Related papers: A discrete and finite approach to past proper time
The present paper is devoted to the relativistic statistical theory, introduced in Phys. Rev. E {\bf 66} (2002) 056125 and Phys. Rev. E {\bf 72} (2005) 036108, predicting the particle distribution function $p(E)= \exp_{\kappa}…
The article examines the distribution of the power series of the function $ w(y) = \left( 1 + \sqrt{1 - y} \right)^{-\frac{1}{2}}. $ The distribution of the considered function into a power series is obtained $ \left(1 + \sqrt{1 -…
A new phase of temporal evolution of the one-dimensional self-gravitating system is numerically discovered. Fractal structure is dynamically created from non-fractal initial conditions. Implication to astrophysics and mathematical physics…
We study an unbiased, discrete time random walk on the nonnegative integers, with the origin absorbing. The process has a history-dependent step length: the walker takes steps of length v while in a region which has been visited before, and…
In this paper, we present a very accurate approximation for gamma function: \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi x}\left( \dfrac{x}{e}\right) ^{x}\left( x\sinh \frac{1}{x}\right) ^{x/2}\exp \left(…
We study analytically the order statistics of a time series generated by the successive positions of a symmetric random walk of n steps with step lengths of finite variance \sigma^2. We show that the statistics of the gap d_{k,n}=M_{k,n}…
Significant absorption of radiation is usually accompanied by refraction. This is not the case for $\gamma$ rays travelling cosmic distances. We show that the real and imaginary parts of the refraction index are indeed commensurable, as…
We study a discrete random walk on a one-dimensional finite lattice, where each state has different probabilities to move one step forward, backward, staying for a moment or being absorbed. We obtain expected number of arrivals and expected…
Let $\omega_0, \omega_1,\ldots, \omega_n$ be a full set of outcomes (letters, symbols) and let positive $p_i$, $i=0,\ldots,n$, be their probabilities ($\sum_{i=0}^n p_i=1$). Let us treat $\omega_0$ as a stop symbol; it can occur in…
Random walk has wide applications in many fields, such as machine learning, biology, physics, and chemistry. Random walk can be discrete or continuous in time and space. Asymmetric random walk could be described by drift-diffusion equation.…
We discuss some applications of the Mittag-Leffler function and related probability distributions in the theory of renewal processes and continuous time random walks. In particular we show the asymptotic (long time) equivalence of a generic…
Consider the extreme value of a Bernoulli random walk on the one-dimensional integer lattice, with reflection at 0, over a finite discrete time interval. Only the asymmetric (biased) case is discussed. Asymptotic mean/variance results are…
The Dirac equation can be modelled as a quantum walk, with the quantum walk being: discrete in time and space (i.e. a unitary evolution of the wave-function of a particle on a lattice); homogeneous (i.e. translation-invariant and…
We refine a remark of Steinerberger (2024), proving that for $\alpha \in \mathbb{R}$, there exists integers $1 \leq b_{1}, \ldots, b_{k} \leq n$ such that \[ \left\| \sum_{j=1}^k \sqrt{b_j} - \alpha \right\| = O(n^{-\gamma_k}), \] where…
Temporal correlations of time series or event sequences in natural and social phenomena have been characterized by power-law decaying autocorrelation functions with decaying exponent $\gamma$. Such temporal correlations can be understood in…
Continuous-time random walks are generalisations of random walks frequently used to account for the consistent observations that many molecules in living cells undergo anomalous diffusion, i.e. subdiffusion. Here, we describe the…
We investigate the random walk process in relativistic flow. In the relativistic flow, photon propagation is concentrated in the directions of the flow velocity due to relativistic beaming effect. We show that, in the pure scattering case,…
Given any $\gamma>0$ and for $\eta=\{\eta_v\}_{v\in \mathbb Z^2}$ denoting a sample of the two-dimensional discrete Gaussian free field on $\mathbb Z^2$ pinned at the origin, we consider the random walk on~$\mathbb Z^2$ among random…
Quantum walks with long-range steps $R^{-\gamma}$ ($R$ being the distance between sites) on a discrete line behave in similar ways for all $\gamma\geq2$. This is in contrast to classical random walks, which for $\gamma >3$ belong to a…
The dynamics of the avalanche width in the evolution model is described using a random walk picture. In this approach the critical exponents for avalanche distribution, $\tau$, and avalanche average time, $\gamma$, are found to be the same…