superstatistical generalization of the work fluctuation theorem abstract we derive a generalized version of the work fluctuation theorem for non equilibrium systems with spatio temporal temperature fluctuations for 2 distributed inverse temperature we obtain a generalized fluctuation theorem based on q exponentials whereas for other temperature distributions more complicated formulae arise since q exponentials have a power law decay the decay rate in this generalized fluctuation theorem is much slower than the conventional exponential decay this implies that work fluctuations can be of relevance for the design of micro or nano structures since the work done on the system is relatively much larger than in the conventional fluctuation theorem

convergence of the iterates of descent methods for analytic cost functions in the early eighties lojasiewicz in seminari di geometria 1982 1983 universita di bologna istituto di geometria dipartimento di matematica 1984 pp 115 117 proved that a bounded solution of a gradient flow for an analytic cost function converges to a well defined limit point in this paper we show that the iterates of numerical descent algorithms for an analytic cost function share this convergence property if they satisfy certain natural descent conditions the results obtained are applicable to a broad class of optimization schemes and strengthen classical weak convergence results for descent methods to strong limit point convergence for a large class of cost functions of practical interest the result does not require that the cost has isolated critical points and requires no assumptions on the convexity of the cost nor any nondegeneracy conditions on the hessian of the cost at critical points

superstatistical distributions from a maximum entropy principle we deal with a generalized statistical description of nonequilibrium complex systems based on least biased distributions given some prior information a maximum entropy principle is introduced that allows for the determination of the distribution of the fluctuating intensive parameter beta of a superstatistical system given certain constraints on the complex system under consideration we apply the theory to three examples the superstatistical quantum mechanical harmonic oscillator the superstatistical classical ideal gas and velocity time series as measured in a turbulent taylor couette flow

finite traces and representations of the group of infinite matrices over a finite field abstract the article is devoted to the representation theory of locally compact infinite dimensional group glb of almost upper triangular infinite matrices over the finite field with q elements this group was defined by s k a v and andrei zelevinsky in 1982 as an adequate n analogue of general linear groups gl n q it serves as an alternative to gl q whose representation theory is poor our most important results are the description of semifinite unipotent traces characters of the group glb via certain probability measures on the borel subgroup b and the construction of the corresponding von neumann factor representations of type ii as a main tool we use the subalgebra a glb of smooth functions in the group algebra l 1 glb this subalgebra is an inductive limit of the finite dimensional group algebras c gl n q under parabolic embeddings as in other examples of the asymptotic representation theory we discover remarkable properties of the infinite case which does not take place for finite groups like multiplicativity of indecomposable characters or connections to probabilistic concepts the infinite dimensional iwahori hecke algebra h q plays a special role in our considerations and allows to understand the deep analogy of the developed theory with the representation theory of infinite symmetric group s which had been intensively studied in numerous previous papers

a quantization of the harmonic analysis on the infinite dimensional unitary group abstract the present work stemmed from the study of the problem of harmonic analysis on the infinite dimensional unitary group u that problem consisted in the decomposition of a certain 4 parameter family of unitary representations which replace the nonexisting two sided regular representation olshanski 31 the required decomposition is governed by certain probability measures on an infinite dimensional space which is a dual object to u a way to describe those measures is to convert them into determinantal point processes on the real line it turned out that their correlation kernels are computable in explicit form they admit a closed expression in terms of the gauss hypergeometric function f 1 2 borodin and olshanski 8 in the present work we describe a nonevident q discretization of the whole construction this leads us to a new family of determinantal point processes we reveal its connection with an exotic finite system of q discrete orthogonal polynomials the so called pseudo big q jacobi polynomials the new point processes live on a double q lattice and we show that their correlation kernels are expressed through the basic hypergeometric function 1 2 a crucial novel ingredient of our approach is an extended version g of the gelfand tsetlin graph the conventional graph describes the gelfand tsetlin branching rule for irreducible representations of unitary groups we find the q boundary of g thus extending previously known results gorin 17

note on two theorems in nonequilibrium statistical mechanics an attempt is made to clarify the difference between a theorem derived by evans and searles in 1994 on the statistics of trajectories in phase space and a theorem proved by the authors in 1995 on the statistics of fluctuations on phase space trajectory segments in a nonequilibrium stationary state

fisher info and thermodynamics first law abstract we show starting from first principles that thermodynamics first law can be microscopically obtained for fisher s information measure without need of invoking the adiabatic theorem further it is proved that enforcing the fisher first law requirements in a process in which the probability distribution is infinitesimally varied is equivalent to minimizing fisher s information measure subject to appropriate constraints

markov processes on the path space of the gelfand tsetlin graph and on its boundary we construct a four parameter family of markov processes on infinite gelfand tsetlin schemes that preserve the class of central gibbs measures any process in the family induces a feller markov process on the infinite dimensional boundary of the gelfand tsetlin graph or equivalently the space of extreme characters of the infinite dimensional unitary group u the process has a unique invariant distribution which arises as the decomposing measure in a natural problem of harmonic analysis on u posed in olshanski 2003 44 as was shown in borodin and olshanski 2005 11 this measure can also be described as a determinantal point process with a correlation kernel expressed through the gauss hypergeometric function

generalized superstatistics of nonequilibrium markovian systems abstract the paper is devoted to the construction of the superstatistical description for nonequilibrium markovian systems it is based on kirchhoff s diagram technique and the assumption on the system under consideration to possess a wide variety of cycles with vanishing probability fluxes the latter feature enables us to introduce equivalence classes called channels within which detailed balance holds individually then stationary probability as well as flux distributions are represented as some sums over the channels the latter construction actually forms the superstatistical description which however deals with a certain superposition of equilibrium subsystems rather than a formal expansion of the nonequilibrium steady state distribution into terms of the boltzmann type

the q gelfand tsetlin graph gibbs measures and q toeplitz matrices abstract the problem of the description of finite factor representations of the infinite dimensional unitary group investigated by voiculescu in 1976 is equivalent to the description of all totally positive toeplitz matrices vershik kerov showed that this problem is also equivalent to the description of the simplex of central i e possessing a certain gibbs property measures on paths in the gelfand tsetlin graph we study a quantum version of the latter problem we introduce a notion of a q centrality and describe the simplex of all q central measures on paths in the gelfand tsetlin graph conjecturally q central measures are related to representations of the quantized universal enveloping algebra u gl we also define a class of q toeplitz matrices and show that extreme q central measures correspond to q toeplitz matrices with non negative minors finally our results can be viewed as a classification theorem for certain gibbs measures on rhombus tilings of the halfplane we use a class of q interpolation polynomials related to schur functions one of the key ingredients of our proofs is the binomial formula for these polynomials proved by okounkov

the boundary of the gelfand tsetlin graph a new approach the gelfand tsetlin graph is an infinite graded graph that encodes branching of irreducible characters of the unitary groups the boundary of the gelfand tsetlin graph has at least three incarnations as a discrete potential theory boundary as the set of finite indecomposable characters of the infinite dimensional unitary group and as the set of doubly infinite totally positive sequences an old deep result due to albert edrei and dan voiculescu provides an explicit description of the boundary it can be realized as a region in an infinite dimensional coordinate space the paper contains a novel approach to the edrei voiculescu theorem it is based on a new explicit formula for the number of semi standard young tableaux of a given skew shape or of gelfand tsetlin schemes of trapezoidal shape the formula is obtained via the theory of symmetric functions and new schur like symmetric functions play a key role in the derivation

on the łojasiewicz simon gradient inequality on submanifolds abstract we provide sufficient conditions for the ojasiewicz simon gradient inequality to hold on a submanifold of a banach space and discuss the optimality of our assumptions our result provides a tool to study asymptotic properties of quasilinear parabolic equations with nonlinear constraints

velocity probability density functions of high reynolds number turbulence abstract this paper deals with the probability density function pdf of velocity differences between two points separated by distance r measurements of pdfs were made for r lying in the inertial range for two different flows in a jet with r 852 and in a wind tunnel with r 2720 these pdfs have a characteristic non gaussian shape with exponential tails following kolmogorov s general ideas of log normality a new model for the pdf is developed which contains two parameters determined by experiments this empirical model agrees with the experimental results that the tails of the pdf deviate from a truly exponential behaviour in particular for small r in addition the model leads to the general scaling law ln e 2 r r 0 differenr from kolmogorov s third hypothesis ln e 2 ln r r 0 restricted to the inertial range only x is x x we develop also a formalism based on an extremum principle which is consistent with both the log normality of e and the above mentioned power law in this formalism can be interpreted as the codimension of dissipative structures and asymptotically varies as 1 ln r

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

constructing a statistical mechanics for beck cohen superstatistics the basic aspects of both boltzmann gibbs bg and nonextensive statistical mechanics can be seen through three different stages first the proposal of an entropic functional s bg k sigma i p i ln p i for the bg formalism with the appropriate constraints sigma i p i 1 and sigma i p i e i u for the bg canonical ensemble second through optimization the equilibrium or stationary state distribution p i e betae i z bg with z bg sigma j e betae j for bg third the connection to thermodynamics e g f bg 1 beta ln z bg and u bg partial differential partial differential beta ln z bg assuming temperature fluctuations beck and cohen recently proposed a generalized boltzmann factor b e integral infinity 0 dbetaf beta e betae this corresponds to the second stage described above in this paper we solve the corresponding first stage i e we present an entropic functional and its associated constraints which lead precisely to b e we illustrate with all six admissible examples given by beck and cohen

measuring nonextensitivity parameters in a turbulent couette taylor flow we investigate probability density functions of velocity differences at different distances r measured in a couette taylor flow for a range of reynolds numbers re there is good agreement with the predictions of a theoretical model based on nonextensive statistical mechanics where the entropies are nonadditive for independent subsystems we extract the scale dependent nonextensitivity parameter q r mathrm re from the laboratory data

log concavity of multiplicities with application to characters ofu the log concavity of the reduction multiplicities for the classical groups of typean bn cnis proved moreover the skew schur functionss are shown to be log concave coefficient by coefficient the results are applied to the calculation of the characters of the infinite dimensional classical groups the log concavity of density of the push forward of the liouville measure on coadjoint orbits under moment map is proved

convergence of solutions to the equation of quasi static approximation of viscoelasticity with capillarity the long time behavior of solutions to the quasi steady approximation of the equation of viscoelasticity with capillarity is investigated we consider the equation with a number of boundary conditions including such that the set of equilibria is necessarily at least one dimensional we prove that any solution converges to an equilibrium point we admit some kind of initial data with infinite energy we use tools based on the assumption that the nonlinear term is real analytic

harmonic analysis on the infinite dimensional unitary group and determinantal point processes the infinite dimensional unitary group u is the inductive limit of growing compact unitary groups u n in this paper we solve a problem of harmonic analysis on u stated in ol3 the problem consists in computing spectral decomposition for a remarkable 4 parameter family of characters of u these characters generate representations which should be viewed as analogs of nonexisting regular representation of u the spectral decomposition of a character of u is described by the spectral measure which lives on an infinite dimensional space of indecomposable characters the key idea which allows us to solve the problem is to embed into the space of point configurations on the real line without two points this turns the spectral measure into a stochastic point process on the real line the main result of the paper is a complete description of the processes corresponding to our concrete family of characters we prove that each of the processes is a determinantal point process that is its correlation functions have determinantal form with a certain kernel our kernels have a special integrable form and are expressed through the gauss hypergeometric function from the analytic point of view the problem of computing the correlation kernels can be reduced to a problem of evaluating uniform asymptotics of certain discrete orthogonal polynomials studied earlier by richard askey and peter lesky one difficulty lies in the fact that we need to compute the asymptotics in the oscillatory regime with the period of oscillations tending to 0 we do this by expressing the polynomials in terms of a solution of a discrete riemann hilbert problem and computing the nonoscillatory asymptotics of this solution from the point of view of statistical physics we study thermodynamic limit of a discrete log gas system an interesting feature of this log gas is that its density function is asymptotically equal to the characteristic function of an interval our point processes describe how different the random particle configuration is from the typical densely packed configuration in simpler situations of harmonic analysis on infinite symmetric groups and harmonic analysis of unitarily invariant measures on infinite hermitian matrices similar results were obtained in our papers bo1 bo2 bo4

willmore blowups are never compact we prove a ojasiewicz simon gradient inequality for the willmore functional near willmore surfaces and we apply this inequality to exclude compact blowups for the willmore flow

stationary and transient work fluctuation theorems for a dragged brownian particle recently wang et al carried out a laboratory experiment where a brownian particle was dragged through a fluid by a harmonic force with constant velocity of its center this experiment confirmed a theoretically predicted work related integrated transient fluctuation theorem itft which gives an expression for the ratio for the probability to find positive or negative values for the fluctuations of the total work done on the system in a given time in a transient state the corresponding integrated stationary state fluctuation theorem issft was not observed using an overdamped langevin equation and an arbitrary motion for the center of the harmonic force all quantities of interest for these theorems and the corresponding nonintegrated ones tft and ssft respectively are theoretically explicitly obtained in this paper while the tft and the itft are satisfied for all times the ssft and the issft only hold asymptotically in time suggestions for further experiments with arbitrary velocity of the harmonic force and in which also the issft could be observed are given in addition a nontrivial long time relation between the itft and the issft was discovered which could be observed experimentally especially in the case of a resonant circular motion of the center of the harmonic force

distributions on partitions point processes and the hypergeometric kernel we study a 3 parametric family of stochastic point processes on the one dimensional lattice originated from a remarkable family of representations of the infinite symmetric group we prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel the kernel can be expressed through the gauss hypergeometric function we call it the hypergeometric kernel in a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles as we showed in previous works the correlation functions of these limit processes also have determinantal form with so called whittaker kernel we show that the scaling limit of the hypergeometric kernel is the whittaker kernel integrable operator as defined by its izergin korepin and slavnov we argue that the hypergeometric kernel can be considered as a kernel defining a discrete integrable operator we also show that the hypergeometric kernel degenerates for certain values of parameters to the christoffel darboux kernel for meixner orthogonal polynomials this fact is parallel to the degeneration of the whittaker kernel to the christoffel darboux kernel for laguerre polynomials

the łojasiewicz gradient inequality in the infinite dimensional hilbert space framework abstract we provide a reasonably optimal answer to the natural question of the conditions under which an analytic function on an infinite dimensional hilbert space satisfies the ojasiewicz gradient inequality

the problem of harmonic analysis on the infinite dimensional unitary group abstract the goal of harmonic analysis on a noncommutative group is to decompose the most natural unitary representations of this group like the regular representation on irreducible ones the infinite dimensional unitary group u is one of the basic examples of big groups whose irreducible representations depend on infinitely many parameters our aim is to explain what the harmonic analysis on u consists of we deal with unitary representations of a reasonable class which are in 1 1 correspondence with characters central positive definite normalized functions on u the decomposition of any representation of this class is described by a probability measure called spectral measure on the space of indecomposable characters the indecomposable characters were found by dan voiculescu in 1976 the main result of the present paper consists in explicitly constructing a 4 parameter family of natural representations and computing their characters we view these representations as a substitute of the nonexisting regular representation of u we state the problem of harmonic analysis on u as the problem of computing the spectral measures for these natural representations a solution to this problem is given in the next paper harmonic analysis on the infinite dimensional unitary group and determinantal point processes math 0109194 to appear in ann math joint with alexei borodin we also prove a few auxiliary general results in particular it is proved that the spectral measure of any character of u can be approximated by a sequence of discrete spectral measures for the restrictions of the character to the compact unitary groups u n this fact is a starting point for computing spectral measures

on the closed form solutions for non extensive value at risk abstract we examine possible closed form solutions for the cumulative distribution function for systems where the probability density function can be adequately described by the generalized non extensive statistics framework application to financial time series as a possible value at risk technique indicates reasonable agreement with the data under consideration including all possible extremes and asymmetries of the returns numerical results to illustrate the efficiency of the method are presented

superstatistics in nanoscale electrochemical systems stochastic electrochemical reaction steps on nanosized electrodes are non markovian when externally driven by an applied voltage we show that compared to the markovian case when external driving is absent nanoscale electrochemical systems obey a superstatistics characterized by a superposition of tsallis q indices the distribution of tsallis q indices along stochastic trajectories can be calculated from the electrochemical master equation and normal distributions from boltzmann gibbs thermostatistics are recovered in the thermodynamic limit the infinite electrode size limit although on the nanoscale the external control makes intricate correlations between the microstates in the superstatistical frame one can still address the microstates as if they were uncorrelated the resulting superstatistical entropic form is additive in this frame and tsallis indices have on the time average values which is indeed an example of a superstatistical system where no ad hoc distribution has to be assumed for the fluctuations rather the distribution is directly calculated from a mesoscopic master equation without freely adjustable parameters

asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory we develop a new method for studying the asymptotics of symmetric polynomials of representation theoretic origin as the number of variables tends to infinity several applications of our method are presented we prove a number of theorems concerning characters of infinite dimensional unitary group and their q deformations we study the behavior of uniformly random lozenge tilings of large polygonal domains and find the gue eigenvalues distribution in the limit we also investigate similar behavior for alternating sign matrices equivalently six vertex model with domain wall boundary conditions finally we compute the asymptotic expansion of certain observables in theo n 1 dense loop model

a simple unified approach to some convergence theorems of l simon abstract we give a new and easier proof for the convergences result of l simon 15 concerning global and bounded solutions of a heat equation with analytic nonlinearity by constructing a new liapunov functional for the elliptic equation we are able to prove the second convergence theorem of 15 in the same way as the first theorem the results are more general and can be applied to other problems such as fourth order equation and vector valued systems of gradient type

fluctuations of particle systems determined by schur generating functions abstract we develop a new toolbox for the analysis of the global behavior of stochastic discrete particle systems we introduce and study the notion of the schur generating function of a random discrete configuration our main result provides a central limit theorem clt for such a configuration given certain conditions on the schur generating function as applications of this approach we prove clt s for several probabilistic models coming from asymptotic representation theory and statistical physics including random lozenge and domino tilings non intersecting random walks decompositions of tensor products of representations of unitary groups

the representation ring of the unitary groups and markov processes of algebraic origin abstract the paper consists of two parts the first part introduces the representation ring for the family of compact unitary groups u 1 u 2 this novel object is a commutative graded algebra r with infinite dimensional homogeneous components it plays the role of the algebra of symmetric functions which serves as the representation ring for the family of finite symmetric groups the purpose of the first part is to elaborate on the basic definitions and prepare the ground for the construction of the second part of the paper the second part deals with a family of markov processes on the dual object to the infinite dimensional unitary group u these processes were defined in a joint work with alexei borodin 2012 5 the main result of the present paper consists in the derivation of an explicit expression for their infinitesimal generators it is shown that the generators are implemented by certain second order partial differential operators with countably many variables initially defined as operators on r

convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity abstract we prove convergence of global bounded and smooth solutions of the wave equation with linear dissipation and analytic nonlinearity a generalization and examples of applications are given at the end of the paper

the łojasiewicz simon gradient inequality for open elastic curves abstract in this paper we consider the elastic energy for open curves in euclidean space subject to clamped boundary conditions and obtain the ojasiewicz simon gradient inequality for this energy functional thanks to this inequality we can prove that a suitably reparametrized solution to the associated l 2 gradient flow converges for large time to an elastica that is to a critical point of the functional

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators