arXiv subjects

E. Pechersky

Publications and source records attributed to E. Pechersky.

10 recordsLinked to original sources

arXiv2019

Hawking-Penrose Black Hole Model. Large Emission Regime

In this paper, we propose a stochastic version of the Hawking-Penrose black hole model. We describe the dynamics of the stochastic model as a continuous-time Markov jump process of quanta out and in the black hole. The average of the random process satisfies the deterministic picture accepted in the physical literature. Assuming that the number of quanta is finite the proposed Markov process consists of two components: the number of the quanta in the black hole and the amount of the quanta outside. The stochastic representation allows us to apply large deviation theory to study the asymptotics of probabilities of rare events when the number of quanta grows to infinity. The theory provides explicitly the rate functional for the process. Its infimum over the set of all trajectories leading to a large emission event is attained on the most probable trajectory. This trajectory is a solution of a highly nonlinear Hamiltonian system of equations. Under the condition of the stationarity of the fraction of quanta in the black hole, we found the most probable trajectory corresponding to a large emission event.

math.PR↗

arXiv2018

Large Emission Regime in Mean Field Luminescence

We study a class of random processes on $N$ particles which can be interpreted as stochastic model of luminescence. Each particle can stay in one of two states: Excited state or ground state. Any particle at ground state is excited with a constant rate (pumping). The number of excited particles decreases by means of photon emission through interactions of the particles. We analyse the rare event of flashes, i.e., the emission of a very large number of photons $B$ during a fixed time interval $T$. We employ the theory of large deviations to provide the asymptotics of the probability of such event when the total number of particles $N$ tends to infinity. This theory gives us also the optimal trajectory of scaled process corresponding to this event. The stationary regime of this process we call the large emission regime. In several cases we prove that in the large emission regime a share of excited particles in a system is stable under the changes of the pumping and emission rates.

math.PR↗

arXiv2018

Stochastic Dynamics of Einstein Matter-Radiation Model with Spikes

Motivated by the Einstein classical description of the matter-radiation dynamics we revise a dynamical system producing spikes of the photon emission. Then we study the corresponding stochastic model, which takes into account the randomness of spontaneous and stimulated atomic transitions supporting by a pumping. Our model reduces to Markovian density dependent processes for the inversion coefficient and the photon specific value driven by small but fast jumps. We analyse the model in three limits: the many-component, the mean-field, and the one-component. In the last case it is a positive recurrent Markov chain with sound spikes.

math.PR↗

arXiv2018

Large Fluctuations in two-level systems with stimulated emission

We consider a system of $N$ identical independent Markov processes, each taking values 0 or 1. The system describes a stochastic dynamics of an ensemble of two-level atoms. The atoms are exposed to a photon flux. Under the photon flux action, every atom changes its state with some intensities either from its ground state (state 0) to the excited state (state 1) or from the excited state to the ground state (the stimulated emission). The atom can also change the state from the excited to the ground one spontaneously. We study rare events when the big cumulative emission occurs on the fixed time interval $[0,T]$. To this end we apply the large deviation theory which allows one to make asymptotic analysis as $N\to\infty$.

math.PR↗

arXiv2017

Large Fluctuations of Radiation in Stochastically Activated Two-Level Systems

We study the large fluctuations of emitted radiations in the system of $N$ non-interacting two-level atoms. Two methods are used to calculate the probability of the large fluctuations and the time dependence of the excitation and emission. The first method is based on the large deviation principle for Markov processes. The second one uses an analogue of the quantum formalism for classical probability problems. Particularly we prove that in a large fluctuation limit approximately a half of the atoms are excited. This fact is independent on the fraction of the excited atoms in the stationary state.

cond-mat.stat-mech↗

arXiv2015

Random walks in a queueing network environment

We propose a class of models of random walks in a random environment where an exact solution can be given for a stationary distribution. The tool is the detailed balance equations.

math.PR↗

arXiv2014

Dynamics of Tectonic Plates

We suggest a model that describes a mutual dynamic of tectonic plates. The dynamic is a sort of stick-slip one which is modeled by a Markov random process. The process defines a microlevel of the dynamic. A macrolevel is obtained by a scaling limit which leads to a system of integro-differential equations which determines a kind of mean field systems. Conditions when Gutenberg-Richter empirical law are presented on the mean field level. These conditions are rather universal and do not depend on features of resistant forces.

math.PR↗

arXiv2013

Large deviations for excursions of non-homogeneous Markov processes

In this paper, the large deviations on trajectory level for ergodic Markov processes are studied. These processes take values in the non-negative quadrant of the two dimension lattice and are concentrated on step-wise functions. The rates of jumps towards the axes (jump down) depend on the position of the process -- the higher the position, the greater the rate. The rates of jumps going in the same direction as the axes (jump up) are constants. Therefore the processes are ergodic. The large deviations are studied under equal scalings of both space and time. The scaled versions of the processes converge to 0. The main result is that the probabilities of long excursions out of 0 tend to 0 exponentially fast with an exponent proportional to the square of the scaling parameter. A proportionality coefficient is an integral of a linear combination of path components. A rate function of the large deviation principle is calculated for continuous functions only.

math.PR↗

arXiv2009

Percolation properties of non-ideal gas

We estimate locations of the regions of the percolation and of the non-percolation in the plane $(λ,β)$: the Poisson rate -- the inverse temperature, for interacted particle systems in finite dimension Euclidean spaces. Our results about the percolation and about the non-percolation are obtained under different assumptions. The intersection of two groups of the assumptions reduces the results to two dimension Euclidean space, $\R^2$, and to a potential function of the interactions having a hard core. The technics for the percolation proof is based on a contour method which is applied to a discretization of the Euclidean space. The technics for the non-percolation proof is based on the coupling of the Gibbs field with a branching process.

math-ph↗

arXiv2007

Markov Process of Muscle Motors

We study a Markov random process describing a muscle molecular motor behavior. Every motor is either bound up with a thin filament or unbound. In the bound state the motor creates a force proportional to its displacement from the neutral position. In both states the motor spend an exponential time depending on the state. The thin filament moves at its velocity proportional to average of all displacements of all motors. We assume that the time which a motor stays at the bound state does not depend on its displacement. Then one can find an exact solution of a non-linear equation appearing in the limit of infinite number of the motors.

math-ph↗