Convergence of resonances on thin branched quantum wave guides

We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family X_\eps of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances on X_\eps approximate those of the Laplacian with ``free'' boundary conditions on $X_0$, the skeleton graph of X_\eps.Comment: 48 pages, 1 figur

Boundary crossing Random Walks, clinical trials and multinomial sequential estimation

A sufficient condition for the uniqueness of multinomial sequential unbiased estimators is provided generalizing a classical result for binomial samples. Unbiased estimators are applied to infer the parameters of multidimensional or multinomial Random Walks which are observed until they reach a boundary. An application to clinical trials is presented

Statistical mechanics of spatial evolutionary games

We discuss the long-run behavior of stochastic dynamics of many interacting players in spatial evolutionary games. In particular, we investigate the effect of the number of players and the noise level on the stochastic stability of Nash equilibria. We discuss similarities and differences between systems of interacting players maximizing their individual payoffs and particles minimizing their interaction energy. We use concepts and techniques of statistical mechanics to study game-theoretic models. In order to obtain results in the case of the so-called potential games, we analyze the thermodynamic limit of the appropriate models of interacting particles.Comment: 19 pages, to appear in J. Phys.

On population extinction risk in the aftermath of a catastrophic event

We investigate how a catastrophic event (modeled as a temporary fall of the reproduction rate) increases the extinction probability of an isolated self-regulated stochastic population. Using a variant of the Verhulst logistic model as an example, we combine the probability generating function technique with an eikonal approximation to evaluate the exponentially large increase in the extinction probability caused by the catastrophe. This quantity is given by the eikonal action computed over "the optimal path" (instanton) of an effective classical Hamiltonian system with a time-dependent Hamiltonian. For a general catastrophe the eikonal equations can be solved numerically. For simple models of catastrophic events analytic solutions can be obtained. One such solution becomes quite simple close to the bifurcation point of the Verhulst model. The eikonal results for the increase in the extinction probability caused by a catastrophe agree well with numerical solutions of the master equation.Comment: 11 pages, 11 figure

Heat bounds and the blowtorch theorem

We study driven systems with possible population inversion and we give optimal bounds on the relative occupations in terms of released heat. A precise meaning to Landauer's blowtorch theorem (1975) is obtained stating that nonequilibrium occupations are essentially modified by kinetic effects. Towards very low temperatures we apply a Freidlin-Wentzel type analysis for continuous time Markov jump processes. It leads to a definition of dominant states in terms of both heat and escape rates.Comment: 11 pages; v2: minor changes, 1 reference adde

Non-Markovian Random Walks and Non-Linear Reactions: Subdiffusion and Propagating Fronts

We propose a reaction-transport model for CTRW with non-linear reactions and non-exponential waiting time distributions. We derive non-linear evolution equation for mesoscopic density of particles. We apply this equation to the problem of fronts propagation into unstable state of reaction-transport systems with anomalous diffusion. We have found an explicit expression for the speed of propagating front in the case of subdiffusion transport.Comment: 7 page

Path integral approach to random motion with nonlinear friction

Using a path integral approach, we derive an analytical solution of a nonlinear and singular Langevin equation, which has been introduced previously by P.-G. de Gennes as a simple phenomenological model for the stick-slip motion of a solid object on a vibrating horizontal surface. We show that the optimal (or most probable) paths of this model can be divided into two classes of paths, which correspond physically to a sliding or slip motion, where the object moves with a non-zero velocity over the underlying surface, and a stick-slip motion, where the object is stuck to the surface for a finite time. These two kinds of basic motions underlie the behavior of many more complicated systems with solid/solid friction and appear naturally in de Gennes' model in the path integral framework.Comment: 18 pages, 3 figure

Scattering solutions in a network of thin fibers: small diameter asymptotics

Small diameter asymptotics is obtained for scattering solutions in a network of thin fibers. The asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph. We calculate the Lagrangian gluing conditions at vertices for the problems on the limiting graph. If the frequency of the incident wave is above the bottom of the absolutely continuous spectrum, the gluing conditions are formulated in terms of the scattering data for each individual junction of the network

Fluctuations of Current in Non-Stationary Diffusive Lattice Gases

We employ the macroscopic fluctuation theory to study fluctuations of integrated current in one-dimensional lattice gases with a step-like initial density profile. We analytically determine the variance of the current fluctuations for a class of diffusive processes with a density-independent diffusion coefficient, but otherwise arbitrary. Our calculations rely on a perturbation theory around the noiseless hydrodynamic solution. We consider both quenched and annealed types of averaging (the initial condition is allowed to fluctuate in the latter situation). The general results for the variance are specialized to a few interesting models including the symmetric exclusion process and the Kipnis-Marchioro-Presutti model. We also probe large deviations of the current for the symmetric exclusion process. This is done by numerically solving the governing equations of the macroscopic fluctuation theory using an efficient iteration algorithm.Comment: Slightly extended version. 12 pages, 6 figure

Numerical simulations versus theoretical predictions for a non-Gaussian noise induced escape problem in application to full counting statistics

A theoretical approach for characterizing the influence of asymmetry of noise distribution on the escape rate of a multistable system is presented. This was carried out via the estimation of an action, which is defined as an exponential factor in the escape rate, and discussed in the context of full counting statistics paradigm. The approach takes into account all cumulants of the noise distribution and demonstrates an excellent agreement with the results of numerical simulations. An approximation of the third-order cumulant was shown to have limitations on the range of dynamic stochastic system parameters. The applicability of the theoretical approaches developed so far is discussed for an adequate characterization of the escape rate measured in experiments