Workshop on Stochastic Processes in Random Media
(4 - 15 May 2015)
## ~ Abstracts ~
Mark Adler, Brandeis University, USAWe study lozenge tiling where both the top and bottom boundary is irregular, and derive a correlation kernel for 2 associated processes and the inverse Kastelyn matrix. This should lead, upon taking appropriate limits to some interesting processes. The derivations required some new machinery, itself of interest.
Jinho Baik, University of Michigan, USAWe consider the spherical spin glass model, also known as the spherical Sherrington-Kirkpatrick (SSK) model. We show that the limit of the law of the fluctuations of the free energy is given by the GOE Tracy-Widom distribution in the low temperature regime. A universality for non-Gaussian interactions is also considered. This is a joint work with Ji Oon Li.
Adrian Baule, Queen Mary University of London, UKA fundamental understanding of the dynamics of systems under the influence of thermal fluctuations is provided by investigating the large deviation properties in the limit of weak noise strength. This approach has provided, for example, theories of activated escape in low temperature regimes and is also intimately linked to the description of quantum mechanical systems within a semiclassical approximation. However, many complex systems are driven by non-thermal (active) fluctuations with non-Gaussian characteristics. Using a path-integral approach we investigate the weak-noise limit of such systems for a general class of non-Gaussian fluctuations. We define a scaling limit that identifies the optimal paths of the dynamics as minima of a stochastic action, while retaining the infinite hierarchy of noise cumulants. We apply this approach to the paradigmatic problem of noise-induced escape from a metastable potential. Exact results for the large deviation asymptotics of Kramer's escape rate and the optimal escape paths are obtained.
Federico Camia, Vrije Universiteit, The NetherlandsStarting from two-dimensional Brownian motion, I will discuss the construction of a family of fields (generalized functions or Schwartz distributions) with interesting properties of conformal covariance. Similar properties are known or expected to be enjoyed by fields obtained in the critical or near-critical scaling limit of various models of statistical mechanics. (This talk is based on joint work with Alberto Gandolfi and Matthew Kleban, and with Marcin Lis.)
Francesco Caravenna, University of Milano-Bicocca, ItalyWe consider statistical mechanics models on a lattice, in which disorder acts as an external random field. Examples include disordered pinning models, directed polymer models and the random field Ising model. We present a unified approach to study the continuum and weak disorder scaling limits of such models, in the so-called "relevant disorder" regime. Our approach relies on some general results of independent interest, providing conditions for a sequence of multi-linear polynomials of independent random variables to converge in distribution, based on a Lindeberg principle. Recent progress has been made in the interesting "marginal disorder" regime, where the standard approach breaks down. A finer analysis shows that in this case the partition function admits an explicit log-normal limiting distribution, which is universal across different models. Connections with the 2d stochastic heat equation will be discussed. (Joint work with Rongfeng Sun and Nikos Zygouras)
Kuldeep Chaudhary, Amity University, IndiaTo depicting and predicting adoption curve for products in different sectors of market and economy, Bass's innovation-diffusion model and its extended forms have been applied successfully. All these models assume the adoption process as a discrete counting process. However, if the potential adopter population is large and product is in the market with greater life cycle length, it is quite likely that adoption process is a stochastic process with continuous state space. In this talk, we discuss an innovation and diffusion model based on type of stochastic differential equation. It also incorporates the change-point concept, where the rate of product adoption per remaining potential adopter might change due shift in marketing/promotional strategy, entry/exit of some of the competitors in the market. The applicability and accuracy of the proposed model are illustrated using new product sales data. Predictive validity and mean squared error have been used to check the validity of the model. It has been shown that SDE-based model with change point performs comparatively better than Bass's innovation- diffusion model.
Yang Chen, University of Macau, ChinaThe generating function of a singular linear statistics is computed in the background of the Laguerre unitary ensembles. Under a double scaling, where n the "size" of the ensemble tends to infinity and a parameter t tends to 0, such that 2nt=:s is finite, is shown to be another Painleve III with smaller number of parameters . Asymptotic expansions for small and large s are obtained
David S Dean, Université Bordeaux, FranceThe effect of a constant applied external force, induced for instance by an electric or gravitational field, on the dispersion of Brownian particles in periodic media with spatially varying diffusivity, and thus mobility, is studied. We show that external forces can greatly enhance dispersion in the direction of the applied force and also modify, to a lesser extent and in some cases non-monotonically, dispersion perpendicular to the applied force. These results suggest the possibility of modulating the dispersive properties of heterogeneous media by using externally applied force fields. The results are obtained via a Kubo formula which can be applied to any periodic advection diffusion system in any spatial dimension.
Frank den Hollander, University of Leiden, The NetherlandsWe describe the metastable behaviour of the Widom-Rowlinson model on a finite two-dimensional box subject to a stochastic dynamics in which particles are randomly created and annihilated inside the box according to an infinite reservoir with a given chemical potential. The particles are viewed as points carrying disks and the energy of a particle configuration is equal to minus the volume of the total overlap of the disks. Consequently, the interaction between the particles is attractive. We are interested in the metastable behaviour of the system at low temperature when the chemical potential is supercritical. In particular, we start with the empty box and are interested in the first time when the box is fully covered by disks. In order to achieve the transition from empty to full, the system needs to create a sufficiently large droplet, called critical droplet, which triggers the crossover. We compute the distribution of the crossover time, identify the size and the shape of the critical droplet, and investigate how the system behaves on its way from empty to full. Joint work with Sabine Jansen (Bochum), Roman Kotecky (Prague/Warwick), Elena Pulvirenti (Leiden).
Victor Dotsenko, Université Pierre et Marie Curie, FranceThe problem of randomly forced Burgers turbulence is considered in terms of the one-dimensional directed polymers model. For the particular Gaussian model using the replica technique the exact result for the two-time velocity distribution function is derived.
Massimiliano Esposito, Université de Luxembourg, LuxembourgI will present a systematic procedure to study the thermodynamics of chemical networks made of coupled elementary reactions satisfying mass action law. Convergence to equilibrium is proved in closed networks. In open networks (i.e. networks where concentrations of some of the species called chemostats are hold fixed), assuming the existence of a steady state, I will demonstrate that the number of independent fluxes and forces constituting the entropy production of the network depends on the topology of the network. The main result is that the number of forces is equal to the number of chemostats minus the number of symmetries broken by adding the chemostats. Applications to the study of polymerization will be provided and the importance of the results for metabolic modelling will be discussed. Reference: M. Polettini and M. Esposito, "Irreversible thermodynamics of open chemical networks I : Emergent cycles and broken conservation laws", J. Chem. Phys. 141, 024117 (2014).
Patrik Ferrari, Universität Bonn, GermanyIn this talk we will present results on the fluctuation of the free energy of two directed polymer models. One is the O'Connell-Yor semidiscrete directed polymer and the second is the universal scaling limit aka continuous directed random polymer. These are at positive temperature, where determinantal structures are lost, but still there is a reasonably nice integrable structure in the background allowing them to be solved. The work is based on joint works with Alexei Borodin, Ivan Corwin and Balint Veto.
Cristian Giardina, University of Modena and Reggio Emilia, ItalySeveral models of non-equilibrium statistical mechanics can be studied by means of stochastic duality. This includes both (symmetric) systems in contact with reservoirs and (asymmetric) systems with bulk driving. In both setting I will show that the existence of a dual process is due to an underlying algebraic structure given by a Lie algebra (in the symmetric case) and its quantum deformation (in the asymmetric case). Joint work with G. Carinci, F. Redig, T. Sasamoto.
Mark Holmes, University of Auckland, New ZealandWe study the asymptotic behaviour of random walks in i.i.d. non-elliptic random environments on Z^d. We prove ballisticity, and a functional central limit theorem for specific models of interest, where the usual methods do not immediately apply. (Joint work with Tom Salisbury.)
Chien-Hao Huang, Academia SInica, TaiwanWe consider a Hamiltonian involving the range of the simple random walk and the Wiener sausage so that the walk tends to stretch itself. This Hamiltonian can be easily extended to the multidimensional cases, since the Wiener sausage is well-defined in any dimension. In dimension one, we give a formula for the speed and the spread of the endpoint of the polymer path. It can be easily showed that if the self-repelling strength is stronger, the end point is going away faster. This monotonicity of speed has not been proven in the literature for the one-dimensional case.
Pierre Illien, Université Pierre et Marie Curie, FranceWe study a minimal model of active transport in a dense environment. We consider a discrete system in which a tracer particle performs a biased random walk in a bath of particles performing symmetric random walks with exclusion interactions. In confined geometries and in the high-density limit, an analytical calculation of the fluctuations of the tracer particle position predicts a long-lived superdiffusive regime and a crossover towards an ultimate diffusive regime. We show that this observation is associated to a velocity anomaly in quasi-one-dimensional systems (such as stripes or capillaries): the velocity of the tracer particle displays a long plateau before reaching an ultimate lower value. Finally, we study the case of one-dimensional systems, which is related to the well-known single-file diffusion.
Alberto Imparato, University of Aarhus, DenmarkMolecular motors work in environments at constant temperature, differently from, e.g., heat engines, whose efficiency is bounded by Carnot's law. Thus, the efficiency of molecular motors is constrained by the thermodynamic limit 1, which can, however, only be achieved in the limitof vanishing output power. It is more interesting, therefore, to understand the behavior of their efficiency as a function of their output power, and in particular their efficiency at maximum power (EMP). Furthermore, in a many motor system, the motor-motor interaction can lead to an increase in EMP with respect to the single motor case. I will first consider the case of motor traffic on linear lattices, and on networks. I will then consider the case of coupled motors, as modelled, for example, by the Kuramoto model.
Oliver Jovanovski, University of Leiden, The NetherlandsWe look at the Ising model set on random graphs that are based on the configuration model. Our interest is focused on properties of these graphs that allow us to describe the occurrence of metastability. This is based on an ongoing project with Frank den Hollander, Francesca Nardi and Sander Dommers.
Minnie Kabra, Indian Institute of Technology, IndiaMy work describes a way to use Machine Learning techniques to predict maliciousness of a process running on a system. Since we have a lot of data (malicious as well as non-malicious samples), we used supervised learning techniques to train a classifier. We have 270 feature vectors in our data. In order to reduce it to its most important features, I calculated statistical correlation between different features using Pearson product-moment correlation coefficient and Spearman's Rank Coefficient. Based on the results of correlation, I used Decision tree classifier and Random forest classifier on training data. Cross validation is being used to estimate the parameters of the classifier. We have improved accuracy by 30% in False Negative data and 75% accuracy in False Positive data.
David A. Kessler, Bar Ilan, IsraelWe investigate the motion of atoms undergoing Sisyphus cooling, so that the momentum performs a random walk with a weakening nonlinear friction. We show that the atomic positions show anomalous diffusion, described by a Levy alpha-stable distribution. This diffusion is related to the area under a Bessel excursion. The dynamics shows ageing behavior, which can be described by a generalized Green-Kubo formula.
Konstantin Khanin, University of Toronto, CanadaWe shall discuss a connection between a problem of global solutions to the random Hamilton-Jacobi equation in dimension 1+1 and KPZ universality. We shall also discuss an analogue of directed polymers in the case of convex non-quadratic Hamiltonians and corresponding stochastic flows.
Katja Lindenberg, University of California at San Diego, USAD. Campos (Grupo de F'sica Estad'stica, Departmament de F'sica, Facultat de Ciencias, Universitat Aut-noma de Barcelona, 08193 Bellaterra, Barcelona, Spain) E. Abad (Departamento de F'sica Aplicada and Instituto de Computaci-n Cient'fica Avanzada (ICCAEX), Centro Universitario de MZrica, Universidad de Extremadura, E-06800, MZrica, Spain) V. MZndez (Grupo de F'sica Estad'stica, Departmament de F'sica, Facultat de Ciencias, Universitat Aut-noma de Barcelona, 08193 Bellaterra, Barcelona, Spain) S. B. Yuste (Departamento de F'sica and Instituto de Computaci-n Cient'fica Avanzada (ICCAEX), Universidad de Extremadura, E-06071, Badajoz, Spain) K. Lindenberg (Department of Chemistry and Biochemistry, and BioCircuits Institute, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0340, USA We present a simple paradigm for detection of an immobile target by a L\'evy walker with a finite lifetime. The motion of the walkerr is characterized by linear displacements at a fixed speed and exponentially distributed duration, interrupted by random changes in the direction of motion and resumption of motion in the new direction with the same speed. The mortal walker may die at any time during its motion according to an exponential decay law characterized by a finite mean death rate $\omega_m$. While still alive, the creeper has a finite mean frequency $\omega$ of change of the direction of motion. In particular, we consider the efficiency of the target search process, characterized by the probability that the creeper will eventually detect the target. Analytic results confirmed by numerical results show that there is an $\omega_m$-dependent optimal frequency $\omega=\omega_{opt}$ that maximizes the probability of eventual target detection. We further consider the survival probability of the target in the presence of many independent walkers beginning their motion at the same location and at the same time. We also consider a version of the standard ``target problem" in which many walkers start at random locations at the same time.
Satya Majumdar, Université de Paris-Sud, FranceConvex hull of a set of points in two dimensions roughly describes the shape of the set. In this talk, I will discuss the statistical properties of the convex hull for two stochastic processes in two dimensions: (i) a set of n independent planar Brownian paths (ii) a branching Brownian motion with death. We show how to compute exactly the mean perimeter and the mean area of the convex hull in these two problems. The first problem has application in estimating the home range of an animal population of size n, while the second will be used to estimate the spatial extent of the outbreak of animal epidemics. Our result also makes an interesting connection between random geometry and extreme value statistics.
Carlos Mejia-Monasterio, Technical University of Madrid, SpainA particle driven by a constant external force through a quiescent bath creates a non equilibrium inhomonegeity around it: the bath particles accumulate in front of it and are depleted behind. Using a stochastic lattice model of particles with exclusion we show that such inhomonegeity produces an attractive force between driven particles mediated by the bath and the external force. When several driven particles are present they cluster and remain together for a time that depends on the size of the cluster. The drift velocity of the cluster's centre of mass, their life time and the fluctuation of the cluster size are studied.
Charles Newman, New York University, USAIn joint work with M Damron, S. M. Eckner, H. Kogan and V. Sidoravicius (see arXiv.1410.0619), we have studied the coarsening (or zero-temperature stochastic Ising) model on ${\bf Z}^d$ with the modification that there is a quenched random environment in which some sites are frozen as either +1 or -1. Other sites update in continuous time to agree with a majority of neighbors, or flip a fair coin in case of a tie. One of our results is that with no frozen -1 sites and an arbitrarily small, but positive, density of frozen +1 sites, all sites eventually become fixed at +1. The main tool for +this and related theorems is the use of known bootstrap percolation results.
Carlos Perez-Espigares, University of Modena and Reggio Emilia, ItalyFor systems of interacting particles and for interacting diffusions in d dimensions, driven out-of- equilibrium by an external field, a fluctuation relation for the generating function of the current is derived as a consequence of spatial symmetries. Those symmetries are in turn associated to transformations on the physical space that leave invariant the path space measure of the system without driving. This shows that in dimension d >= 2 new fluctuation relations arise beyond the Gallavotti-Cohen fluctuation theorem related to the time-reversal symmetry.
Nicolas Petrelis, University of Nantes, FranceIn this talk, we will consider a model for a 1+1 dimensional self-interacting and partially directed self-avoiding walk, usually referred to as IPDSAW. The IPDSAW is known to undergo a collapse transition at some critical temperature and had been studied until recently with combinatorics techniques exclusively. We will present here a new method that provides a probabilistic representation of the partition function and allows us to push forward the investigation of the model. For instance, we will provide the precise asymptotic of the free energy close to criticality and we will establish some path properties of the random walk inside the collapsed phase, that is we will show that the geometric conformation adopted by the polymer is made of a succession of long vertical stretches that attract each other to form a unique macroscopic bead, whose upper and lower envelopes, once properly rescalled, converge to a deterministice Wulff shape (joint work with Philippe Carmona and Gia boa Nguyen).
Alberto Rosso, Université de Paris-Sud, FranceDynamic Nuclear Polarization (DNP) is to date the most effective technique to increase the nuclear polarization up to a factor $100,000$ opening disruptive perspectives for medical applications. In DNP, the nuclear spins are driven to an - out of equilibrium - hyperpolarized state by microwave saturation of the electron spins in interaction with them. Here we show that the electron dipolar interactions compete with the local magnetic fields resulting in two distinct dynamical phases: for strong interactions the electron spins equilibrate to an extremely low effective temperature that boosts DNP efficiency. For weak interaction this spin temperature is not defined and the polarisation profile has an 'hole burning' shape characteristic of the non interacting case. The study of the many-body eigenstates reveals that these two phases are intimately related to the problem of thermalization in closed quantum systems where breaking of ergodicity is expected varying the strength of the interactions.
Akira Sakai, Hokkaido University, JapanThe lattice $\varphi^4$ model is a scalar field-theoretical model that is known to exhibit a phase transition. It is believed to be in the same universality class as Ising ferromagnets. In fact, we can construct the $\varphi^4$ model as the large-$N$ limit of the sum of $N$ Ising systems (with the right scaling of spin-spin couplings). Using this Griffiths-Simon construction and applying the lace expansion for the Ising model, we can prove mean-field asymptotic behavior for the critical $\varphi^4$ two-point function in dimensions higher than the upper-critical dimension. For the case of finite-range spin-spin couplings, the mean-field asymptotic behavior is Newtonian, and the upper-critical dimension is 4. I will explain the key ideas of the proof and discuss extension of the results to the case of power-law decaying spin-spin couplings.
Deepak Sanjel, Minnesota State University, USASeveral methods of analyzing and modeling extreme events are proposed in the literature. However, most of the proposed methods are based on the assumption of extreme value limit distributions or some related family of distributions. We purpose Bayesian approach using Markov chain Monte Carlo method to analyze extreme events and used time series statistical models for prediction.
Tomohiro Sasamoto, Tokyo Institute of Technology, JapanFor the Gaussian unitary ensemble (GUE), it is well known that the eigenvalues are determinantal because the probability density of the eigenvalues is written in the form of two determinants. For the O'Connell-Yor(OY) polymer model, a generating function of the partition function can be written as a Fredholm determinant but the underlying determinant structure is not well understood. We discuss a determinantal structure associated with the OY polymer model, which is related to the above-mentioned Freholm determinant and has a generalization of Warren's Brownian motion in the Gelfand-Tsetlin cone. This is based on a collaboration with T. Imamura.
Grégory Schehr, Université de Paris-Sud, FranceWe consider the system of $N$ one-dimensional free fermions confined by a harmonic well $V(x) = m\omega^2 {x^2}/{2}$ at finite inverse temperature $\beta = 1/T$. The average density of fermions $\rho_N(x,T)$ at position $x$ is derived. For $N \gg 1$ and $\beta \sim {\cal O}(1/N)$, $\rho_N(x,T)$ is given by a scaling function interpolating between a Gaussian at high temperature, for $\beta \ll 1/N$, and the Wigner semi-circle law at low temperature, for $\beta \gg N^{-1}$. In the latter regime, we unveil a scaling limit, for $\beta {\hbar \omega}= b N^{-1/3}$, where the fluctuations close to the edge of the support, at $x \sim \pm \sqrt{2\hbar N/(m\omega)}$, are described by a limiting kernel $K^{\rm ff}_b(s,s')$ that depends continuously on $b$ and is a generalization of the Airy kernel, found in the Gaussian Unitary Ensemble of random matrices. Remarkably, exactly the same kernel $K^{\rm ff}_b(s,s')$ arises in the exact solution of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions at finite time $t$, with the correspondence {$t= b^3$}.
Jan Swart, Academy of Sciences of the Czech Republic, Czech RepublicIn this talk, we will take a look at some systems of interacting particles on the real line, where the only spatial structure that is relevant for the dynamics is the relative order of the particles. Examples of such systems are the modified Bak-Sneppen model, introduced (as a variation of the original 1993 model) by Meester and Sarkar (2012), Barabási's (2005) queueing system and a variation on the latter due to Gabrielli and Caldarelli (2009), a model for the evolution of the state of an order book on a stock market, introduced by Stigler (1964) and independently by Luckock (2003), and a two models for canyon formation introduced by me (2014). All these systems employ a version of the rule "kill the lowest particle" and seem to exhibit self-organized criticality at a critical point that marks the boundary between an interval where all particles are eventually removed and an interval where particle stay in the system forever.
Julian Talbot, Université Pierre et Marie Curie, FranceImagine a flimsy bridge conveying vehicular traffic. Let us assume that if more than a certain number of cars are on the bridge at the same time, it will collapse. If we have some information about the arrival statistics, we can ask what is the survival probability and related questions such as the average time before failure and the number of cars that successfully traverse the bridge before this happens. This is a paradigmatic example of systems where blockage or failure occurs when the carrying capacity of a channel is exceeded. Other examples include filtration, traffic flow, granular materials and the flow of macromolecules in biological and artificial channels. More generally, we are considering processes where a change in the state of the system is triggered by the occurrence of a certain number of events in a given time interval. The simplest model for concurrent flow considers particles that arrive at the entrance of a channel according to a homogeneous Poisson process. A single particle exits the channel after a transit time $\tau$. If, however, N=2 particles are simultaneously present in the channel, blockage occurs instantaneously and the flow ceases. We consider various generalizations of the basic model including counterflowing particulate streams, blocking of finite duration, requiring N>2 particles to trigger blocking, as well as an inhomogeneous flux of entering particles.
Niccolò Torri, University of Milano-Biccoca, Italy and University of Lyon, FranceWe study pinning models, i.e. we consider discrete Markov chains and perturb their behavior through a privileged interaction with a distinguished state. The return time to this state has a polynomial decay with tail exponent $\alpha \in (1/2, 1)$. The interaction depends on an external and independent source of randomness, called disorder, which can attract or repel the Markov chain path. This can produce concentration and localization phenomena of the Markov chain path around the distinguished state, leading to a phase transition. Inspired by [Caravenna, Sun and Zygouras, J. Eur. Math. Soc. (to appear)] we prove that the behavior of the pinning models in the weak coupling regime is universal. To be more precisely, we show that the free energy and critical phase transition of discrete pinning models, suitably rescaled, converge to the analogous quantities of related continuum models. This is obtained by a subtle coarse-graining procedure, which generalizes and refines [Bolthausen and den Hollander, Ann. Probab. 25 (1997), 1334-1366] and [Caravenna and Giacomin, Ann. Probab. 38 (2010), 2322-2378]. Joint work with Francesco Caravenna and Fabio Toninelli.
Pierre van Moerbeke, Université Catholique de Louvain, BelgiumI will discuss the surprising connections between combinatorial questions related to domino tilings and the spectrum of random matrices. References: 1. M. Adler, K. Johansson, P. van Moerbeke: Double Aztec diamonds and the tacnode process. Adv. Math. 252, 518 - 571 (2014). 2. M. Adler, S. Chhita, K. Johansson, P. van Moerbeke: Tacnode GUE-minor Processes and Double Aztec Diamonds, Probability Theory and Related Fields (2015) (arXiv:1303.5279) 3. M. Adler, P. van Moerbeke: Coupled GUE-minor Processes, Intern. Math. Res. Notices (2015) (arXiv:1312.3859)
Oleg Vasilyev, Max-Planck Institute for Intelligent systems, GermanyLet us consider a system of colloidal particles immersed in a viscous solvent. Each of these particles demonstrate a Brownian motion. Let us now apply a constant force acting on some selected (the fraction of selected particles is small) particles. These driven particles start to drift in the direction of applied force pushing surrounding particles aside. Therefore behind any of driven particles the region free of inert particles or with reduced concentration of inert particles is formed. Such variations in concentration produces effective attractive forces acting between driven particles. This type of behavior is numerically investigated for the system of Brownian disks. The driven particles (disks) form clusters which increases the average velocity of driven particles. We also demonstrate, that in the thin channel clusters of driven disks form traffic jams. In this case we observe the paradoxical situation: after increasing the driving force the average velocity of driven particles may decrease.
Pascal Viot, Université Pierre et Marie Curie, FranceHamiltonian systems with long-range interactions give rise to long-lived out-of-equilibrium macroscopic states, so-called quasistationary states. We show here that, in a suitably generalized form, this result remains valid for many such systems in the presence of dissipation Simulation of oneone-dimensional self-gravitating systems confirms the relevance of these solutions and gives indications of their regime of validity in line with theoretical predictions.
Makisim Zubkov, Nanyang Technological UniversityWe use simplified Hamiltonian model to investigate the propagation of mechanical vibrations through a ladder chain with correlated disorder. We calculate the localization length of mechanical modes in the chain in presence of short and long-range correlated disorder by the aid of transfer matrix method. Our result shows that, existence of correlation in the chain responsible to delocalization of mechanical modes. Our analytical results in prediction of extended modes are confirmed by numerical simulations. For quantification of disorder and correlation we introduce the concept of entropy and we use the entropy for explain the propagation of mechanical modes in the ladder chain. Due to similarity of our model and DNA, our result may be applicable to the DNA chain. |
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