Universal Large-Deviation Function of the Kardar–Parisi–Zhang Equation in One Dimension

Using the Bethe ansatz, we calculate the whole large-deviation function of the displacement of particles in the asymmetric simple exclusion process (ASEP) on a ring. When the size of the ring is large, the central part of this large deviation function takes a scaling form independent of the density of particles. We suggest that this scaling function found for the ASEP is universal and should be characteristic of all the systems described by the Kardar–Parisi–Zhang equation in 1+1 dimension. Simulations done on two simple growth models are in reasonable agreement with this conjecture.     

Weakly Asymmetric Non-Simple Exclusion Process and the Kardar–Parisi–Zhang Equation

We analyze a class of non-simple exclusion processes and the corresponding growth models by generalizing the discrete Cole–Hopf transformation of Gärtner (Stoch Process Appl, 27:233–260, 1987). We identify the main non-linearity and eliminate it by imposing a gradient type condition. For hopping range at most 3, using the generalized transformation, we prove the convergence of the exclusion process toward the Kardar–Parisi–Zhang (kpz) equation. This is the first universality result under the weak asymmetry concerning interacting particle systems. While this class of exclusion processes are not explicitly solvable, under the weak asymmetry we obtain the exact one-point limiting distribution for the step initial condition by using the previous result of Amir et al. (Commun Pure Appl Math, 64(4): 466–537, 2011) and our convergence result.     

Kardar–Parisi–Zhang Equation and Large Deviations for Random Walks in Weak Random Environments

We consider the transition probabilities for random walks in \(1+1\) dimensional space-time random environments (RWRE). For critically tuned weak disorder we prove a sharp large deviation result: after appropriate rescaling, the transition probabilities for the RWRE evaluated in the large deviation regime, converge to the solution to the stochastic heat equation (SHE) with multiplicative noise (the logarithm of which is the KPZ equation). We apply this to the exactly solvable Beta RWRE and additionally present a formal derivation of the convergence of certain moment formulas for that model to those for the SHE.     

Determination of the Kardar–Parisi–Zhang equation from experimental data with a small number of configurations

We introduce an inverse method to determine the parameters of the Kardar–Parisi–Zhang equation corresponding to an evolving interface which requires a small number of configurations as input data. Our approach presents advantages for applications in real world scenarios since it does not require small time intervals between fronts. The method is applied to a restricted solid-on-solid model and a stochastic cellular automata model for fire front propagation.     

On Fractional Duffin–Kemmer–Petiau Equation

In this paper we treat a fractional bosonic, scalar and vectorial, time equation namely Duffin–Kemmer–Petiau Equation. The fractional variational principle was used, the fractional Euler–Lagrange equations were presented. The wave functions were determined and expressed in terms of Mittag–Leffler function.     

Eigenstate Distribution Fluctuation of a Quenched Disordered Bose–Hubbard System in Thermal-to-Localized Transitions

We study the thermalization of a quenched disordered Bose–Hubbard system. By considering the eigenstate distribution fluctuation, we show that the thermal to many-body localized transition is always connected to a minimum of this distribution fluctuation. We also observe a Mott-localized regime, where the system fails to thermalize due to the strong on-site repulsion. Lastly, we show how to detect this eigenstate distribution fluctuation in a cold atom system, which is equivalent to measure the Loschmidt echo of the system. Our work suggests a way to measure the thermal-to-localized transitions in experiments, especially for a large system.     

Equation of state and liquid–vapour equilibrium in a triangle-well fluid

The second-order thermodynamic perturbation theory formulation of Barker and Henderson is used to derive the equation of state of the triangle-well fluid. This is combined with the rational function approximation to the radial distribution function of the hard-sphere fluid. Results are obtained for the critical parameters and the liquid–vapour coexistence curve for various values of the range of the potential. A comparison with available simulation data is presented.     

Time Asymptotics for Solutions of Vlasov–Poisson Equation in a Circle

We prove that there exists a class of solutions of the nonlinear Vlasov–Poisson equation (VPE) on a circle that converges weakly, as t , to a stationary homogeneous solution of VPE. This behavior is called, in the linear case, Landau damping. The result is obtained by constructing a suitable scattering problem for the solutions of the Vlasov–Poisson problem. A consequence of this result is that a class of stationary solutions of the Vlasov–Poisson equation is unstable in a weak topology.     

Investigation of Conformable Fractional Schr¨odinger Equation in Presence of Killingbeck and Hyperbolic Potentials

In this article, conformable fractional form of Schrdinger equation has been presented. Then in this formalism two different and well-known potential have been come in. Wave function of these potential are obtained in terms of Heun function and energy eigen values of each case is determined as well.     

Critical Study of Hierarchical Lattice Renormalization Group in Magnetic Ordered and Quenched Disordered Systems: Ising and Blume–Emery–Griffiths Models

Renormalization group based on the Migdal–Kadanoff bond removal approach is often considered a simple and valuable tool to understand the critical behavior of complicated statistical mechanical models. In presence of quenched disorder, however, predictions obtained with the Migdal–Kadanoff bond removal approach quite often fail to quantitatively and qualitatively reproduce critical properties obtained in the mean-field approximation or by numerical simulations in finite dimensions. In an attempt to overcome this limitation we analyze the behavior of Ising and Blume–Emery–Griffiths models on more structured hierarchical lattices. We find that, apart from some exceptions, the failure is not limited to Midgal–Kadanoff cells but originates right from the hierarchization of Bravais lattices on small cells, and shows up also when in-cell loops are considered.     

Scaling Law of Exponents in Cosmological Clustering

The density of N-clusters(the clusters which contain N galaxies)in the Universe is shown from observations to scale with N as nN ∝ N^-τ with τ=3/γ 1 and the correlation exponent γ≈1.8.Correspondingly,a acaling relation τ=29-9γ/12-4γ between the two exponents,which agrees with the observations,is found analytically in our naive clustering model.     

Critical behavior of a stochastic anisotropic Bak–Sneppen model

In this paper we present our study on the critical behavior of a stochastic anisotropic Bak–Sneppen (saBS) model, in which a parameter α is introduced to describe the interaction strength among nearest species. We estimate the threshold fitness f _c and the critical exponent τ _r by numerically integrating a master equation for the distribution of avalanche spatial sizes. Other critical exponents are then evaluated from previously known scaling relations. The numerical results are in good agreement with the counterparts yielded by the Monte Carlo simulations. Our results indicate that all saBS models with nonzero interaction strength exhibit self-organized criticality, and fall into the same universality class, by sharing the universal critical exponents.     

Analysis of Energy Eigenvalue in Complex Ginzburg–Landau Equation

In this paper, we consider the two-dimensional complex Ginzburg–Landau equation(CGLE) as the spatiotemporal model, and an expression of energy eigenvalue is derived by using the phase-amplitude representation and the basic ideas from quantum mechanics. By numerical simulation, we find the energy eigenvalue in the CGLE system can be divided into two parts, corresponding to spiral wave and bulk oscillation. The energy eigenvalue of spiral wave is positive, which shows that it propagates outwardly; while the energy eigenvalue of spiral wave is negative, which shows that it propagates inwardly. There is a necessary condition for generating a spiral wave that the energy eigenvalue of spiral wave is greater than bulk oscillation. A wave with larger energy eigenvalue dominates when it competes with another wave with smaller energy eigenvalue in the space of the CGLE system. At the end of this study, a tentative discussion of the relationship between wave propagation and energy transmission is given.     

Langevin Equation of Collective Modes of Bose–Einstein Condensates in Traps

A quantum Langevin equation for the amplitudes of the collective modes in Bose–Einstein condensate is derived. The collective modes are coupled to a thermal reservoir of quasi-particles, whose elimination leads to the quantum Langevin equation. The dissipation rates are determined via the correlation function of the fluctuating force and are evaluated in the local-density approximation for the spectrum of quasi-particles and the Thomas–Fermi approximation for the condensate.I take great pleasure in dedicating this paper to Gregoire Nicolis on the occasion of his sixtieth birthday.     

Critical number of solitons in the Bose–Einstein condensate

The dynamics of solitons in the Bose–Einstein condensate under the effect of the fluctuation interaction of condensate atoms is studied. A system of equations of motion describing changes in the parameters of the soliton wave function is obtained using the method of the averaged Lagrangian. The minimum critical number of solitons is found, and the influence of the fluctuation interaction on the dynamics of solitons near their critical width is studied.     

A Rigorous Derivation of a Linear Kinetic Equation of Fokker–Planck Type in the Limit of Grazing Collisions

We rigorously derive a linear kinetic equation of Fokker–Planck type for a 2-D Lorentz gas in which the obstacles are randomly distributed. Each obstacle of the Lorentz gas generates a potential V( ), where V is a smooth radially symmetric function with compact support, and >0. The density of obstacles diverges as ^– , where >0. We prove that when 0< <1/8 and =2+1, the probability density of a test particle converges as 0 to a solution of our kinetic equation.     

Quasiparticles in the multicomponent Zhang–Hansson–Kivelson model

We study the vortex solutions in a multicomponent Zhang–Hansson–Kivelson model for the fractional quantum Hall effect, at the self-dual point. Vortices with minimal free energy represent Laughlin quasiholes. We find at least two classes of solutions, distinguished by their global invariance, or by the number of conserved charges.