Mean first passage time (MFPT)

We theoretically study the trapping time distribution and the efficiency of the excitation energy transport in dendritic systems. Trapping of excitations, created at the periphery of the dendrimer, on a trap located at its core, is used as a probe of the efficiency of the energy transport across the dendrimer. The transport process is treated as incoherent hopping of excitations between nearest-neighbor dendrimer units and is described using a rate equation. We account for radiative and nonradiative decay of the excitations while diffusing across the dendrimer. We derive exact expressions for the Laplace transform of the trapping time distribution and the efficiency of trapping, and analyze those for various realizations of the energy bias, number of dendrimer generations, and relative rates for decay and hopping. We show that the essential parameter that governs the trapping efficiency is the product of the on-site excitation decay rate and the trapping time (mean first passage tim...

We develop techniques to obtain rigorous bounds on the behaviour of random walks on combs. Using these bounds we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but finite length. We also calculate exactly the spectral dimension of some fixed non-translationally invariant combs. We relate the spectral dimension to the critical exponent of the mass of the two-point function for random walks on random combs, and compute mean displacements as a function of walk duration. We prove that the mean first passage time is generally infinite for combs with anomalous spectral dimension.

We consider a LÃ evy yer of order that starts from a point x0 on an interval [0; L] with absorbing boundaries. We ÿnd a closed-form expression for an arbitrary average quantity, characterizing the trajectory of the yer, such as mean ÿrst passage time, average total path length, probability to be absorbed by one of the boundaries. Using fractional di erential equations with a Riesz kernel, we ÿnd exact analytical expressions for these quantities in the continuous limit. We ÿnd numerically the eigenfunctions and the eigenvalues of these equations. We study how the results of Monte-Carlo simulations of the LÃ evy ights with di erent ight length distributions converge to the continuous approximations. We show that if x0 is placed in the vicinity of absorbing boundaries, the average total path length has a minimum near = 1, corresponding to the Cauchy distribution. We discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere.

The problems of escape from metastable state in randomly flipping potential and of diffusion in fast fluctuating periodic potentials are considered. For the overdamped Brownian particle moving in a piecewise linear dichotomously fluctuating metastable potential we obtain the mean first-passage time (MFPT) as a function of the potential parameters, the noise intensity and the mean rate of switchings of the dichotomous noise. We find noise enhanced stability (NES) phenomenon in the system investigated and the parameter region of the fluctuating potential where the effect can be observed. For the diffusion of the overdamped Brownian particle in a fast fluctuating symmetric periodic potential we obtain that the effective diffusion coefficient depends on the mean first-passage time, as discovered for fixed periodic potential. The effective diffusion coefficients for sawtooth, sinusoidal and piecewise parabolic potentials are calculated in closed analytical form.

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A novel approach to evaluate characteristic times of general relaxation processes is presented. The decay of the unstable state of a prototype model is studied. Our results are compared with thosegiven by mean first passage time techniques. The advantages and possibilities of the new approach are briefly discussed.

We use stochastic dynamics to develop the patterned attractor of a non-local extended system. This is done analytically using the stochastic path perturbation approach scheme, where a theory of perturbation in the small noise parameter is introduced to analyze the random escape of the stochastic field from the unstable state. Emphasis is placed on the specific mode selection that these types of systems exhibit. Concerning the stochastic propagation of the front we have carried out Monte Carlo simulations which coincide with our theoretical predictions.

The two-dimensional backward Fokker-Planck equation is used to calculate the mean first-passage times (MFPTs) of the magnetic moment of a nanoparticle driven by a rotating magnetic field. It is shown that a magnetic field that is rapidly rotating in the plane perpendicular to the easy axis of the nanoparticle governs the MFPTs just in the same way as a static magnetic field that is applied along the easy axis. Within this framework, the features of the magnetic relaxation and net magnetization of systems composed of ferromagnetic nanoparticles arising from the action of the rotating field are revealed.

The transition rate of a non-Markovian Brownian particle in a double well potential is determined analytically by means of asymptotic methods and compared with both current theories and numerical simulations by Straub, Borkovec, and Berne [J. Chem. Phys. 83, 3172 (1985)]. We obtain good agreement with these simulations. The ranges of validity for the different current theories which we find do, however, not exhaust the complete parameter range. In particular, for large static friction we identify a region of bath correlation times in which the rate differs grossly from the value predicted by either Grote–Hynes theory or non-Markovian energy diffusion. Additionally, corrections to the Grote–Hynes rate are determined and an analytical expression for the non-Markovian energy diffusion rate is obtained.

The Ehrenfest urn process, also known as the dogs and fleas model, is realistically simulated by molecular dynamics of the Lennard-Jones fluid. The key variable is ∆z, i.e. the absolute value of the difference between the number of particles in one half of the simulation box and in the other half. This is a pure-jump stochastic process induced, under coarse graining, by the deterministic time evolution of the atomic coordinates. We discuss the Markov hypothesis by analyzing the statistical properties of the jumps and of the waiting times between the jumps. In the limit of a vanishing integration time-step, the distribution of waiting times becomes closer to an exponential and, therefore, the continuous-time jump stochastic process is Markovian. The random variable ∆z behaves as a Markov chain and, in the gas phase, the observed transition probabilities follow the predictions of the Ehrenfest theory.

The dynamics of the standard integrate-fire model and a simpler model (that reproduces the important features of the integrate-fire model under certain conditions) of neural dynamics are studied in the presence of a deterministic external driving force, taken to be time-periodic, and white background noise. Both models possess resonant phenomena in the first passage probability distribution and mean first passage time, arising from the interplay of characteristic time scales in the system.

Stationary distributions of perturbed finite irreducible discrete time Markov chains are intimately connected with the behaviour of associated mean first passage times. This interconnection is explored through the use of generalized matrix inverses. Some interesting qualitative results regarding the nature of the relative and absolute changes to the stationary probabilities are obtained together with some improved bounds.

We study a thermochemical gaseous system in the vicinity of the bifurcation related to the emergence of bistability. Corrections to the standard deterministic dynamics induced by the perturbation of the particle velocity distribution are obtained from the solution of the Boltzmann equation. Using these results, analytical expressions including the nonequilibrium effects are derived for the ignition time in the explosive regime and mean first passage time in the bistable regime. It is demonstrated that a departure from partial equilibrium can shift the bifurcation point. The system which was bistable according to the standard deterministic approach, can become monostable and explosive in the presence of nonequilibrium effects. Even when the system remains in the bistable regime, the mean first passage time can be changed by several orders of magnitude. In the monostable domain, the ignition time can be about ten times smaller than the unperturbed value. These analytical predictions agree well with the results of the microscopic simulations of the dilute gas system.

We consider the e ect of subdividing the potential barrier along the reaction coordinate on Kramer's escape rate for a model potential. Using the known supersymmetric potential approach, we show the existence of an optimal number of subdivisions that maximizes the rate. We cast the problem as a mean ÿrst passage time problem of a biased random walker and obtain equivalent results. We brie y summarize the results of our investigation on the increase in the escape rate by placing a blowtorch in the unstable part of one of the potential wells.

We study a Langevin equation derived from the Michaelis-Menten (MM) phenomenological scheme for catalysis accompanying a spontaneous replication of molecules, which may serve as a simple model of cell-mediated immune surveillance against cancer. We examine how two different and statistically independent sources of noise-dichotomous multiplicative noise and additive Gaussian white noise-influence the population's extinction time. This quantity is identified as the mean first passage time of the system across the zero population state. We observe the effects of resonant activation (RA) and noise-enhanced stability (NES) and we report the evidence for competitive co-occurrence of both phenomena in a given regime of noise parameters. We discuss the statistics of first passage times in this regime and the role of different pseudo-potential profiles on the RA and NES phenomena. The RA/NES coexistence region brings an interesting interpretation for the growth kinetics of cancer cells population, as the NES effect enhancing the stability of the tumoral state becomes strongly reduced by the RA phenomenon.

We analyze the colored-noise problem from the point of view of consistent Markovian approximations. We extend the "uni6ed colored-noise approximation" of P. Hanggi et al. through its interpretation as an interpolation procedure between the zero correlation time limit (white noise) and the infinite correlation time limit. We consider other interpolating functions, obtaining stationary probability distributions and the mean first passage time (associated with the lowest nouzero eigenvalue) and compare with exact numerical results. The potential of this scheme to represent adequately the results of the colored-noise problem through a convenient choice of the interpolating function is also discussed.

We present an analytical method of calculating the mean first-passage times ͑MFPTs͒ for the magnetic moment of a uniaxial nanoparticle which is driven by a rapidly rotating, circularly polarized magnetic field and interacts with a heat bath. The method is based on the solution of the equation for the MFPT derived from the two-dimensional backward Fokker-Planck equation in the rotating frame. We solve these equations in the high-frequency limit and perform precise, numerical simulations which verify the analytical findings. The results are used for the description of the rates of escape from the metastable domains, which in turn determine the magnetic relaxation dynamics. A main finding is that the presence of a rotating field can cause a drastic decrease of the relaxation time and a strong magnetization of the nanoparticle system. The resulting stationary magnetization along the direction of the easy axis is compared with the mean magnetization following from the stationary solution of the Fokker-Planck equation.

Master equation approach is used to study the influence of fluctuations on the dynamics of a model thermochemical system. For appropriate values of parameters, the deterministic description of the system gives the subcritical or supercritical Hopf bifurcations. For small systems ͑containing 100 000 particles͒ close to the supercritical Hopf bifurcation, the stochastic trajectories obtained from numerical simulations do not allow to distinguish between damped oscillations around a stable focus and sustained oscillations around a small stable limit cycle. This uncertainty disappears if the number of particles in the system is increased ͑up to 1 000 000͒. Close to subcritical Hopf bifurcation the stochastic trajectory of the system jumps from the basin of attraction of a stable focus to the basin of attraction of a stable limit cycle. In this case the time dependencies of temperature and concentration of reactant in the system are apparently similar to intermittent chaotic oscillations. The mean first passage time for the transitions from the stable focus to the stable limit cycle show the characteristic exponential dependence on the number of particles. This passage time depends very strongly on the bifurcation parameter ͑reaction heat͒, which determines the distance between the stable focus and an unstable limit cycle.

The transition rate of a non-Markovian Brownian particle in a double well potential is determined analytically by means of asymptotic methods and compared with both current theories and numerical simulations by Straub, Borkovec, and Berne [J. Chem. Phys. 83, 3172 (1985)]. We obtain good agreement with these simulations. The ranges of validity for the different current theories which we find do, however, not exhaust the complete parameter range. In particular, for large static friction we identify a region of bath correlation times in which the rate differs grossly from the value predicted by either Grote–Hynes theory or non-Markovian energy diffusion. Additionally, corrections to the Grote–Hynes rate are determined and an analytical expression for the non-Markovian energy diffusion rate is obtained.

The study of the noise induced effects on the dynamics of a chain molecule crossing a potential barrier, in the presence of a metastable state, is presented. A two-dimensional stochastic version of the Rouse model for a flexible polymer has been adopted to mimic the molecular dynamics and to take into account the interactions between adjacent monomers. We obtain a nonmonotonic behavior of the mean first passage time and its standard deviation, of the polymer center of inertia, with the noise intensity. These findings reveal a noise induced effect on the mean crossing time. The role of the polymer length is also investigated.

Let m ij be the mean first passage time from state i to state j in an n-state ergodic homogeneous Markov chain with transition matrix T. Let G be the weighted digraph without loops whose vertex set coincides with the set of states of the Markov chain and arc weights are equal to the corresponding transition probabilities. We show that m ij = f ij /q j , if i = j, 1/q j , if i = j, where f ij is the total weight of 2-tree spanning converging forests in G that have one tree containing i and the other tree converging to j, q j is the total weight of spanning trees converging to j in G, andq j = q j / n k=1 q k. The result is illustrated by an example.

We present exact, analytic results for the mean time to trapping of a random walker on the class of deterministic Sierpinski graphs embedded in d 2 Euclidean dimensions, when both nearest-neighbor (NN) and next-nearestneighbor (NNN) jumps are included. Mean first-passage times are shown to be modified significantly as a consequence of the fact that NNN transitions connect fractals of two consecutive generations.

Real networks can be classified into two categories: fractal networks and non-fractal networks. Here we introduce a unifying model for the two types of networks. Our model network is governed by a parameter q. We obtain the topological properties of the network including the degree distribution, average path length, diameter, fractal dimensions, and betweenness centrality distribution, which are controlled by parameter q. Interestingly, we show that by adjusting q, the networks undergo a transition from fractal to non-fractal scalings, and exhibit a crossover from 'large' to small worlds at the same time. Our research may shed some light on understanding the evolution and relationships of fractal and non-fractal networks.

PACS 05.40.Fb-Random walks and Levy flights PACS 89.75.Hc-Networks and genealogical trees PACS 05.60.Cd-Classical transport Abstract.-Explicit determination of the mean first-passage time (MFPT) for trapping problem on complex media is a theoretical challenge. In this paper, we study random walks on the Apollonian network with a trap fixed at a given hub node (i.e. node with the highest degree), which are simultaneously scale-free and small-world. We obtain the precise analytic expression for the MFPT that is confirmed by direct numerical calculations. In the large system size limit, the MFPT approximately grows as a power-law function of the number of nodes, with the exponent much less than 1, which is significantly different from the scaling for some regular networks or fractals, such as regular lattices, Sierpinski fractals, T-graph, and complete graphs. The Apollonian network is the most efficient configuration for transport by diffusion among all previously studied structure.

We develop techniques to obtain rigorous bounds on the behaviour of random walks on combs. Using these bounds we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but finite length. We also calculate exactly the spectral dimension of some fixed non-translationally invariant combs. We relate the spectral dimension to the critical exponent of the mass of the two-point function for random walks on random combs, and compute mean displacements as a function of walk duration. We prove that the mean first passage time is generally infinite for combs with anomalous spectral dimension.

The mean first passage time of a Brownian particle from an initial unstable state in a metastable system with damping is investigated. The system is analyzed in the low to high damping regime, and the role played by the damping parameter is studied. We observe the noise enhanced stability effect for all the initial unstable states under study and for all values of the damping parameter ␥ investigated. The curves show a behavior of the mean first passage time vs ␥ very close to that observed for an overdamped particle in the presence of colored noise as a function of the correlation time.

The passage of ions through membrane channels plays an important role in many fields of biology. An earlier paper [M. Bogun˜a´, A.M. Berezhkovskii, G.H. Weiss, Phys. Rev. E 62 (2000) 3250] developed a toy model for statistical properties of the occupancy of a single site by different numbers of lattice random walkers chosen from an infinite set. It was assumed there that the residence time in one sojourn at the origin differed from the residence time of points elsewhere. In this paper we derive some properties of the corresponding first-passage time to the occupancy of the special site by kð41Þ random walkers in one or two dimensions. Results of our study were obtained from an extensive set of simulations.

Autonomous Underwater Vehicles (AUVs) operating near the surface are subject to significant disturbances due to wave motion. In order to counteract the oscillatory effect of the waves in the actuators system, it is critical to remove it from the feedback loop. In this paper, a linear passive observer is designed to estimate shallow water wave motion and low frecuency motion by using depth measurement. A set of simulation is presented to show the performance of the proposed observer for depth control of an AUV by using both simulated and real data. The results based on collected data confirm the suitable filtering and the navigation response expected, reducing control actions and thus vibrations of the rudders

In this work we study the noise induced effects on the dynamics of short polymers crossing a potential barrier, in the presence of a metastable state. An improved version of the Rouse model for a flexible polymer has been adopted to mimic the molecular dynamics by taking into account both the interactions between adjacent monomers and introducing a Lennard-Jones potential between all beads. A bending recoil torque has also been included in our model. The polymer dynamics is simulated in a two-dimensional domain by numerically solving the Langevin equations of motion with a Gaussian uncorrelated noise. We find a nonmonotonic behaviour of the mean first passage time and the most probable translocation time, of the polymer centre of inertia, as a function of the polymer length at low noise intensity. We show how thermal fluctuations influence the motion of short polymers, by inducing two different regimes of translocation in the molecule transport dynamics. In this context, the role played by the length of the molecule in the translocation time is investigated.

The one-dimensional overdamped Brownian motion in a symmetric periodic potential modulated by external time-reversible noise is analyzed. The calculation of the effective diffusion coefficient is reduced to the mean first passage time problem. We derive general equations to calculate the effective diffusion coefficient of Brownian particles moving in arbitrary supersymmetric potential modulated: (i) by external white Gaussian noise and (ii) by Markovian dichotomous noise. For both cases the exact expressions for the effective diffusion coefficient are derived. We obtain acceleration of diffusion in comparison with the free diffusion case for fast fluctuating potentials with arbitrary profile and for sawtooth potential in case (ii). In this case the parameter region where this effect can be observed is given. We obtain also a finite net diffusion in the absence of thermal noise. For rectangular potential the diffusion slows down in comparison with the case when particles diffuse freely, for all parameters of noise and of potential.

We calculate the activation rates of metastable states of general one-dimensional Markov jump processes by calculating mean first-passage times. We employ methods of singular perturbation theory to derive expressions for these rates, utilizing the full Kramers-Moyal expansions for the forward and backward operators in the master equation. We discuss various boundary conditions for the first-passage-tine problem, and present some examples. We also discuss the validity of various diffusion approximations to the master equation, and their limitations.

The diffusion dynamics of a Yukawa fluid confined in an attractive planar slit pore was studied by solving the Smoluchowski/Hypernnetedchain equation for the mean first passage times. We show how the time it takes for a particle to get adsorbed at the wall surface is modulated by the effective potential barrier originated by the fluid local density profile. The effect of wetting layering on the apparent constant for the transverse diffusion towards an attractive pore wall is evaluated.

Cellular transport machinery, such as channels and pumps, is working against the background of unassisted material transport through membranes. The permeation of a blocked tryptophan through a 1,2-Dioleoyl-sn-glycero-3-phosphocholine (DOPC) membrane is investigated to probe unassisted or physical transport. The transport rate is measured experimentally and modeled computationally. The time scale measured by Parallel Artificial Membrane Permeation Assay (PAMPA) experiments is ~8 h. Simulations with the Milestoning algorithm suggest Mean First Passage Time (MFPT) of ~4 h and the presence of a large barrier at the center of the bilayer. A similar calculation with the solubility-diffusion model yields MFPT of ~15 min. This permeation rate is nine orders of magnitude slower than the permeation rate of only a tryptophan side chain (computed by us and others). This difference suggests critical dependence of transport time on permeant size and hydrophilicity. Analysis of the simulation results suggests that the permeant partially preserves hydrogen bonding of the peptide backbone to water and lipid molecules even when it is moving closer to the bilayer center. As a consequence, defects of the membrane structure are developed to assist permeation.

We consider the escape of a planar diffusion process from the domain of attraction Ω of a stable focus of the drift in the limit of small diffusion. The boundary ∂Ω of Ω is an unstable limit cycle of the drift, and the focus is close to the limit cycle. A new phenomenon of oscillatory decay of the peaks of the survival probability of the process in Ω emerges for a specific distance of the focus to the boundary which depends on the amplitude of the diffusion coefficient. We show that the phenomenon is due to the complex-valued second eigenvalue λ 2 (Ω) of the Fokker-Planck operator with Dirichlet boundary conditions. The dominant oscillation frequency, 1/Im{λ 2 (Ω)}, is independent of the relative noise strength. The density of exit points on ∂Ω is concentrated in a small arc of ∂Ω closest to the focus. When the focus approaches the boundary, the principal eigenvalue, the reciprocal of which is the mean escape time, does not necessarily decay exponentially as the amplitude of the noise strength decays. The oscillatory escape is manifested in a mathematical model of neural networks with synaptic depression. Oscillation peaks are identified in the density of the time the network spends in a specific state. This observation explains the oscillations of stochastic trajectories around a focus prior to escape and also the non-Poissonian escape times. This phenomenon has been observed and reported in experiments and simulations of neural networks.

We study the actual and important problem of Mean First Passage Time (MFPT) for diffusion in fluctuating media. We exploit van Kampen's technique of composite stochastic processes, obtaining analytical expressions for the MFPT for a general system, and focus on the two state case where the transitions between the states are modelled introducing both Markovian and non-Markovian processes. The comparison between the analytical and simulations results show an excellent agreement.

We study the dynamics of a system of particles diffusing in a fluctuating medium: a lattice which can be in two states, with transitions among them induced by a combination of a periodic deterministic and a stochastic process. We characterize the problem studying the mean first passage time (MFPT), the probability of return to the origin (P s 0), and the number of visited lattice sites. We observe a double resonant activation-like phenomenon when we plot the MFPT and P s 0 as functions of the intensity of the transition rate stochastic component.

An analytical soluble model based on a Continuous Time Random Walk (CTRW) scheme for the adsorption-desorption processes at interfaces, called bulk-mediated surface diffusion, is presented. The time evolution of the effective probability distribution width on the surface is calculated and analyzed within an anomalous diffusion framework. The asymptotic behavior for large times shows a sub-diffusive regime for the effective surface diffusion but, depending on the observed range of time, other regimes may be obtained. Montecarlo simulations show excellent agreement with analytical results. As an important byproduct of the indicated approach, we present the evaluation of the time for the first visit to the surface.

Based on a coherent state representation of noise operator and an ensemble averaging procedure we have recently developed [Phys. Rev. E 65, 021109 (2002); ibid. 051106 (2002)] a scheme for quantum Brownian motion to derive the equations for time evolution of true probability distribution functions in c-number phase space. We extend the treatment to develop a numerical method for generation of c-number noise with arbitrary correlation and strength at any temperature, along with the solution of the associated generalized quantum Langevin equation. The method is illustrated with the help of a calculation of quantum mean first passage time in a cubic potential to demonstrate quantum Kramers turnover and quantum Arrhenius plot.

We calculate the activation rates of metastable states of general one-dimensional Markov jump processes by calculating mean first-passage times. We employ methods of singular perturbation theory to derive expressions for these rates, utilizing the full Kramers-Moyal expansions for the forward and backward operators in the master equation. We discuss various boundary conditions for the first-passage-tine problem, and present some examples. We also discuss the validity of various diffusion approximations to the master equation, and their limitations.

The main scope of this paper is to give some explicit classes of examples of L 1-optimal couplings. Optimal transportation w.r.t. the Kantorovich metric 1 (resp. the Wasserstein metric W 1) between two absolutely continuous measures is known since the basic papers of Kantorovich and Rubinstein (1957) and Sudakov (1979) to occur on rays induced by a decomposition of the basic space (and more generally to higher dimensional decompositions in the case of general measures) induced by the corresponding dual potentials. Several papers have given this kind of structural result and established existence and uniqueness of solutions in varying generality. Since the dual problems pose typically too strong challenges to be solved in explicit form, these structural results have so far been applied for the solution of few particular instances. First, we give a self-contained review of some basic optimal coupling results and we propose and investigate in particular some basic principles for the construction of L 1-optimal couplings given by a reduction principle and some usable forms of the decomposition method. This reduction principle, together with symmetry properties of the reduced measures, gives a hint to the decomposition of the space into sectors and via the non crossing property of optimal transport leads to the choice of transportation rays. The optimality of the induced transports is then a consequence of the characterization results of optimal couplings. Then, we apply these principles to determine in explicit form L 1-optimal couplings for several classes of examples of elliptical distributions. In particular, we give for the first time a general construction of L 1-optimal couplings between two bivariate Gaussian distributions. We also discuss optimality of special constructions like shifts and scalings, and provide an extended class of dual functionals allowing for the closed-form computation of the 1-metric or of accurate lower bounds of it in a variety of examples.

The main object of this work is the vehicular traffic simulation based on macroscopic as well as microscopic approaches. First of them is one-dimensional and uses the quasi-gas-dynamic system of equations. The second one simulates the multilane movement using the cellular automata theory. Different types of driver behavior are employed in the model. The self-dependent problem is to apply the created software to the high-performance computer systems to simulate the traffic flow on the real city network. K E Y W O R D S cellular automata, mathematical modeling of traffic flows, parallel computing 1 INTRODUCTION There are two main approaches used for traffic flow modeling nowadays: macroscopic approach and microscopic approach. 1 The macroscopic approach is useful only when the macroscopic values, such as average velocity and density of a traffic flow, are of interest. Based on these values, depending on the time, it is possible to assess the state of the traffic and make predictions for the future. Previously, the authors proposed a multilane model of traffic flows on the basis of the quasi-gas-dynamic (QGD) system of equations. 2 This system has proven to be efficient for simulating a wide range of gas-dynamic flows and has shown its adequacy for modeling traffic flows. 3 However, many road situations may be described with sufficient accuracy using one-dimensional macroscopic models, especially when simulating traffic at crossroads and other road intersections, while the two-dimensional description can be too bulky. This way, one can save computational time and get qualitatively and possibly quantitatively correct results. The one-dimensional continuum model based on QGD-system and calculations based on it are presented in this article. Microscopic approach deals with separate vehicles and, therefore, is dependent on the amount of cars in the system. One of the ways to decrease the computational cost is to use cellular automata (CA) theory. Models based on it are flexible in depicting the peculiarities of vehicular traffic, and, unlike macroscopic models, do not have any restrictions regarding traffic flow density. Also, modern computer systems allow simulating traffic on city-scale networks in real time, that is why microscopic CA models become popular means of traffic simulation these days. 4-7

Kinetic aspects of Landauer's blow torch effect is investigated for a simple double well potential using the supersymmetric approach. We find that the presence of a hot region enhances the escape rate which increases as a function of its strength and width. We derive an approximate analytical expression for the escape rate as a function of the strength and width of the hot region and also as a function of height of the barrier, which agrees well with the numerical results. We also find that the presence of the hot region close to the bottom of the potential enhances the escape rate more efficiently than when it is close to the top of the barrier.