Statistical Mechanics Conference Short Talk Schedule

A string of hard-core point particles undergoing elastic collisions in 1D can form an integrable system even when they are of unequal masses. This talk will introduce a simple example of such an integrable system that consists of three particles with two unequal mass species. A2: Ori Hirschberg, Technion Coauthors: David Mukamel, Gunter M. Schutz Emergent Motion Of Condensates In Mass Transport Models Abstract: Condensation phenomena, in which a finite fraction of the "mass" in a macroscopic system is concentrated in a microscopic fraction of its volume, are rather common in nature. Examples include the formation of traffic jams in transportation systems, the clustering of particles in shaken granular gases, the emergence of macroscopically-linked hubs in complex networks and many others. In this talk I will examine the emergent dynamics of the condensate in such systems and report that quite generically the condensate in asymmetric systems develops a drift motion. The mechanism driving this motion is explained using a simplified toy model. Condensation phenomena, in which a finite fraction of the "mass" in a macroscopic system is concentrated in a microscopic fraction of its volume, are rather common in nature. Examples include the formation of traffic jams in transportation systems, the clustering of particles in shaken granular gases, the emergence of macroscopically-linked hubs in complex networks and many others. In this talk I will examine the emergent dynamics of the condensate in such systems and report that quite generically the condensate in asymmetric systems develops a drift motion. The mechanism driving this motion is explained using a simplified toy model. A3: Ohad Shpielberg, Technion Coauthors: Eric Akkermans Le Chatelier Principle For Out Of Equilibrium And Boundary Driven Systems: Application To Dynamical Phase Transitions. Abstract: A stability analysis of out of equilibrium and boundary driven systems is presented. It is performed in the framework of the hydrodynamic macroscopic fluctuation theory and assuming the additivity principle whose interpretation is discussed with the help of a Hamiltonian description. An extension of Le Chatelier principle for out of equilibrium situations is presented which allows to formulate the conditions of validity of the additivity principle. Examples of application of these results in the realm of classical and quantum systems are provided. A stability analysis of out of equilibrium and boundary driven systems is presented. It is performed in the framework of the hydrodynamic macroscopic fluctuation theory and assuming the additivity principle whose interpretation is discussed with the help of a Hamiltonian description. An extension of Le Chatelier principle for out of equilibrium situations is presented which allows to formulate the conditions of validity of the additivity principle. Examples of application of these results in the realm of classical and quantum systems are provided. A4: Misha Shvartsman, University of St. Thomas Coauthors: Pavel Bělík, Douglas P. Dokken, Kurt Scholz On Thermodynamic Balance In Tornado Theory Abstract: We explore the energy balance in a thunderstorm, in particular, how energy is redistributed on a local level inside a tornado-like flow. We explore the energy balance in a thunderstorm, in particular, how energy is redistributed on a local level inside a tornado-like flow. A5: Simon Thalabard, Umass Amherst Coauthors: Bruce Turkington An Optimal Closure for Turbulent Flows Abstract: I will describe an ``optimal closure framework’’ to describe the large scale dynamics of decaying turbulence, that does not rely on a phenomenological modeling of the viscous damping. The statistical closure employs a Gaussian model for the turbulent scales, with corresponding vanishing third cumulant, and yet it captures an intrinsic damping. The key to this apparent paradox lies in a clear distinction between true ensemble averages and their proxies, most easily grasped when one works directly with the Liouville equation rather than the cumulant hierarchy. I will describe an ``optimal closure framework’’ to describe the large scale dynamics of decaying turbulence, that does not rely on a phenomenological modeling of the viscous damping. The statistical closure employs a Gaussian model for the turbulent scales, with corresponding vanishing third cumulant, and yet it captures an intrinsic damping. The key to this apparent paradox lies in a clear distinction between true ensemble averages and their proxies, most easily grasped when one works directly with the Liouville equation rather than the cumulant hierarchy. A6: David Wolpert, Santa Fe Institute Extending Landauer’s Bound to Arbitrary Computation Abstract: Recently there has been great progress in bounding the thermodynamic work required to perform any computation whose output is independent of its input, e.g., bit erasure. These bounds depend on fine-grained details of the physical computer that implements the computation. Here I extend these results to bound the work required for any computation, even one whose output depends on its input. I use this extension to show that if the computer implementing the computation will be reused, then the work bound depends only on the dynamics of the logical variable under the computation, with no dependence on the physical details of that computer. This establishes a formal identity between the thermodynamics of (re-usable) computers and theoretical computer science. As an illustration of this identity, I use it to prove that the work needed to compute a bit string σ on a Turing machine M is kBT ln(2) × [Kolmogorov complexity of σ + log of the (Bernoulli) measure of the set of strings that compute σ + log of the halting probability of M]. Recently there has been great progress in bounding the thermodynamic work required to perform any computation whose output is independent of its input, e.g., bit erasure. These bounds depend on fine-grained details of the physical computer that implements the computation. Here I extend these results to bound the work required for any computation, even one whose output depends on its input. I use this extension to show that if the computer implementing the computation will be reused, then the work bound depends only on the dynamics of the logical variable under the computation, with no dependence on the physical details of that computer. This establishes a formal identity between the thermodynamics of (re-usable) computers and theoretical computer science. As an illustration of this identity, I use it to prove that the work needed to compute a bit string σ on a Turing machine M is kBT ln(2) × [Kolmogorov complexity of σ + log of the (Bernoulli) measure of the set of strings that compute σ + log of the halting probability of M]. A7: Ben Webb, Brigham Young University Coauthors: E.G.D. Cohen Rigorous Results for a New Class of Nonplanar Lorentz Lattice Gas Abstract: In the study of Lorentz Lattice Gas (LLG) most rigorous results are restricted to 1-d and 2-d planar lattices. As an intermediate step to understanding higher dimensional LLGs, we consider the motion of a particle on a number of nonplanar 2-d lattices whose sites are randomly occupied by scatterers that deterministically rotate the particle to its left or right depending on the rotator's orientation. We find that the particle's motion is qualitatively similar to the motion observed on certain planar lattices, suggesting there may be ways to adapt, in certain In the study of Lorentz Lattice Gas (LLG) most rigorous results are restricted to 1-d and 2-d planar lattices. As an intermediate step to understanding higher dimensional LLGs, we consider the motion of a particle on a number of nonplanar 2-d lattices whose sites are randomly occupied by scatterers that deterministically rotate the particle to its left or right depending on the rotator's orientation. We find that the particle's motion is qualitatively similar to the motion observed on certain planar lattices, suggesting there may be ways to adapt, in certain cases, the theory of low-dimensional LLGs to higher dimensional systems. A8: Dimitri Petritis, University of Rennes 1 Coauthors: Massimo Campanino Type Transition Of Simple Random Walks On Randomly Directed Regular Lattices Abstract: Simple random walks on a partially directed version of $\mathbb{Z}^2$ are considered. More precisely, vertical edges between neighbouring vertices of $\mathbb{Z}^2$ can be traversed in both directions (they are undirected) while horizontal edges are one-way. The horizontal orientation is prescribed by a random perturbation of a periodic function, the perturbation probability decays according to a Simple random walks on a partially directed version of $\mathbb{Z}^2$ are considered. More precisely, vertical edges between neighbouring vertices of $\mathbb{Z}^2$ can be traversed in both directions (they are undirected) while horizontal edges are one-way. The horizontal orientation is prescribed by a random perturbation of a periodic function, the perturbation probability decays according to a power law in the absolute value of the ordinate. We study the type of the simple random walk, i.e. its being recurrent or transient, and show that there exists a critical value of the decay power, above which it is almost surely recurrent and below which it is almost surely transient. A9: Yao Li, University of Massachusetts Amherst Coauthors: Lili Hu A Fast Kinetic Monte Carlo Simulation Method Abstract: Kinetic Monte Carlo method is used to simulate a large class of Markov jump processes arising in statistical mechanics and other scientific/engineering fields. In this talk, I will introduce a novel exact simulation method, called the HashingLeaping method (HLM), for the exact simulation of these Markov process. Under suitable condition, the computational cost of this method is a constant, independent of the scale of the Markov jump process. The main idea of the HLM is to repeatedly implement a hash-table-like bucket sort algorithm for all times of occurrence covered by a time step with length τ. Kinetic Monte Carlo method is used to simulate a large class of Markov jump processes arising in statistical mechanics and other scientific/engineering fields. In this talk, I will introduce a novel exact simulation method, called the HashingLeaping method (HLM), for the exact simulation of these Markov process. Under suitable condition, the computational cost of this method is a constant, independent of the scale of the Markov jump process. The main idea of the HLM is to repeatedly implement a hash-table-like bucket sort algorithm for all times of occurrence covered by a time step with length τ. A10: Flora Koukiou, Cergy-Pontoise University The Freezing Property For Gaussian Models Abstract: We discuss the freezing property for Gaussian Models in relation with the entropy of the associated Gibbs measures. We discuss the freezing property for Gaussian Models in relation with the entropy of the associated Gibbs measures. A11: Robert Ziff, University of Michigan Coauthors: Hao Hu and Yougin Deng, University of Science and Technology of China Holes In Percolation Clusters Abstract: For different percolation models, we did simulations to observe holes in the largest cluster, and holes in the largest backbone cluster. We find that the distribution of the size of the holes follows a power law distribution, the dimension of the holes is 2, and a general hyperscaling relation is satisfied. For site percolation on both the triangular and the square lattice, and bond percolation on the square lattice, it is found that the largest hole occupies exactly half of the lattice sites, which follows by a symmetry argument. We also define a percolation model on sites in the holes -a kind of recursive percolation. We find a critical line, on which the dimension of clusters formed in the holes is also 2. For different percolation models, we did simulations to observe holes in the largest cluster, and holes in the largest backbone cluster. We find that the distribution of the size of the holes follows a power law distribution, the dimension of the holes is 2, and a general hyperscaling relation is satisfied. For site percolation on both the triangular and the square lattice, and bond percolation on the square lattice, it is found that the largest hole occupies exactly half of the lattice sites, which follows by a symmetry argument. We also define a percolation model on sites in the holes -a kind of recursive percolation. We find a critical line, on which the dimension of clusters formed in the holes is also 2. A12: Eugene Kolomeisky, University of Virgina Coauthors: J. P. Straley and D. L. Abrams Atomic Collapse, Screening In Bilayer Graphene And Flat Thomas-Fermi Atom Abstract: Undoped bilayer graphene is a two-dimensional semimetal with hyperbolic in the momentum low-energy excitation spectrum. As a result screening of an external charge exceeding a critical value Z_{c}e is accompanied by a reconstruction of the ground state: valence band electrons (for Z>0) are promoted to form space charge around the external charge while the holes leave the physical picture. This resembles the situation in the monolayer graphene, but unlike the latter, the bilayer screening response for Z >> Z_{c} is described by strictly linear Thomas-Fermi theory which predicts that material's static dielectric response is identical to that of the twodimensional electron gas in the long-wavelength limit. As a byproduct we also solve Undoped bilayer graphene is a two-dimensional semimetal with hyperbolic in the momentum low-energy excitation spectrum. As a result screening of an external charge exceeding a critical value Z_{c}e is accompanied by a reconstruction of the ground state: valence band electrons (for Z>0) are promoted to form space charge around the external charge while the holes leave the physical picture. This resembles the situation in the monolayer graphene, but unlike the latter, the bilayer screening response for Z >> Z_{c} is described by strictly linear Thomas-Fermi theory which predicts that material's static dielectric response is identical to that of the twodimensional electron gas in the long-wavelength limit. As a byproduct we also solve the problem of flat Thomas-Fermi atom embedded into three-dimensional space. A13: Irina Navrotskaya, University of Pittsburgh Differentiability of the inverse (from density to potential) map and its applications to theory of liquids Abstract: Let us consider a system of identical particles with the total energy W + U, where W is a fixed scalar function, and U is an additional internal or external potential in the form of a sum of m-particle interactions u. The inverse conjecture states that any positive, integrable function of m particle coordinates ρ is the equilibrium mparticle density corresponding to some unique potential u. This conjecture has now been proved for m ≥ 1 for both canonical and grand canonical ensembles [1, 2, 3, 4]. In the grand canonical formulation, the inverse map is also differentiable for m ≥ 1. (The problem of differentiability in the canonical formulation is much more subtle Let us consider a system of identical particles with the total energy W + U, where W is a fixed scalar function, and U is an additional internal or external potential in the form of a sum of m-particle interactions u. The inverse conjecture states that any positive, integrable function of m particle coordinates ρ is the equilibrium mparticle density corresponding to some unique potential u. This conjecture has now been proved for m ≥ 1 for both canonical and grand canonical ensembles [1, 2, 3, 4]. In the grand canonical formulation, the inverse map is also differentiable for m ≥ 1. (The problem of differentiability in the canonical formulation is much more subtle because the integral of ρ is a constant [5].) Existence and differentiability of the inverse map for m ≥ 2 provides the basis for the variational principle on which generalizations to density functional theory can be formulated. For example, a generalized Ornstein-Zernike equat ion (one for every m) connecting the 2m-particle direct correlation function and 2m-,...,m-particle densities can be constructed. (This generalization is different from the one considered by Stell. A14: Bruce Miller, Texas Christian University Coauthors: Cameron Langer Regular and Chaotic Motion of a Gravitational Billiard in a Cone Abstract: We study the nonlinear dynamics of a three dimensional billiard in a constant gravitational field colliding elastically with a linear cone. We derive a twodimensional Poincare map with two parameters, the half angle of the cone and the component of the billiard’s angular momentum parallel to the symmetry axis. We demonstrate several integrable cases of the parameter values, and analytically compute the system’s fixed point, analyzing the stability of this orbit as a function of the parameters as well as its relation to the physical trajectory of the billiard. Next, we explore the phase space of the system numerically. We find that for small values of the angular momentum the conic billiard exhibits behavior characteristic of twodegree-of-freedom Hamiltonian systems with a discontinuity, and the dynamics is qualitatively similar to that of the wedge billiard, although the correspondence is not exact. As we increase the angular momentum the dyn amics becomes, on the whole, less chaotic and the correspondence with the wedge billiard is lost. In common with the wedge billiard, we anticipate that modifications of the cone system will prove valuable for experimental investigation, both with atoms at low temperature and driven billiards. arxiv 1507.06693 We study the nonlinear dynamics of a three dimensional billiard in a constant gravitational field colliding elastically with a linear cone. We derive a twodimensional Poincare map with two parameters, the half angle of the cone and the component of the billiard’s angular momentum parallel to the symmetry axis. We demonstrate several integrable cases of the parameter values, and analytically compute the system’s fixed point, analyzing the stability of this orbit as a function of the parameters as well as its relation to the physical trajectory of the billiard. Next, we explore the phase space of the system numerically. We find that for small values of the angular momentum the conic billiard exhibits behavior characteristic of twodegree-of-freedom Hamiltonian systems with a discontinuity, and the dynamics is qualitatively similar to that of the wedge billiard, although the correspondence is not exact. As we increase the angular momentum the dyn amics becomes, on the whole, less chaotic and the correspondence with the wedge billiard is lost. In common with the wedge billiard, we anticipate that modifications of the cone system will prove valuable for experimental investigation, both with atoms at low temperature and driven billiards. arxiv 1507.06693 A15: Alex Blumenthal, Courant Institute, NYU Coauthors: Jinxin Xue and Lai-Sang Young Lyapunov Exponents For Random Perturbations of Some Prototypical 2D Maps Abstract: I will present results showing that the introduction of sufficiently large random perturbations can be used to drastically simplify the estimate of Lyapunov exponents. Outside the uniformly hyperbolic setting, estimating Lyapunov exponents for deterministic maps is profoundly difficult. Even when a given map is I will present results showing that the introduction of sufficiently large random perturbations can be used to drastically simplify the estimate of Lyapunov exponents. Outside the uniformly hyperbolic setting, estimating Lyapunov exponents for deterministic maps is profoundly difficult. Even when a given map is “predominantly hyperbolic”, in the sense that it exhibits strong expansion on a large (noninvariant) subset of phase space, there is no guarantee that Lyapunov exponents will reflect this expansion on any positive Lebesgue measure subset of phase space. On the other hand, randomizations can be used to steer trajectories with high probability: we apply this principle to random perturbations of a broad class of “prototypical” two-dimensional maps exhibiting strong expansion. Our results are applicable to many well-known examples of two-dimensional systems, e.g. Henon-like maps and the Chirikov Standard Map. A16: Nandor Simanyi, University of Alabama at Birmingham Coauthors: Caleb C. Moxley Non-commutative rotation vectors for toroidal billiards Abstract: We give a brief presentation of the homotopical rotation vectors that we introduced for toroidal billiards. Some basic properties of these rotation vectors will be shown for two cylindrical billiards that are known to be chaotic. We give a brief presentation of the homotopical rotation vectors that we introduced for toroidal billiards. Some basic properties of these rotation vectors will be shown for two cylindrical billiards that are known to be chaotic. A17: Jonathan Mattingly, Duke University Coauthors: David Herzog, Iowa State Stabilization By Noise, Nonequilibrium steadystates and Intermiticy Abstract: I will consider a family of simple planer ODEs which have unstable trajectories, namely \dot z = z^{n} + L.O.T on the complex plane. Adding noise stabilizes the systems and produces a unique stationary state. The delicate balance of the instability and the noise produces a steady state with a number of interesting properties. (1) It has polynomial tails (2) it mixes exponentially fast at a rate uniform in the initial condition (3) the steadystate has a nontrivial flux and asymptotically understandable intermittent behavior. The analysis turns on a systematic construction of a Lyapunov funtion in a principled (sharp) way. This generalizes and extends some work of Krzysztof Gawȩdzki, David Herzog, and Jan Wehr and of Athreya, Kolba, and Mattinglyon a special case of this model \dot z = z^2 I will consider a family of simple planer ODEs which have unstable trajectories, namely \dot z = z^{n} + L.O.T on the complex plane. Adding noise stabilizes the systems and produces a unique stationary state. The delicate balance of the instability and the noise produces a steady state with a number of interesting properties. (1) It has polynomial tails (2) it mixes exponentially fast at a rate uniform in the initial condition (3) the steadystate has a nontrivial flux and asymptotically understandable intermittent behavior. The analysis turns on a systematic construction of a Lyapunov funtion in a principled (sharp) way. This generalizes and extends some work of Krzysztof Gawȩdzki, David Herzog, and Jan Wehr and of Athreya, Kolba, and Mattinglyon a special case of this model \dot z = z^2