Shock fluctuations in the two-dimensional asymmetric simple exclusion process

We study via computer simulations (using various serial and parallel updating techniques) the time evolution of shocks, particularly the shock width(t), in several versions of the two-dimensional asymmetric simple exclusion process (ASEP). The basic dynamics of this process consists of particles jumping independently to empty neighboring lattice sites with ratesp _up=p _down=p andp _left<p _right. If the system is initially divided into two regions with densities _left< _right, the boundary between the two regions corresponds to a shock front. Macroscopically the shock remains sharp and moves with a constant velocityv _shock=(p _right– _left)(1–p _left–p _right). We find that microscopic fluctuations cause to grow ast , 1/4. This is consistent with theoretical expectations. We also study the nonequilibrium stationary states of the ASEP on a periodic lattice, where we break translation invariance by reducing the jump rates across the bonds between two neighboring columns of the system by a factorr. We find that for fixed overall density _avg and reduction factorr sufficiently small (depending on _avg and the jump rates) the system segregates into two regions with densities ₁ and ₂=1– ₁, where these densities do not depend on the overall density _avg. The boundary between the two regions is again macroscopically sharp. We examine the shock width and the variance in the shock position in the stationary state, paying particular attention to the scaling of these quantities with system size. This scaling behavior shows many of the same features as the time-dependent scaling discussed above, providing an alternate determination of the result1/4.     

Shock profiles for the asymmetric simple exclusion process in one dimension

The asymmetric simple exclusion process (ASEP) on a one-dimensional lattice is a system of particles which jump at ratesp and 1-p (herep > 1/2) to adjacent empty sites on their right and left respectively. The system is described on suitable macroscopic spatial and temporal scales by the inviscid Burgers’ equation; the latter has shock solutions with a discontinuous jump from left density ρ- to right density ρ+, ρ-< ρ +, which travel with velocity (2p−1 )(1−ρ₊−p ₋). In the microscopic system we may track the shock position by introducing a second class particle, which is attracted to and travels with the shock. In this paper we obtain the time-invariant measure for this shock solution in the ASEP, as seen from such a particle. The mean density at lattice siten, measured from this particle, approachesp _± at an exponential rate asn→ ±∞, witha characteristic length which becomes independent ofp when . For a special value of the asymmetry, given byp/(1−p)=p ₊(1−p ₋)/p ₋(1−p ₊), the measure is Bernoulli, with densityρ ₋ on the left andp ₊ on the right. In the weakly asymmetric limit, 2p−1 → 0, the microscopic width of the shock diverges as (2p+1)^-1. The stationary measure is then essentially a superposition of Bernoulli measures, corresponding to a convolution of a density profile described by the viscous Burgers equation with a well-defined distribution for the location of the second class particle.     

Zone inhomogeneity with the random asymmetric simple exclusion process in a one-lane system

In this paper we use theoretical analysis and extensive simulations to study zone inhomogeneity with the random asymmetric simple exclusion process (ASEP). In the inhomogeneous zone, the hopping probability is less than 1. Two typical lattice geometries are investigated here. In case A, the lattice includes two equal segments. The hopping probability in the left segment is equal to 1, and in the right segment it is equal to p, which is less than 1. In case B, there are three equal segments in the system; the hopping probabilities in the left and right segments are equal to 1, and in the middle segment it is equal to p, which is less than 1. Through theoretical analysis, we can discover the effect on these systems when p is changed.     

Exact solution of the totally asymmetric simple exclusion process: Shock profiles

The microscopic structure of macroscopic shocks in the one-dimensional, totally asymmetric simple exclusion process is obtained exactly from the complete solution of the stationary state of a model system containing two types of particles-first and second class. This nonequilibrium steady state factorizes about any second-class particle, which implies factorization in the one-component system about the (random) shock position. It also exhibits several other interesting features, including long-range correlations in the limit of zero density of the second-class particles. The solution also shows that a finite number of second-class particles in a uniform background of first-class particles form a weakly bound state.     

Exact results for the asymmetric simple exclusion process with a blockage

We present new results for the current as a function of transmission rate in the one-dimensional totally asymmetric simple exclusion process (TASEP) with a blockage that lowers the jump rate at one site from one tor<1. Exact finitevolume results serve to bound the allowed values for the current in the infinite system. This proves the existence of a nonequilibrium phase transition, corresponding to an immiscibility gap in the allowed values of the asymptotic densities which the infinite system can have in a stationary state. A series expansion inr, derived from the finite systems, is proven to be asymptotic for all sufficiently large systems. Padé approximants based on this series, which make specific assumptions about the nature of the singularity atr=1, match numerical data for the infinite system to 1 part in 10⁴.     

Modified shock waves for extracorporeal shock wave lithotripsy: a simulation based on the Gilmore formulation

Extracorporeal shock wave lithotripsy (SWL) is a reliable therapy for the treatment of urolithiasis. Nevertheless, improvements to enhance stone fragmentation and reduce tissue damage are still needed. During SWL, cavitation is one of the most important stone fragmentation mechanisms. Bubbles with a diameter between about 7 and 55 μm have been reported to expand and collapse after shock wave passage, forming liquid microjets at velocities of up to 400 m/s that contribute to the pulverization of renal calculi. Several authors have reported that the fragmentation efficiency may be improved by using tandem shock waves. Tandem SWL is based on the fact that the collapse of a bubble can be intensified if a second shock wave arrives tenths or even a few hundredths of microseconds before its collapse. The object of this study is to determine if tandem pulses consisting of a conventional shock wave (estimated rise time between 1 and 20 ns), followed by a slower second pressure profile (0.8 μs rise time), have advantages over conventional tandem SWL. The Gilmore equation was used to simulate the influence of the modified pressure field on the dynamics of a single bubble immersed in water and compare the results with the behavior of the same bubble subjected to tandem shock waves. The influence of the delay between pulses on the dynamics of the collapsing bubble was also studied for both conventional and modified tandem waves. For a bubble of 0.07 mm, our results indicate that the modified pressure profile enhances cavitation compared to conventional tandem waves at a wide range of delays (10-280 μs). According to this, the proposed pressure profile could be more efficient for SWL than conventional tandem shock waves. Similar results were obtained for a ten times smaller bubble.     

二次曝光全息术在圆柱形冲击波会聚过程观测中的应用

本文对全息曝光技术进行了分析,设计了一种全息干涉仪,对圆柱形冲击波会聚过程的不稳定性进行了观察,并得出了几个有重要价值的结论。     

Interactions of dispersive shock waves

Collisions and interactions of dispersive shock waves in defocusing (repulsive) nonlinear Schrödinger type systems are investigated analytically and numerically. Two canonical cases are considered. In one case, two counterpropagating dispersive shock waves experience a head-on collision, interact and eventually exit the interaction region with larger amplitudes and altered speeds. In the other case, a fast dispersive shock overtakes a slower one, giving rise to an interaction. Eventually the two merge into a single dispersive shock wave. In both cases, the interaction region is described by a modulated, quasi-periodic two-phase solution of the nonlinear Schrödinger equation. The boundaries between the background density, dispersive shock waves and their interaction region are calculated by solving the Whitham modulation equations. These asymptotic results are in excellent agreement with full numerical simulations. It is further shown that the interactions of two dispersive shock waves have some qualitative similarities to the interactions of two classical shock waves.     

On the viscous and inviscid stability of magnetohydrodynamic shock waves

We study the viscous and inviscid stability of shock waves in barotropic and full magnetohydrodynamics. We show that there are magnetohydrodynamic shock waves that are one-dimensionally stable as viscous shock profiles while they are multidimensionally strongly unstable as planar shock discontinuities.     

Effects of bi-kappa distributed electrons on dust-ion-acoustic shock waves in dusty superthermal plasmas

The basic properties of dust-ion-acoustic （DIA） shock waves in an unmagnetized dusty plasma （containing inertial ions, kappa distributed electrons with two distinct temperatures, and negatively charged immobile dust grains） are investi- gated both numerically and analytically. The hydrodynamic equation for inertial ions has been used to derive the Burgers equation. The effects of superthermal bi-kappa electrons and ion kinematic viscosity, which are found to modify the basic features of DIA shock waves significantly, are briefly discussed.     

Exact solution of the master equation for the asymmetric exclusion process

Using the Bethe ansatz, we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one-dimensional lattice. We derive explicit expressions for the conditional probabilitiesP(x₁,...,x_N;t/y ₁,...,y_N; 0) of findingN particles on lattices sitesx ₁,...,x_N at timet with initial occupationy ₁,...,y_N at timet=0.     

Power spectra of the total occupancy in the totally asymmetric simple exclusion process

As a solvable and broadly applicable model system, the totally asymmetric exclusion process enjoys iconic status in the theory of nonequilibrium phase transitions. Here, we focus on the time dependence of the total number of particles on a 1-dimensional open lattice and its power spectrum. Using both Monte Carlo simulations and analytic methods, we explore its behavior in different characteristic regimes. In the maximal current phase and on the coexistence line (between high and low density phases), the power spectrum displays algebraic decay, with exponents -1.62 and -2.00, respectively. Deep within the high and low density phases, we find pronounced oscillations, which damp into power laws. This behavior can be understood in terms of driven biased diffusion with conserved noise in the bulk.     

Non-stationary probabilities for the asymmetric exclusion process on a ring

A solution of the master equation for a system of interacting particles for finite time and particle density is presented. By using a new form of the Bethe ansatz, the totally asymmetric exclusion process on a ring is solved for arbitrary initial conditions and time intervals.     

Successive defects asymmetric simple exclusion processes with particles of arbitrary size

This paper uses various mean-field approaches and the Monte Carlo simulation to calculate asymmetric simple exclusion processes with particles of arbitrary size in the successive defects system. In this system, the hopping probability p (p<1) and the size d of particles are not constant. Through theoretical calculation and computer simulation, it obtains the exact theoretical results and finds that the theoretical results are in agreement with the computer simulation. These results are helpful in analysing the effect of traffic with different hopping probabilities p and sizes d of particle.     

Anisotropic sound and shock waves in dipolar Bose-Einstein condensate

We study the propagation of anisotropic sound and shock waves in dipolar Bose-Einstein condensate in three dimensions (3D) as well as in quasi-two (2D, disk shape) and quasi-one (1D, cigar shape) dimensions using the mean-field approach. In 3D, the propagation of sound and shock waves are distinct in directions parallel and perpendicular to dipole axis with the appearance of instability above a critical value corresponding to attraction. Similar instability appears in 1D and not in 2D. The numerical anisotropic Mach angle agrees with theoretical prediction. The numerical sound velocity in all cases agrees with that calculated from Bogoliubov theory. A movie of the anisotropic wave propagation in a dipolar condensate is made available as supplementary material.     

Large deviation function for the current in the open asymmetric simple exclusion process

We consider the one-dimensional asymmetric exclusion process with particle injection and extraction at two boundaries. The model is known to exhibit four distinct phases in its stationary state. We analyze the current statistics at the first site in the low and high density phases. In the limit of infinite system size, we conjecture an exact expression for the current large deviation function.     

Nonlinear interaction phenomenon of dust acoustic solitary and shock waves in dusty plasma

The effects of head-on collision on dust acoustic (DA) solitary and shock waves in dusty plasma are investigated considering positively charged inertial dust, Boltzmann distributed negatively charged heavy ions, positively charged light ions, and superthermal electrons in the plasma system. The nonlinear Korteweg-de-Vries (KdV) Burger equations are derived taking the extended Poincaré-Lighthill-Kuo method into account to study the characteristic properties of nonlinearity and production of solitary shock due to collisions. The study reveals that the amplitudes and widths of the DA shock waves are decreasing with increasing viscosity, electron to dust density ratio, and dust to ion temperature ratio, while they are increasing due to the presence of superthermal electrons. The nonlinearity of DA waves are enhanced with increasing density ratio of electron to dust and temperature ratio of dust to ion and electron, respectively, but it is reducing with superthermal electrons. The phase shifts of DA solitary waves are found to decrease with rising superthermality of electrons and increase with the density ratio of electron to dust.     

Obliquely propagating ion-acoustic shock waves in a degenerate quantum plasma

A theoretical investigation has been carried out on the propagation of non-linear ion-acoustic shock waves (IASHWs) in a magnetized degenerate quantum plasma system composed of inertial non-relativistic positively charged light and heavy ions, inertialess non-relativistically or ultra-relativistically degenerate electrons and positrons. The reductive perturbation method has been employed to derive the Burgers' equation. It has been observed that under consideration, our plasma model supports only positive potential shock structure. It is also found that the amplitude and steepness of the IASHWs have been significantly modified by the variation of ion kinematic viscosity, oblique angle, number density, and charge state of the plasma species. The results of our present investigation will be helpful for understanding the propagation of IASHWs in white dwarfs and neutron stars.     

Measurement of shock waves using phase shifting pulsed holographic interferometer

A phase shifting pulsed holographic interferometer was applied to the experimental study of the propagation of laser-induced shock waves over metal plates. A double-pulsed ruby laser was used to generate the shock waves and to make a holographic interferogram of the wave fields. The phase shifting method with a dual-reference beam solved the sign ambiguity problem in holographic fringe patterns and allowed a quantitative evaluation of the phase of the interference patterns. The transient surface profile and propagation behavior of the shock wave over plates were investigated from the holographic fringe patterns.