on the layering transition of an sos surface interacting with a wall i equilibrium results we consider the model of a 2d surface above a fixed wall and attracted toward it by means of a positive magnetic field h in the solid on solid sos approximation when the inverse temperature fl is very large and the external field h is exponentially small in ft we improve considerably previous results by dinaburg and mazel on the competition between the external field and the entropic repulsion with the wall leading in this case to the phenomenon of layering phase transitions in particular we show using the pirogov sinai scheme as given by zahradnik that there exists a unique critical value h fl in the interval 88 4 k 4e 4 k such that for all h h i h and fl large enough there exists a unique infinite volume gibbs state the typical configurations are small perturbations of the ground state represented by a surface at height k 1 above the wall moreover for the same choice of the thermodynamic parameters the influence of the boundary conditions of the gibbs measure in a finite cube decays exponentially fast with the distance from the boundary when h h fl we prove instead the convergence of the cluster expansion for both k and k 1 boundary conditions this fact signals the presence of a phase transition in the second paper of this series we will consider a glauber dynamics for the above model and we will study the rate of approach to equilibrium in a large finite cube with arbitrary boundary conditions as a function of the external field h using the results proven in this paper we will show that there is a dramatic slowing down in the approach to equilibrium when the magnetic field takes one of the critical values and the boundary conditions are free absent

a travelling wave solution to the kdv burgers equation in the present work by introducing a new potential function and by using the hyperbolic tangent method and an exponential rational function approach a travelling wave solution to the kdv burgers kdvb equation is presented it is observed that both methods lead to the same type of solution the solution method we introduced here is less restrictive and comprises some solutions existing in the current literature wave motion 11 1989 559 wave motion 14 1991 559

the droplet in the tube a case of phase transition in the canonical ensemble we consider the 2 dimensional ising model with ferromagnetic nearest neighbour interaction at inverse temperature lets n t be the total magnetization inside ann n square box per be the gibbs state in with periodic b c andm be the spontaneous magnetization we show the existence of the limit psi varrho mathop lim limits n to infty left frac 1 beta n right ln mu lambda per s n n varrho for m provided is large enough it turns out that the quantity is closely related to the wulf construction and the dependence of the function on is singular

the logarithmic sobolev inequality for discrete spin systems on a lattice for finite range lattice gases with a finite spin space it is shown that the dobrushin shlosman mixing condition is equivalent to the existence of a logarithmic sobolev inequality for the associated unique gibbs state in addition implications of these considerations for the ergodic properties of the corresponding glauber dynamics are examined

improved perturbation expansion for disordered systems beating griffiths singularities we introduce a new expansion to prove exponential clustering of connected correlations in a large class of disordered systems our expansion converges for values of the temperature and magnetic field where standard cluster expansions diverge due to the presence of griffiths type singularities it is organized inductively over an infinite sequence of increasing distance scales in each induction step one redefines what is means by the unperturbed system a procedure somewhat reminiscent of k a m theory our techniques may be useful in dealing with the so called large field problem in real space renormalization group schemes

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

relaxation properties of 1 1 dimensional driven interfaces in disordered media we use the kardar parisi zhang equation with quenched noise in order to study the relaxation properties of driven interfaces in disordered media for 0 this equation belongs to the directed percolation depinning universality class and for 0 it belongs to the quenched edwards wilkinson universality class we study the fourier transform of the two time autocorrelation function of the interface height ck t t these functions depend on the difference of times t t in the steady state regime we find a two step relaxation decay in this regime for both universality classes the long time tail can be fitted by a stretched exponential function where the exponent depends on the universality class the relaxation time and the wavelength of the fourier transform where the two step relaxation is lost are related to the length of the pinned regions the stretched exponential relaxation is caused by the existence of pinned regions which is a direct consequence of the quenched noise

relaxation to equilibrium for two dimensional disordered ising systems in the griffiths phase we consider glauber type dynamics for two dimensional disordered magnets of ising type we prove that if the disorder averaged influence of the boundary condition is sufficiently small in the equilibrium system then the corresponding glauber dynamics is ergodic with probability one and the disorder average c t of time autocorrelation function satisfies for large t for the standard two dimensional dilute ising ferromagnet with i i d random nearest neighbor couplings taking the values 0 or j 0 0 our results apply even if the active bonds percolate and j 0 is larger than the critical value j c of the corresponding pure ising model for the same model we also prove that in the whole griffiths phase the previous upper bound is optimal this implies the existence of a dynamical phase transition which occurs when j crosses j c

cluster expansion for d dimensional lattice systems and finite volume factorization properties we consider classical lattice systems with finite range interactions ind dimensions by means of a block decimation procedure we transform our original system into a polymer system whose activity is small provided a suitable factorization property of finite volume partition functions holds in this way we extend a result of olivieri

percolation techniques in disordered spin flip dynamics relaxation to the unique invariant measure we consider lattice spin systems with short range but random and unbounded interactions we give criteria for ergodicity of spin flip dynamics and estimate the speed of convergence to the unique invariant measure we find for this convergence a stretched exponential in time for a class of directed dynamics such as in the disordered toom or stavskaya model for the general case we show that the relaxation is faster than any power in time no assumptions of reversibility are made the methods are based on relating the problem to an oriented percolation problem contact process and for the general case using a slightly modified version of the multiscale analysis of e g klein 1993

spectral gap and logarithmic sobolev inequality for kawasaki and glauber dynamics we prove that the spectral gap of the kawasaki dynamics shrink at the rate of 1 l2 for cubes of sizel provided that some mixing conditions are satisfied we also prove that the logarithmic sobolev inequality for the glauber dynamics in standard cubes holds uniformly in the size of the cube if the dobrushin shlosman mixing condition holds for standard cubes

disagreement percolation in the study of markov fields recently one of the authors van den berg has obtained a uniqueness condition for gibbs measures in terms of disagreement percolation involving two independent realizations in the present paper we study the dependence of markov fields on boundary conditions by taking a more suitable coupling this coupling leads to a new uniqueness condition which improves the one mentioned above we also compare it with the dobrushin uniqueness condition in the case of the ising model our coupling shares certain properties with the fortuin kasteleyn representation it gives an explicit expression of the boundary effect on a certain vertex in terms of percolation like probabilities

on the geometrically exact beam model a consistent effective and simple derivation from three dimensional finite elasticity the goal of the present work is to emphasize some intrinsic features of the three dimensional beam model originally proposed by simo simo j c 1985 a finite strain beam formulation the three dimensional dynamic problem part i comput methods appl mech eng 49 55 70 and to derive the model equations for the general finite deformation case as well as for the finite deformation small strain case in a consistent but simpler way with respect to what it is generally done in the literature

jacobi elliptic wave solutions for two variable coefficients cylindrical korteweg de vries models arising in dusty plasmas by using direct reduction method abstract in this paper generalized models for both 2 1 dimensional cylindrical modified korteweg de vries cmkdv equation with variable coefficients and 3 1 dimensional variable coefficients cylindrical korteweg de vries ckdv equation are studied by direct reduction method a direct reduction to nonlinear ordinary differential equations in the form of riccati equations obtained for the considered equations under some integrability conditions the search for solutions for the reduced riccati equations has yielded many jacobi elliptic wave solutions solitary and periodic wave solutions for both 2 1 dimensional cmkdv and 3 1 dimensional ckdv equations physical application for the obtained solutions as dust ion acoustic waves in plasma physics is given

on the two dimensional dynamical ising model in the phase coexistence region we consider a glauber dynamics reversible with respect to the two dimensional ising model in a finite square of sidel in the absence of an external field and at large inverse temperature we first consider the gap in the spectrum of the generator of the dynamics in two different cases with plus and open boundary conditions we prove that when the symmetry under global spin flip is broken by the boundary conditions the gap is much larger than the case in which the symmetry is present for this latter we compute exactly the asymptotics of 1 l log gap asl and show that it coincides with the surface tension along one of the coordinate axes as a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spin spin time correlation in the infinite volume plus phase our results establish a connection between the dynamical large deviations and those of the equilibrium gibbs measure studied by shlosman in the framework of the rigorous description of the wulff shape for the ising model finally we show that in the case of open boundary conditions it is possible to rescale the time withl in such a way that asl the finite dimensional distributions of the time rescaled magnetization converge to those of a symmetric continuous time markov chain on the two state space m m m being the spontaneous magnetization our methods rely upon a novel combination of techniques for bounding from below the gap of symmetric markov chains on complicated graphs developed by jerrum and sinclair in their markov chain approach to hard computational problems and the idea of introducing block glauber dynamics instead of the standard single site dynamics in order to put in evidence more effectively the effect of the boundary conditions in the approach to equilibrium

relaxation of disordered magnets in the griffiths regime we study the relaxation to equilibrium of discrete spin systems with random finite range not necessarily ferromagnetic interactions in the griffiths regime we prove that the speed of convergence to the unique reversible gibbs measure is almost surely faster than any stretched exponential at least if the probability distribution of the interaction decays faster than exponential e g gaussian furthermore if the interac tion is uniformly bounded the average over the disorder of the time autocorrelation function goes to equilibrium as exp k log t d d 1 in d 1 in agreement with previous results obtained for the dilute ising model

slow droplet driven relaxation of stochastic ising models in the vicinity of the phase coexistence region we consider the stochastic ising models glauber dynamics corresponding to the infinite volume basic ising model in arbitrary dimensiond 2 with nearest neighbor interaction and under a positive external magnetic fieldh under minimal assumptions on the rates of flip so that all the common choices are included we obtain results which state that when the system is at low temperaturet the relaxation time when the evolution is started with all the spins down blows up whenh 0 as exp t hd 1 the precise results are lower and upper bounds of this form moreover after a time which does not scale withh and before a time which also grows as an exponential of a multiple of 1 hd 1 ash 0 the law of the state of the process stays whenh is small close to the minus phase of the same ising model without an external field these results may be considered as a partial vindication of a conjecture raised by aizenman and lebowitz in connection to the metastable behavior of these stochastic ising models

the effect of a bump in an elastic tube on wave propagation in a viscous fluid of variable viscosity in the present work treating the arteries as a thin walled prestressed elastic tube with a bump and the blood as a newtonian fluid of variable viscosity we have studied the propagation of weakly nonlinear waves in such a medium by employing the reductive perturbation method in the longwave approximation korteweg devries burgers equation with variable coefficients is obtained as the evolution equation seeking a progressive wave type of solution to this evolution equation it is observed that the wave speed is variable the numerical calculations show that the wave speed reaches to its maximum value at the center of the bump but it gets smaller and smaller as we go away from the center of the bump such a result seems to be reasonable from physical considerations

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

stretched exponential relaxation in the biased random voter model we study the relaxation properties of the voter model with i i d random bias we prove under mild conditions that the disorder averaged relaxation of this biased random voter model is faster than a stretched exponential with exponent d d where 0 2 depends on the transition rates of the non biased voter model under an additional assumption we show that the above upper bound is optimal the main ingredient of our proof is a result of donsker and varadhan

upper and lower bounds on dynamic correlations in the griffiths phase dilute ferromagnetic spin systems are considered in the griffiths phase below the percolation threshold it is shown that for relaxational dynamics the spin autocorrelation functions are bounded both above and below by functions of the form exp a 1n t d d 1 ising systems and exp bt12 heisenberg systems establishing these as the correct asymptotic forms

the phase boundary in dilute and random ising and potts ferromagnets the authors present an approach to dilute ising and potts models based on the fortuin kasteleyn random cluster representation which is simultaneously rigorous intuitive and surprisingly simple their analysis yields with no dimensional restrictions or other caveats the following asymptotic form of the phase boundary for the regular dilute model in which bonds have constant ferromagnetic coupling j with probability p and are vacant with probability 1 p the critical temperature scales as exp j ktc p approximately mod p pc mod implying that the crossover exponent is phi 1 if the constant couplings are replaced by a distribution f j with mass near j 0 quite different crossover behaviour is observed for example if f j approximately jalpha then for p near pc tc p approximately mod p pc mod 1 alpha

strong decay to equilibrium in one dimensional random spin systems we consider a spin system on a lattice with finite range possibly unbounded random interactions we show that for such systems the glauber dynamics cannot decay to equilibrium exponentially fast inl2 even at high temperatures additionally for one dimensional systems with unbounded random couplings we prove that with probability one the corresponding glauber dynamics has a fast subexponential decay to equilibrium in the uniform norm provided that the distribution of random couplings satisfies some exponential bound

uniform and l 2 convergence in one dimensional stochastic ising models we study the rate of convergence to equilibrium of one dimensional stochastic ising models with finite range interactions we donot assume that the interactions are ferromagnetic or that the flip rates are attractive the infinitesimal generators of these processes all have gaps between zero and the rest of their spectra we prove that if one of these processes is observed by means of local observables then the convergence is seen to be exponentially fast with an exponent that is any number less than the spectral gap moreover this exponential convergence is uniform in the initial configuration

approach to equilibrium of glauber dynamics in the one phase region ii the general case we develop a new method based on renormalization group ideas block decimation procedure to prove under an assumption of strong mixing in a finite cube 0 a logarithmic sobolev inequality for the gibbs state of a discrete spin system as a consequence we derive the hypercontractivity of the markov semigroup of the associated glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes multiples of the cube 0

on the two dimensional stochastic ising model in the phase coexistence region near the critical point we consider the two dimensional stochastic ising model in finite square with free boundary conditions at inverse temperature 0 and zero external field using duality and recent results of ioffe on the wulff construction close to the critical temperature we extend some of the results obtained by martinelli in the low temperature regime to any temperature below the critical one in particular we show that the gap in the spectrum of the generator of the dynamics goes to zero in the thermodynamic limit as an exponential of the side length of with a rate constant determined by the surface tension along one of the coordinate axes we also extend to the same range of temperatures the result due to shlosman on the equilibrium large deviations of the magnetization with free boundary conditions

the spectral gap for the kawasaki dynamics at low temperature in this paper we analyze the convergence to equilibrium of kawasaki dynamics for the ising model in the phase coexistence region first we show in strict analogy with the nonconservative case that in any lattice dimension for any boundary condition and any positive temperature and particle density the spectral gap in a box of side l does not shrink faster than a negative exponential of the surface ld 1 then we prove that in two dimensions and for free boundary condition the spectral gap in a box of side l is smaller than a negative exponential of l provided that the temperature is below the critical one and the particle density satisfies where represents the particle density of the plus and minus phase respectively

taming griffiths singularities infinite differentiability of quenched correlation functions we prove infinite differentiability of the magnetization and of all quenched correlation functions for disordered spin systems at high temperature or strong magnetic field in the presence of griffiths singularities we also show uniqueness of the gibbs state and exponential decay of truncated correlation functions with probability one our results are obtained through new simple modified high temperature or low activity expansions whose convergence can be displayed by elementary probabilistic arguments our results require no assumptions on the probability distributions of the random parameters except for the obvious one of no percolation of infinite couplings and in the strong field situation for the also obvious requirement that zero magnetic fields do not percolate

convergence to equilibrium of random ising models in the griffiths phase we consider glauber type dynamics for disordered ising spin systems with nearest neighbor pair interactions in the griffiths phase we prove that in a nontrivial portion of the griffiths phase the system has exponentially decaying correlations of distant functions with probability exponentially close to 1 this condition has in turn been shown elsewhere to imply that the convergence to equilibrium is faster than any stretched exponential and that the average over the disorder of the time autocorrelation function goes to equilibrium faster than exp k log t d d 1 we then show that for the diluted ising model these upper bounds are optimal

a constructive mixing condition for 2 d gibbs measures with random interactions for spin systems with random finite range interactions we define an analog of the usual weak mixing property which we call weak mixing in expectation wme this property implies almost sure uniqueness of the gibbs measure we concentrate on the two dimensional case for which we present finite volume conditions which are sufficient for wme we also show the reverse if the system is wme then the condition is satisfied for some sufficiently large volume simultaneously we obtain an extension to random interactions of the result by martinelli olivieri and schonmann that weak mixing implies strong mixing our method is based on a rescaled version of the disagreement percolation approach of van den berg and maes combined with ideas and techniques of gielis and maes and martinelli olivieri and schonmann however apart from some general results on coupling stated in section 2 this paper is self contained

dynamics at a smeared phase transition we investigate the effects of rare regions on the dynamics of ising magnets with planar defects i e disorder perfectly correlated in two dimensions in these systems the magnetic phase transition is smeared because static long range order can develop on isolated rare regions we first study an infinite range model by numerically solving local dynamic mean field equations then we use extremal statistics and scaling arguments to discuss the dynamics beyond mean field theory in the tail region of the smeared transition the dynamics is even slower than in a conventional griffiths phase the spin autocorrelation function decays like a stretched exponential at intermediate times before approaching the exponentially small equilibrium value following a power law at late times