitting times for the states of a Markov chain can be ordered, i.e., they do not overlap after a certain finite moment of time. If one considers instead each state of a Markov chain as a source rather than a sink then again the states can generically be ordered according to their efficiency. The mechanisms underlying these two orderings are essentially different though.Comment: 12 pages, 1 figur

First Passage Distributions in a Collective Model of Anomalous Diffusion with Tunable Exponent

We consider a model system in which anomalous diffusion is generated by superposition of underlying linear modes with a broad range of relaxation times. In the language of Gaussian polymers, our model corresponds to Rouse (Fourier) modes whose friction coefficients scale as wavenumber to the power $2-z$. A single (tagged) monomer then executes subdiffusion over a broad range of time scales, and its mean square displacement increases as $t^\alpha$ with $\alpha=1/z$. To demonstrate non-trivial aspects of the model, we numerically study the absorption of the tagged particle in one dimension near an absorbing boundary or in the interval between two such boundaries. We obtain absorption probability densities as a function of time, as well as the position-dependent distribution for unabsorbed particles, at several values of $\alpha$. Each of these properties has features characterized by exponents that depend on $\alpha$. Characteristic distributions found for different values of $\alpha$ have similar qualitative features, but are not simply related quantitatively. Comparison of the motion of translocation coordinate of a polymer moving through a pore in a membrane with the diffusing tagged monomer with identical $\alpha$ also reveals quantitative differences.Comment: LaTeX, 10 pages, 8 eps figure

Preservation of information in a prebiotic package model

The coexistence between different informational molecules has been the preferred mode to circumvent the limitation posed by imperfect replication on the amount of information stored by each of these molecules. Here we reexamine a classic package model in which distinct information carriers or templates are forced to coexist within vesicles, which in turn can proliferate freely through binary division. The combined dynamics of vesicles and templates is described by a multitype branching process which allows us to write equations for the average number of the different types of vesicles as well as for their extinction probabilities. The threshold phenomenon associated to the extinction of the vesicle population is studied quantitatively using finite-size scaling techniques. We conclude that the resultant coexistence is too frail in the presence of parasites and so confinement of templates in vesicles without an explicit mechanism of cooperation does not resolve the information crisis of prebiotic evolution.Comment: 9 pages, 8 figures, accepted version, to be published in PR