asymptotics in asep with step initial condition in previous work the authors considered the asymmetric simple exclusion process on the integer lattice in the case of step initial condition particles beginning at the positive integers there it was shown that the probability distribution for the position of an individual particle is given by an integral whose integrand involves a fredholm determinant here we use this formula to obtain three asymptotic results for the positions of these particles in one an apparently new distribution function arises and in another the distribution function f2 arises the latter extends a result of johansson on tasep to asep and hence proves kpz universality for asep with step initial condition

localized patterns for the quintic complex swift hohenberg equation abstract we show using numerical simulations that a variety of localized patterns arise in a model equation the quintic swift hohenberg equation with complex coefficients we demonstrate that various sizes of localized standing wave patterns are possible when the imaginary part of the complex coefficient is small localized traveling waves as well as localized standing waves with a fixed size are observed when the imaginary part is rather large we also present stable localized patterns in two spatial dimensions and study their interaction

various equations for gap probabilities of coupled gaussian matrices versions of tracy widom tw and adler shiota van moerbeke asvm approaches are applied to derive various partial differential equations pde satisfied by joint eigenvalue spacing probabilities of two coupled gaussian hermitian matrices coupled gue all the lowest third order pde satisfied by the probabilities for the largest eigenvalues of two coupled gue are found and the results of both approaches are compared the tw approach allows us to derive all pde at once while in the asvm one starting with different bilinear identities leads to different subsets of the full set of equations an interesting result is that the joint probability for the largest eigenvalues of coupled gaussian matrices satisfies a number of different pde and the previously known adler van moerbeke equation avm 1999 ann math 149 921 76 is only one of them some of the new equations appear like coupled painlev iv and have the usual painlev iv equation as the one matrix limit i e when the spectral endpoint of one of the matrices goes to infinity this is in contrast to the avm equation which becomes trivial in this limit moreover the new pde which stem from the matrix kernel approach of tracy and widom 2004 commun math phys 252 7 41 do not contain derivatives w r t the strength of coupling unlike the avm equation in other words they contain fewer independent variables and in this sense are simpler

the correspondence between tracy widom and adler shiota van moerbeke approaches in random matrix theory the gaussian case two approaches to the derivation of integrable differential equations for random matrix probabilities are compared both methods are rewritten in such a form that simple and explicit relations between all tw dependent variables and functions of asvm are found for the example of finite size gaussian matrices orthogonal function systems and toda lattice are seen as the core structure of both approaches and their relationship

scale invariance of the png droplet and the airy process we establish that the static height fluctuations of a particular growth model the png droplet converges upon proper rescaling to a limit process which we call the airy process a y the airy process is stationary it has continuous sample paths its single time fixed y distribution is the tracy widom distribution of the largest eigenvalue of a gue random matrix and the airy process has a slow decay of correlations as y 2 roughly the airy process describes the last line of dyson s brownian motion model for random matrices our construction uses a multi layer version of the png model which can be analyzed through fermionic techniques specializing our result to a fixed value of y one reobtains the celebrated result of baik deift and johansson on the length of the longest increasing subsequence of a random permutation

the spectrum of coupled random matrices the study of the spectrum of coupled random matrices has received rather little attention to the best of our knowledge coupled random matrices have been studied to some extent by mehta in this work we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled hermitean random matrices and the equations the associated probabilities satisfy in particular the two toda lattice its algebra of symmetries and its vertex operators will play a prominent role in this interaction namely the method is to introduce time parameters in an artificial way and to dress up a certain matrix integral with a vertex integral operator for which we find virasoro like differential equations these methods lead to very simple nonlinear third order partial differential equations for the joint statistics of the spectra of two coupled gaussian random matrices comment 56 pages published version abstract added in migration

stochastic rotation dynamics i formalism galilean invariance and green kubo relations a detailed analytical and numerical analysis of a recently introduced stochastic model for fluid dynamics with continuous velocities and efficient multi particle collisions is presented it is shown how full galilean invariance can be achieved for arbitrary mach numbers and how other low temperature anomalies can be removed the relaxation towards thermal equilibrium is investigated numerically and the relaxation time is measured equations of motions for the correlation functions of coarse grained hydrodynamic variables are derived using a discrete time projection operator technique and the green kubo relations for all relevant transport coefficients are given in the following paper part 2 analytic expressions for the transport coefficients are derived and compared with simulation results long time tails in the velocity and stress autocorrelation functions are measured and shown to be in good agreement with previous mode coupling theories for continuous systems

analytical approach to localized structures in a simple reaction diffusion system we study from an analytical point of view a simple reaction diffusion model which admits stable oscillating localized structures as a consequence of the coexistence between a stable limit cycle and a stable fixed point using a generalized matching approach we are able to find approximate analytical expressions for localized oscillating structures in this reaction diffusion model capturing all the essential ingredients of these breathing particlelike solutions

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

hydrodynamics of confined colloidal fluids in two dimensions we apply a hybrid molecular dynamics and mesoscopic simulation technique to study the dynamics of two dimensional colloidal disks in confined geometries we calculate the velocity autocorrelation functions and observe the predicted t 1 long time hydrodynamic tail that characterizes unconfined fluids as well as more complex oscillating behavior and negative tails for strongly confined geometries because the t 1 tail of the velocity autocorrelation function is cut off for longer times in finite systems the related diffusion coefficient does not diverge but instead depends logarithmically on the overall size of the system the langevin equation gives a poor approximation to the velocity autocorrelation function at both short and long times

generalized green kubo formulas for fluids with impulsive dissipative stochastic and conservative interactions we present a generalization of the green kubo expressions for thermal transport coefficients mu in complex fluids of the generic form mu mu infinity integral infinity 0 dtv 1 0 i e a sum of an instantaneous transport coefficient mu infinity and a time integral over a time correlation function in a state of thermal equilibrium between a current j and its conjugate current j epsilon the streaming operator exp tl generates the trajectory of a dynamical variable j t exp tl j when used inside the thermal average 0 these formulas are valid for conservative impulsive hard spheres stochastic and dissipative forces langevin fluids provided the system approaches a thermal equilibrium state in general mu infinity not equal 0 and j epsilon not equal j except for the case of conservative forces where the equality signs apply the most important application in the present paper is the hard sphere fluid

on the stability of localized structures in the complex ginzburg landau equation abstract we study analytically the asymptotic linear stability of fixed modulus dissipative dispersive localized solutions of the one dimensional quintic complex ginzburg landau gl equation in the region where there exists a coexistence of homogeneous attractors the linear analysis gives an indication for the existence of pulses with an oscillating modulus

matrices coupled in a chain i eigenvalue correlations the general correlation function for the eigenvalues of p complex hermitian matrices coupled in a chain is given as a single determinant for this we use a slight generalization of a theorem of dyson

on the stable hole solutions in the complex ginzburg landau equation abstract we show numerically that the one dimensional quintic complex ginzburg landau equation admits four different types of stable hole solutions we present a simple analytic method which permits to calculate the region of existence and approximate shape of stable hole solutions in this equation the analytic results are in good agreement with numerical simulations

pdes for the joint distributions of the dyson airy and sine processes in a celebrated paper dyson shows that the spectrum of an n x n random hermitian matrix diffusing according to an ornstein uhlenbeck process evolves as n noncolliding brownian motions held together by a drift term the universal edge and bulk scalings for hermitian random matrices applied to the dyson process lead to the airy and sine processes in particular the airy process is a continuous stationary process describing the motion of the outermost particle of the dyson brownian motion when the number of particles gets large with space and time appropriately rescaled in this paper we answer a question posed by kurt johansson to find a pde for the joint distribution of the airy process at two different times similarly we find a pde satisfied by the joint distribution of the sine process this hinges on finding a pde for the joint distribution of the dyson process which itself is based on the joint probability of the eigenvalues for coupled gaussian hermitian matrices the pde for the dyson process is then subjected to an asymptotic analysis consistent with the edge and bulk rescalings the pdes enable one to compute the asymptotic behavior of the joint distribution and the correlation for these processes at different times t 1 and t 2 when t 2 t 1 infinity as illustrated in this paper for the airy process this paper also contains a rigorous proof that the extended hermite kernel governing the joint probabilities for the dyson process converges to the extended airy and sine kernels after the appropriate rescalings

pulse solutions of the cubic quintic complex ginzburg landau equation in the case of normal dispersion time localized solitary wave solutions of the one dimensional complex ginzburg landau equation cgle are analyzed for the case of normal group velocity dispersion exact soliton solutions are found for both the cubic and the quintic cgle the stability of these solutions is investigated numerically the regions in the parameter space in which stable pulselike solutions of the quintic cgle exist are numerically determined these regions contain subspaces where analytical solutions may be found an investigation of the role of group velocity dispersion changes in magnitude and sign on the spectral and temporal characteristics of the stable pulse solutions is also carried out

brownian motion ensembles correlation functions of determinantal processes we calculate the n point correlation function for a large class of brownian motion ensembles of random matrix theory the corresponding fokker planck equation describes a determinantal process in the theory of matrix valued stochastic differential equations and can be mapped onto a schodinger equation of non interacting electrons in one dimension the correlation functions are obtained explicitly via a suitable generalization of the method of biorthogonal functions

differential equations for dyson processes we call a dyson process any process on ensembles of matrices in which the entries undergo diffusion we are interested in the distribution of the eigenvalues or singular values of such matrices in the original dyson process it was the ensemble of n n hermitian matrices and the eigenvalues describe n curves given sets x1 xm the probability that for each k no curve passes through xk at time k is given by the fredholm determinant of a certain matrix kernel the extended hermite kernel for this reason we call this dyson process the hermite process similarly when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the laguerre process for which there is a corresponding extended laguerre kernel scaling the hermite process at the edge leads to the airy process which was introduced by prahofer and spohn as the limiting stationary process for a polynuclear growth model and in the bulk to the sine process scaling the laguerre process at the edge leads to the bessel process

computational steering of cluster formation in brownian suspensions we simulate cluster formation of model colloidal particles interacting via dlvo derjaguin landau vervey overbeek potentials the interaction potentials can be related to experimental conditions defined by the ph value the salt concentration and the volume fraction of solid particles suspended in water the system shows different structural properties for different conditions including cluster formation a glass like repulsive structure or a liquid suspension typically many simulations are required to explore the parameter space in order to reduce the computational effort and data storage requirements we developed a steering approach to control a running simulation and to detect interesting transitions from one region in the configuration space to another the advantages of the steering approach and the restrictions of its applicability due to physical constraints are illustrated by several example cases

on the moving pulse solutions in systems with broken parity abstract we study analytically a system sustaining stable moving localized structures namely the one dimensional quintic complex ginzburg landau g l equation with non linear gradients we obtain approximate solutions for the stable moving pulse and its velocity the results are in excellent agreement with direct numerical simulations

saddle node bifurcation appearance mechanism of pulses in the subcritical complex ginzburg landau equation we study stationary localized solutions in the complex subcritical ginzburg landau equation in the region where there exists coexistence of homogeneous attractors using a matching approach we report on the fact that the appearance of pulses are related to a saddle node bifurcation numerical simulations are in good agreement with our theoretical predictions

universal structure and universal equations pde for unitary ensembles random matrix ensembles with unitary invariance of measure ue are described in a unified way using a combination of tracy widom tw and adler shiota van moerbeke approaches to the derivation of partial differential equations pdes for spectral gap probabilities first general three term recurrence relations for ue restricted to subsets of real line or in other words for functions in the resolvent kernel are obtained using them simple universal relations between all tw dependent variables and one dimensional toda lattice functions are found a universal system of pde for ue is derived from previous relations which leads also to a single independent pde for spectral gap probability of various ue thus orthogonal function bases and toda lattice are seen at the core of correspondence of different approaches moreover toda akns system provides a common structure of pde for unitary ensembles interestingly this structure can be seen in two very different forms one arises from orthogonal functi