Fluctuation susceptibility relations for classical spin systems

We prove identities between integrated Ursell functions and derivatives of the pressure in the thermodynamic limit, for multicomponent classical spin systems which obey the Lee-Yang theorem and some form of Gaussian domination, when the susceptibility is finite (T>T _c). Following Refs. 3 and 4, we view the moment generating function of the magnetization as the inverse of an infinitely divisible characteristic function. Fluctuation susceptibility relations of all orders then follow by bounding the corresponding cumulants, taken in zero external field. High-order cumulants are bounded in terms of the susceptibility using Gaussian and Simon's inequalities for short-range interactions.     

Stochastic Dissipative PDE's and Gibbs Measures

We study a class of dissipative nonlinear PDE's forced by a random force η^{omega( t} , ^x ), with the space variable x varying in a bounded domain. The class contains the 2D Navier–Stokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistical hydrodynamics: they are random fields smooth in t and stationary, short-correlated in time t. In this paper, we confine ourselves to “kick forces” of the form

A Limit Result for a System of Particles in Random Environment

We consider an infinite system of particles in one dimension, each particle performs independent Sinai’s random walk in random environment. Considering an instant t, large enough, we prove a result in probability showing that the particles are trapped in the neighborhood of well defined points of the lattice depending on the random environment, t and the starting points of the particles. Supported by GREFI-MEFI and Departimento di Mathematica, Universita di Roma II “Tor Vergata”, Italy.     

Generation of complete events containing very high-<Emphasis Type="Italic">p</Emphasis><Subscript>T</Subscript> jets

The study of very high transverse-momentum jets will be an important issue at the LHC, in particular since the corresponding cross sections will be considerably larger than at RHIC energies. Jets are expected to provide information on QGP formation, due to the energy loss of fast partons in the medium. Jet cross sections can in principle be compared to simple pQCD calculations, based on the hypothesis of factorization. But often it is useful or even necessary to not only compute the production rate of the very high-p _T jets, but in addition the “rest of the event”. The proposed talk is based on recent work, where we try to construct an event generator—fully compatible with pQCD—which allows one to compute complete events, consisting of high-p _T jets plus all the other low p _T particles produced at the same time. Whereas in “generators of inclusive spectra” like Pythia one may easily trigger on high-p _T phenomena, this is not so obvious for “generators of physical events”, where in principle one has to generate a very large number of events in order to finally obtain rare events (like those with a very high-p _T jet). We shall discuss how we overcome these difficulties in the framework of the EPOS model.     

On the Distribution of Surface Extrema in Several One- and Two-dimensional Random Landscapes

We study here a standard next-nearest-neighbor (NNN) model of ballistic growth on one-and two-dimensional substrates focusing our analysis on the probability distribution function P(M,L) of the number M of maximal points (i.e., local “peaks”) of growing surfaces. Our analysis is based on two central results: (i) the proof (presented here) of the fact that uniform one-dimensional ballistic growth process in the steady state can be mapped onto “rise-and-descent” sequences in the ensemble of random permutation matrices; and (ii) the fact, established in Ref. [G. Oshanin and R. Voituriez, J. Phys. A: Math. Gen. 37:6221 (2004)], that different characteristics of “rise-and-descent” patterns in random permutations can be interpreted in terms of a certain continuous-space Hammersley-type process. For one-dimensional system we compute P(M,L) exactly and also present explicit results for the correlation function characterizing the enveloping surface. For surfaces grown on 2d substrates, we pursue similar approach considering the ensemble of permutation matrices with long-ranged correlations. Determining exactly the first three cumulants of the corresponding distribution function, we define it in the scaling limit using an expansion in the Edgeworth series, and show that it converges to a Gaussian function as L → ∞.     

A Noncommutative de Finetti Theorem: Invariance under Quantum Permutations is Equivalent to Freeness with Amalgamation

We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen “exchangeability” (i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to invariance under the action of the quantum permutation group. More precisely, for an infinite sequence of noncommutative random variables , we prove that invariance of the joint distribution of the x _i ’s under quantum permutations is equivalent to the fact that the x _i ’s are identically distributed and free with respect to the conditional expectation onto the tail algebra of the x _i ’s. Research supported by Discovery and LSI grants from NSERC (Canada) and by a Killam Fellowship from the Canada Council for the Arts.     

Experimental study of boiling-up kinetics and superheat limit for n-hexane on solid powder-like structures

The experimental setup is described and results are presented for measuring average boiling-up lag time for superheated n-hexane mixed with solid structures (activated coal, cellulose, silica gel) as function of temperature under atmospheric pressure. The “aging” of the cell with the filler was carried out before measurements: this was about 600–1000 boiling events. We developed a new method for analysing of “aging” procedure: comparison of average flux (frequency) of boiling-up events (processing of experimental data) and the frequency of nucleation obtained from exponential model. By the end of aging of the cell with silica gel the average empirical flux reduces by factor of four relative to the “exponential” value. But for activated coal and cellulose the difference in these fluxes is about 20 %. In all experiments, the event flux was nonstationary. For n-hexane in tested systems, the margin of superheating was T _n/T _cr ≅ 0.873–0.875, although it was T _n/T _cr ≅ ≅ 0.883 for n-pentane in systems filled by nickel powder (sintered porous nickel with grains of 1.5 or 5.0 micron size) and in the presence of a smooth copper plate. The average time of boiling-up lag in n-hexane at low normalized temperatures was also smaller than for n-pentane. For all systems, the lag time is almost the same for the temperature range T _n/T _cr ≅ 0.860–0.874 (plateau). Thus, a smaller amount of superheated liquid or its division into smaller liquid elements does not result in longer liquid lifetime for superheat liquid and the maximal superheat temperature, as one could expect from the classical theory of homogeneous nucleation. Research was supported by the RF Presidential Foundation (NS-905.2003.2) and Russian Foundation for Basic Research (Grant No. 04-02-16285).     

A comparison of μSR and thermopower in Hg 1:2:0:1 high‐Tc cuprates

We present the results of μSR measurements of the transverse field relaxation rate below T_c for seven samples of HgBa₂ CuO_4+\delta, spanning the underdoped, optimally doped, and nominally overdoped regions of the superconducting phase. On comparison with the “anomalous” thermopower data for these compounds, we find that nominally “overdoped” Hg 1:2:0:1 resembles underdoped compounds in certain respects. This revised version was published online in August 2006 with corrections to the Cover Date.     

Superconducting triplet spin valve

We study the critical temperature T _c of SFF trilayers (S is a singlet superconductor, F is a ferromagnetic metal), where the long-range triplet superconducting component is generated at noncollinear magnetizations of the F layers. We demonstrate that T _c can be a nonmonotonic function of the angle α between the magnetizations of the two F layers. The minimum is achieved at an intermediate α, lying between the parallel (P, α = 0) and antiparallel (AP, α = π) cases. This implies a possibility of a “triplet” spin-valve effect: at temperatures above the minimum T _c ^Tr but below T _c ^P and T _c ^AP, the system is superconducting only in the vicinity of the collinear orientations. At certain parameters, we predict a reentrant T _c (α) behavior. At the same time, considering only the P and AP orientations, we find that both the “standard” (T _c ^P < T _c ^AP) and “inverse” (T _c ^P > T _c ^AP) switching effects are possible depending on parameters of the system.     

Experimental determination of superheated liquids boiling-up expectancy time probabilities distribution

Results of statistical measurement of n-pentane and n-hexane boiling-up expectancy time near the boundary of attainable superheating are presented. Experiments were carried out in glass capillaries with various volumes of superheated liquid. Several samples with the volume from 100 to 200 measurements of life time for preset metastable state have been obtained (p, T = const). Their histograms contain small empty initial section, maximum and long “tail” in the area of large times. Non-monotonous dependence of probability distribution density on time proves non-stationary character of the random process resulting in the production of supercritical embryo. Two simple approximations of non-stationary nucleation flow well describing experimental data have been considered. For exponential distribution, the probabilities of experimentally found peculiarities of boiling-up expectancy time distribution density have been evaluated; they prove incompatibility of this distribution with the experimental one.     

Prevalent Dynamics at the First Bifurcation of Hénon-like Families

We study the dynamics of strongly dissipative Hénon-like maps, around the first bifurcation parameter a* at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We prove that a* is a full Lebesgue density point of the set of parameters for which Lebesgue almost every initial point diverges to infinity under positive iteration. A key ingredient is that a* corresponds to the “non-recurrence of every critical point”, reminiscent of Misiurewicz parameters in one-dimensional dynamics. Adapting on the one hand Benedicks & Carleson’s parameter exclusion argument, we construct a set of “good parameters” having a* as a full density point. Adapting Benedicks & Viana’s volume control argument on the other, we analyze Lebesgue typical dynamics corresponding to these good parameters.     

Features of the melting dynamics of a vortex lattice in a high-T c superconductor in the presence of pinning centers

The phase transition “triangular lattice-vortex liquid” in layered high-T _c superconductors in the presence of pinning centers is studied. A two-dimensional system of vortices simulating the superconducting layers in a high-T _c Shubnikov phase is calculated by the Monte Carlo method. It was found that in the presence of defects the melting of the vortex lattice proceeds in two stages: First, the ideal triangular lattice transforms at low temperature (≃3 K)into islands which are pinned to the pinning centers and rotate around them and then, at a higher temperature (≃8 K for T _c 584 K), the boundaries of the “islands” become smeared and the system transforms into a vortex liquid. As the pinning force increases, the temperatures of both phase transitions shift: The temperature of the point “triangular lattice-rotating lattice” decreases slightly (to ≃2 K)and the temperature of the phase transition “rotating lattice-vortex liquid” increases substantially (≃70 K). Pis’ma Zh. éksp. Teor. Fiz. 66, No. 4, 269–274 (25 August 1997)     

Globules of annealed amphiphilic copolymers: Surface structure and interactions

A mean-field theory of globules of random amphiphilic copolymers in selective solvents is developed for the case of an annealed copolymer sequence: each unit can be in one of two states, H (insoluble) or P (soluble or less insoluble). The study is focussed on the regime when H and P units tend to form long blocks, and when P units dominate in the dilute phase, but are rare in the globule core. A first-order coil-to-globule transition is predicted at some T = T _cg. The globule core density at the transition point increases as the affinity of P units to the solvent, ˜, is increased. Two collapse transitions, coil → “loose” globule and “loose” globule → “dense” globule, are predicted if ˜ is high enough and P units are marginally soluble or weakly insoluble. H and P concentration profiles near the globule surface are obtained and analyzed in detail. It is shown that the surface excess of P units rises as ˜ is increased. The surface tension decreases in parallel. Considering the interaction between close enough surfaces of two globules, we show that they always attract each other at a complete equilibrium. It is pointed out, however, that such equilibrium may be difficult to reach, so that partially equilibrium structures (defined by the condition that a chain forming one globule does not penetrate into the core of the other globule) are relevant. It is shown that at such partial equilibrium the interaction is repulsive, so the globules may be stabilized from aggregation. The strongest repulsion is predicted at the coil-to-globule transition point T _cg: the repulsion force decreases with the distance between the surfaces according to a power law. In the general case (apart from T _cg) the force vs. distance decay becomes exponential; the decay length ξ diverges as T → T _cg. The developed theory explains certain anomalous properties observed for globules of amphiphilic homopolymers.     

Brownian rotation of classical spins: dynamical equations for non-bilinear spin-environment couplings

Given two strings X and Y of N and M characters respectively, the Longest Common Subsequence (LCS) Problem asks for the longest sequence of (non-contiguous) matches between X and Y. Using extensive Monte-Carlo simulations for this problem, we find a finite size scaling law of the form for the average LCS length of two random strings of size N over S letters. We provide precise estimates of for .We consider also a related Bernoulli Matching model where the different entries of an array are occupied with a match independently with probability 1/S. On the basis of a cavity-like analysis we find that the length of a longest sequence of matches in that case behaves as where r=M/N and . This formula agrees very well with our numerical computations. It provides a very good approximation for the Random String model, the approximation getting more accurate as S increases. The question of the “universality class” of the LCS problem is also considered. Our results for the Bernoulli Matching model show very good agreement with the scaling predictions of [#!HwaLassig96_PRL!#] for Needleman-Wunsch sequence alignment. We find however that the variance of the LCS length has a scaling different from Var in the Random String model, suggesting that long-ranged correlations among the matches are relevant in this model. We finally study the “ground state” properties of this problem. We find that the number of solutions typically grows exponentially with N. In other words, this system does not satisfy “Nernst's principle”. This is also reflected at the level of the overlap between two LCSs chosen at random, which is found to be self averaging and to approach a definite value q _S <1 as . Received: 23 April 1998 / Revised: 30 July 1998 / Accepted: 14 August 1998     

Three-Point Functions for MN/SN Orbifolds¶with?= 4 Supersymmetry

The D1–D5 system is believed to have an “orbifold point” in its moduli space where its low energy theory is a ?=4 supersymmetric sigma model with target space M ^N /S _N , where M is T ⁴ or K3. We study correlation functions of chiral operators in CFTs arising from such a theory. We construct a basic class of chiral operators from twist fields of the symmetric group and the generators of the superconformal algebra. We find explicitly the 3-point functions for these chiral fields at large N; these expressions are “universal” in that they are independent of the choice of M. We observe that the result is a significantly simpler expression than the corresponding expression for the bosonic theory based on the same orbifold target space. Received: 29 March 2001 / Accepted: 20 January 2002     

Higher derivative corrections in holographic Zamolodchikov–Polchinski theorem

We study higher derivative corrections in holographic dual of Zamolodchikov–Polchinski theorem that states the equivalence between scale invariance and conformal invariance in unitary d-dimensional Poincaré invariant field theories. From the dual holographic perspective, we find that a sufficient condition to show the holographic theorem is the generalized strict null-energy condition of the matter sector in effective (d+1)-dimensional gravitational theory. The same condition has appeared in the holographic dual of the “c-theorem” and our theorem suggests a deep connection between the two, which was manifested in two-dimensional field theoretic proof of the both.     

Fluctuations of Power Injection in Randomly Driven Granular Gases

We investigate the large deviation function π_∞(w) for the fluctuations of the power W(t) = wt, integrated over a time t, injected by a homogeneous random driving into a granular gas, in the infinite time limit. Our analytical study starts from a generalized Liouville equation and exploits a Molecular Chaos-like assumption. We obtain an equation for the generating function of the cumulants μ(λ) which appears as a generalization of the inelastic Boltzmann equation and has a clear physical interpretation. Reasonable assumptions are used to obtain μ(λ) in a closed analytical form. A Legendre transform is sufficient to get the large deviation function π_∞(w). Our main result, apart from an estimate of all the cumulants of W(t) at large times t, is that π_∞ has no negative branch. This immediately results in the inapplicability of the Gallavotti-Cohen Fluctuation Relation (GCFR), that in previous studies had been suggested to be valid for injected power in driven granular gases. We also present numerical results, in order to discuss the finite time behavior of the fluctuations of W (t) . We discover that their probability density function converges extremely slowly to its asymptotic scaling form: the third cumulant saturates after a characteristic time τ larger than ∼50 mean free times and the higher order cumulants evolve even slower. The asymptotic value is in good agreement with our theory. Remarkably, a numerical check of the GCFR is feasible only at small times (at most τ/10), since negative events disappear at larger times. At such small times this check leads to the misleading conclusion that GCFR is satisfied for π_∞(w). We offer an explanation for this remarkable apparent verification. In the inelastic Maxwell model, where a better statistics can be achieved, we are able to numerically observe the “failure” of GCFR.     

Random Walk on the Range of Random Walk

We study the random walk X on the range of a simple random walk on ℤ ^d in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin.     

Random Walk Weakly Attracted to a Wall

We consider a random walk X _n in ℤ₊, starting at X ₀=x≥0, with transition probabilities

Casimir amplitudes in a quantum spherical model with long-range interaction

A d-dimensional quantum model system confined to a general hypercubical geometry with linear spatial size L and “temporal size” 1/T ( T - temperature of the system) is considered in the spherical approximation under periodic boundary conditions. For a film geometry in different space dimensions , where is a parameter controlling the decay of the long-range interaction, the free energy and the Casimir amplitudes are given. We have proven that, if , the Casimir amplitude of the model, characterizing the leading temperature corrections to its ground state, is . The last implies that the universal constant of the model remains the same for both short, as well as long-range interactions, if one takes the normalization factor for the Gaussian model to be such that . This is a generalization to the case of long-range interaction of the well-known result due to Sachdev. That constant differs from the corresponding one characterizing the leading finite-size corrections at zero temperature which for is . Received 3 June 1999 and Received in final form 16 August 1999