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Title of Journal: J Stat Phys

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Abbravation: Journal of Statistical Physics

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Springer US

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DOI

10.1002/bbpc.19600641033

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1572-9613

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New Horizons in Multidimensional Diffusion The Lo

Authors: Carl P Dettmann
Publish Date: 2011/11/19
Volume: 146, Issue: 1, Pages: 181-204
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Abstract

The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers In the two dimensional finite horizon case in which all trajectories involve collisions with the scatterers displacements scaled by the usual diffusive factor sqrtt are normally distributed as shown by Bunimovich and Sinai in 1981 In the infinite horizon case motion is superdiffusive however the normal distribution is recovered when scaling by sqrt tln t with an explicit formula for its variance Here we explore the infinite horizon case in arbitrary dimensions giving explicit formulas for the mean square displacement arguing that it differs from the variance of the limiting distribution making connections with the Riemann Hypothesis in the small scatterer limit and providing evidence for a critical dimension d=6 beyond which correlation decay exhibits fractional powers The results are conditional on a number of conjectures and are corroborated by numerical simulations in up to ten dimensions

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Other Papers In This Journal:
  1. Mean Field Dilute Ferromagnet: High Temperature and Zero Temperature Behavior
  2. Probabilistic Cellular Automata for Low-Temperature 2-d Ising Model
  3. Random Birth-and-Death Networks
  4. Fourth Moment Sum Rule for the Charge Correlations of a Two-Component Classical Plasma
  5. On Asymptotic Behavior of Multilinear Eigenvalue Statistics of Random Matrices
  6. Products of Random Matrices and Generalised Quantum Point Scatterers
  7. Symmetry of the Linearized Boltzmann Equation and Its Application
  8. Activity of Excitatory Neuron with Delayed Feedback Stimulated with Poisson Stream is Non-Markov
  9. Replica Symmetry Breaking Condition Exposed by Random Matrix Calculation of Landscape Complexity
  10. Entropy-Driven Phase Transition in a Polydisperse Hard-Rods Lattice System
  11. Knudsen Gas in a Finite Random Tube: Transport Diffusion and First Passage Properties
  12. Clustering of Fermionic Truncated Expectation Values Via Functional Integration
  13. Large Deviations for Stationary Probabilities of a Family of Continuous Time Markov Chains via Aubry–Mather Theory
  14. Revisiting the Glansdorff–Prigogine Criterion for Stability Within Irreversible Thermodynamics
  15. Ground State Energy of the One-Dimensional Discrete Random Schrödinger Operator with Bernoulli Potential
  16. A Pearson Random Walk with Steps of Uniform Orientation and Dirichlet Distributed Lengths
  17. The More the Merrier?
  18. Non-equilibrium Phase Diagram for a Model with Coalescence, Evaporation and Deposition
  19. Real Spin Glasses Relax Slowly in the Shade of Hierarchical Trees
  20. Galton–Watson Trees with Vanishing Martingale Limit
  21. A Maximum Entropy Method Based on Piecewise Linear Functions for the Recovery of a Stationary Density of Interval Mappings
  22. Uniform Approximation of the Integrated Density of States for Long-Range Percolation Hamiltonians
  23. Dynamics of Infinitely Many Particles Mutually Interacting in Three Dimensions via a Bounded Superstable Long-Range Potential
  24. Statistical Mechanics of Nucleosomes Constrained by Higher-Order Chromatin Structure
  25. Large Deviations for the Branching Brownian Motion in Presence of Selection or Coalescence
  26. Clustering Bounds on n -Point Correlations for Unbounded Spin Systems
  27. Dissipative Diamagnetism—A Case Study for Equilibrium and Nonequilibrium Statistical Mechanics
  28. On the Transfer of Energy Towards Infinity in the Theory of Weak Turbulence for the Nonlinear Schrödinger Equation
  29. Finite-Time Fluctuations in the Degree Statistics of Growing Networks
  30. Exotic Ground States of Directional Pair Potentials via Collective-Density Variables
  31. Strong Asymmetric Coupling of Two Parallel Exclusion Processes
  32. Repeated Interaction Quantum Systems: Van Hove Limits and Asymptotic States
  33. On the Dynamics of the Glass Transition on Bethe Lattices
  34. Coarsening Dynamics on $$\mathbb {Z}^d$$ with Frozen Vertices
  35. A Class of Random Walks in Reversible Dynamic Environments: Antisymmetry and Applications to the East Model
  36. Power-law Decay and the Ergodic–Nonergodic Transition in Simple Fluids
  37. Rigorous Treatment of the Liquid-Vapor Transition in a Polydisperse System with Kac Interaction
  38. R -Local Delaunay Inhibition Model
  39. Patterns and Symmetries in the Visual Cortex and in Natural Images
  40. Parameter Estimation for Multiscale Diffusions
  41. Onsager and Kaufman’s Calculation of the Spontaneous Magnetization of the Ising Model: II
  42. Injected Power Fluctuations in 1D Dissipative Systems
  43. Anomalies in Conductance and Localization Length of Disordered Ladders
  44. Finite-Size Scaling in the Energy-Entropy Plane for the 2D ± Ising Spin Glass
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