self diffusion in a system of interacting langevin particles the behavior of the self diffusion constant of langevin particles interacting via a pairwise interaction is considered the diffusion constant is calculated approximately within a perturbation theory in the potential strength about the bare diffusion constant it is shown how this expansion leads to a systematic double expansion in the inverse temperature beta and the particle density rho the one loop diagrams in this expansion can be summed exactly and we show that this result is exact in the limit of small beta and rhobeta constants the one loop result can also be resummed using a semiphenomenological renormalization group method which has proved useful in the study of diffusion in random media in certain cases the renormalization group calculation predicts the existence of a diverging relaxation time signaled by the vanishing of the diffusion constant possible forms of divergence coming from this approximation are discussed finally at a more quantitative level the results are compared with numerical simulations in two dimensions of particles interacting via a soft potential recently used to model the interaction between coiled polymers

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

perturbation theory for the effective diffusion constant in a medium of random scatterers we develop perturbation theory and physically motivated resummations of the perturbation theory for the problem of a tracer particle diffusing in a random medium the random medium contains point scatterers of density uniformly distributed throughout the material the tracer is a langevin particle subjected to the quenched random force generated by the scatterers via our perturbative analysis we determine when the random potential can be approximated by a gaussian random potential we also develop a self similar renormalization group approach based on thinning out the scatterers this scheme is similar to that used with success for diffusion in gaussian random potentials and agrees with known exact results to assess the accuracy of this approximation scheme its predictions are confronted with results obtained by numerical simulation

the quasi bi hamiltonian formulation of the lagrange top starting from the tri hamiltonian formulation of the lagrange top lt in a six dimensional phase space we discuss the possible reductions of the poisson tensors the vector field and its hamiltonian functions on a four dimensional space we show that the vector field of the lt possesses on the reduced phase space a quasi bi hamiltonian formulation which provides a set of separation variables for the corresponding hamilton jacobi equation

dynamical transition for a particle in a squared gaussian potential we study the problem of a brownian particle diffusing in finite dimensions in a potential given by 2 2 where is gaussian random field exact results for the diffusion constant in the high temperature phase are given in one and two dimensions and it is shown to vanish in a power law fashion at the dynamical transition temperature our results are confronted with numerical simulations where the gaussian field is constructed in a standard way as a sum over random fourier modes we show that when the number of fourier modes is finite the low temperature diffusion constant becomes non zero and has an arrhenius form thus we have a simple model with a fully understood finite size scaling theory for the dynamical transition in addition we analyse the nature of the anomalous diffusion in the low temperature regime and show that the anomalous exponent agrees with that predicted by a trap model

sectional operators new integrable systems and semidirect lie algebras integrable dynamical systems are constructed on semi direct non compact lie algebra so 6 nu 6 where nu 6 is the topological vector space on which the representation of so 6 is realized the construction is done with the help of the sectional operators introduced by trofimov and fomenko 1983 our equations generalize those of a rigid body immersed in a fluid and those of manakov 1976 on so 6 our equations contain the system of two interacting rigid bodies immersed in a fluid as a special case

perturbation theory for effective diffusivity in random gradient flows we investigate a result for the effective diffusivity of particles in a random gradient flow previously obtained by an intuitively plausible renormalization group argument and very accurately verified by numerical simulation we show that to two loop order the result is consistent with a direct perturbation theory calculation to the same order in perturbation theory we also derive a ward identity which guarantees the equality of the ratio of the effective diffusivity to the renormalized coupling with the ratio of the corresponding bare values the invariance of this ratio under renormalization was an important feature of the successful renormalization group calculation

continuum derrida approach to drift and diffusivity in random media by means of rather general arguments based on an approach due to derrida that makes use of samples of finite size we analyse the effective diffusivity and drift tensors in certain types of random media in which the motion of the particles is controlled by molecular diffusion and a local flow field with known statistical properties the power of the derrida method is that it uses the equilibrium probability distribution which exists for each finite sample to compute asymptotic behaviour at large times in the infinite medium in certain cases where this equilibrium situation is associated with a vanishing microcurrent our results demonstrate the equality of the renormalization processes for the effective drift and diffusivity tensors this establishes for those cases a ward identity previously verified only to two loop order in perturbation theory in certain models the technique can also be applied to media in which the diffusivity exhibits spatial fluctuations we derive a simple relationship between the effective diffusivity in this case and that for an associated gradient drift problem that provides an interesting constraint on previously conjectured results

effective permittivity of log normal isotropic random media a method of deriving the effective permittivity epsilon eff of log normal d dimensional isotropic media is developed the approach is based on applying the regularized green function technique for the laplace operator the feasibility of exact determining of the effective permittivity is analysed by deriving epsilon eff at third order in the log permittivity variance it is shown that the 3d effective permittivity is a functional of the spectrum of inhomogeneities and therefore cannot be expressed by a closed formula in particular the formula for the effective permittivity discussed previously by landau and lifshitz 1960 and matheron 1967 deviates from the exact one in the third order term

lipschitz constants and logarithmic sobolev inequality abstract we revisit two results of 3 they are a logarithmic sobolev inequality on r n with lipschitz constants and an expression of lipschitz constants as the limit of a functional by the entropy we have two goals in this paper the first goal is to clarify when the strict inequality holds in this inequality the second goal is to investigate the asymptotic behavior of this functional by the abelian and tauberian theorems of laplace transforms

non holonomic geodesic flows on lie groups and the integrable suslov problem on so 4 we study geodesic flows on lie groups with the left invariant non holonomic constraint in the case of the existence of an invariant measure we find new integrable non hamiltonian systems on so 4 and other six dimensional lie groups

systems of hess appel rot type we construct higher dimensional generalizations of the classical hess appel rot rigid body system we give a lax pair with a spectral parameter leading to an algebro geometric integration of this new class of systems which is closely related to the integration of the lagrange bitop performed by us recently and uses mumford relation for theta divisors of double unramified coverings based on the basic properties satisfied by such a class of systems related to bi poisson structure quasi homogeneity and conditions on the kowalevski exponents we suggest an axiomatic approach leading to what we call the class of systems of hess appel rot type

an optimal logarithmic sobolev inequality with lipschitz constants abstract in this paper we give an optimal logarithmic sobolev inequality on r n with lipschitz constants this inequality is a limit case of the l p logarithmic sobolev inequality of gentil 2003 7 as p as a result of our inequality we show that if a lipschitz continuous function f on r n fulfills some condition then its lipschitz constant can be expressed by using the entropy of f we also show that a hypercontractivity of exponential type occurs in the heat equation on r n this is due to the lipschitz regularizing effect of the heat equation

the use of field theoretic methods for the study of flow in a heterogeneous porous medium the effects of heterogeneities on the steady state flow of a single fluid in a porous medium are examined it is argued that incomplete knowledge of the permeability requires the use of a stochastic model of the system it is shown that the problem may be written as a field theory which allows a perturbation series to be expressed by diagrammatic means this allows the calculation of effective permeability the mean pressure and the pressure variance the method as well as recovering familiar results gives a formal means of improving the approximation and approaching more complex systems

diffusion in periodic correlated random forcing landscapes we study the dynamics of a brownian particle in a strongly correlated quenched random potential defined as a periodically extended with period l finite trajectory of a fractional brownian motion with arbitrary hurst exponent h in 0 1 while the periodicity ensures that the ultimate long time behavior is diffusive the generalised sinai potential considered here leads to a strong logarithmic confinement of particle trajectories at intermediate times these two competing trends lead to dynamical frustration and result in a rich statistical behavior of the diffusion coefficient d l although one has the typical value d rm typ l sim exp beta l h we show via an exact analytical approach that the positive moments k 0 scale like langle d k l rangle sim exp c k beta l h 1 1 h and the negative ones as langle d k l rangle sim exp a k beta l h 2 c and a being numerical constants and beta the inverse temperature these results demonstrate that d l is strongly non self averaging we further show that the probability distribution of d l has a log normal left tail and a highly singular one sided log stable right tail reminiscent of a lifshitz singularity

lower estimates of l norm of gradients for cauchy problems abstract in this paper we consider the cauchy problems for a uniformly parabolic equation and a hamilton jacobi equation our aim is to show the optimality of an upper estimate of the l norm of the gradient of solution to each cauchy problem in order to achieve this aim we investigate a lower estimate of this l norm

classical diffusion in strong random media we study classical diffusion of particles in random media although many of our results are general we focus on the case of an ion in a three dimensional medium with random quenched charge centers obeying bulk charge neutrality within a functional integral framework we calculate the effective diffusion coefficients by first order and second order self consistent perturbation theory with a gaussian reference in both cases we also carry out a one loop order momentum space renormalization group calculation the self consistent methods are complicated numerically and fail beyond intermediate disorder strengths in contrast the renormalization group calculation gives an analytical result that appears valid even to high disorder strengths the methodology generally applicable to a quantitative calculation of effective diffusion coefficients in disordered media resolves deficiencies in self consistent perturbation theory approaches to this class of problems

nonholonomic left and right flows on lie groups nonholonomic systems on lie groups with a left invariant lagrangian and a right invariant nonintegrable distribution are studied a method for construction of the first integrals is given it is applied to three dimensional lie groups and to the four dimensional nonholonomic rigid body motion new integrable problems with an invariant measure are obtained

moser inequalities in gauss space the sharp constants in a family of exponential sobolev type inequalities in gauss space are exhibited they constitute the gaussian analogues of the moser inequality in the borderline case of the sobolev embedding in the euclidean space interestingly the gaussian results have features in common with the euclidean ones but also reveal marked diversities

perturbation schemes for flow in random media we utilize a many body style formalism to compute the long range effective diffusivity kappa e for dispersal of a passive scalar field in a random irrotational flow with molecular diffusivity by comparing the theoretical results with the outcome of computer simulations we show that a renormalization group approach that includes an appropriate vertex renormalization is superior to low order perturbation theory and to the hartree fock approximation and gives a rather accurate method of computing kappa e for a range of parameter values

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

euler poisson equations and existence of quasi periodic motions of heavy tops abstract we study the existence of quasi periodic motions of an asymmetric heavy top by using two different approaches the standard moser weinstein theorem and an extended liapounov center theorem we take advantage of the multiple reversibility of the euler poisson equations and are able to show that the extended liapounov center theorem yields more normal modes than does the moser weinstein theorem

effective diffusivity in non isotropic gradient flows we study the effective diffusivity tensor for particles in a random gradient flow that statistically lacks rotational symmetry the effective diffusivity tensor is computed up to two loop order in perturbation theory and the ward identity that relates this tensor to the effective coupling is verified to the same order we re examine the renormalization group calculation that produced a very accurate numerical value for the effective diffusivity in the rotationally symmetric case and formulate two versions based on distinct divisions of the random potential field that gives rise to the flow both types of renormalization group calculation give good results when compared with numerical simulations however at two loop order in perturbation theory the two methods differ in detail from each other and from the exact perturbation calculation this is in contrast to the corresponding results in the isotropic case

generalizations of classical integrable nonholonomic rigid body systems nonholonomic systems with an invariant measure the suslov chaplygin and veselov veselova problem are considered new families of integrable potential perturbations of these non hamiltonian systems are constructed we also obtain a similar result for a classical hamiltonian system of the motion of a rigid body fixed at a point

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

the effective permeability of a random medium the authors investigate the effective permeability of a heterogeneous random medium by numerical simulation and by self consistent perturbation theory for a class of models including the log normal model for a reasonable range of parameters they find close agreement between the simulation and the perturbative calculation this suggests that both methods may be useful for appropriate models however the simulation approach is widely applicable in circumstances where a perturbation calculation is not possible