Paul Wright's Publications

• Escape rates: A variational approach, in preparation. (with M. Demers and L.-S. Young) [ | ]

• The diffusion coefficient for piecewise expanding maps of the interval with metastable states, to be submitted soon. (with D. Dolgopyat) [ | ]

  Abstract: Consider a piecewise smooth expanding map of the interval possessing two invariant intervals and exactly two ergodic absolutely continuous invariant probability measures (ACIMs). After this system is perturbed slightly to make the invariant sets merge in such a way that there is a unique ACIM, we show how to approximate the diffusion coefficient for an observable of bounded variation by the diffusion coefficient of a related continuous time Markov chain.

• Approximating invariant densities of metastable systems, submitted. (with C. Gonzalez Tokman and B. Hunt) [link to arXiv.org | ]

  Abstract: We consider a piecewise smooth expanding map of the interval possessing two invariant subsets of positive Lebesgue measure and exactly two ergodic absolutely continuous invariant probability measures (ACIMs). When this system is perturbed slightly to make the invariant sets merge, we describe how the unique ACIM of the perturbed map can be approximated by a convex combination of the two initial ergodic ACIMs.

• Escape rates and physically relevant measures for billiards with small holes, to appear in Communications in Mathematical Physics. (with M. Demers and L.-S. Young) [link to arXiv.org | link to journal]

  Abstract: We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map.

• The periodic oscillation of an adiabatic piston in two or three dimensions, Communications in Mathematical Physics 275 (2007), no. 2, 553-580. [link to arXiv.org | link to journal]

  Abstract: We study a heavy piston of mass M that separates finitely many ideal, unit mass gas particles moving in two or three dimensions. Neishtadt and Sinai previously determined a method for finding this system's averaged equation and showed that its solutions oscillate periodically. Using averaging techniques, we prove that the actual motions of the piston converge in probability to the predicted averaged behavior on the time scale M^ {1/2} when M tends to infinity while the total energy of the system is bounded and the number of gas particles is fixed.

• Rigorous Results for the Periodic Oscillation of an Adiabatic Piston, Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University (2007). [link to arXiv.org]

• A simple piston problem in one dimension, Nonlinearity 19 (2006), 2365-2389. [link to arXiv.org | link to journal]

  Abstract: We study a heavy piston that separates finitely many ideal gas particles moving inside a one-dimensional gas chamber. Using averaging techniques, we prove precise rates of convergence of the actual motions of the piston to its averaged behavior. The convergence is uniform over all initial conditions in a compact set. The results extend earlier work by Sinai and Neishtadt, who determined that the averaged behavior is periodic oscillation. In addition, we investigate the piston system when the particle interactions have been smoothed. The convergence to the averaged behavior again takes place uniformly, both over initial conditions and over the amount of smoothing.

• Semiclassical generalization of the Darboux-Christoffel formula, J. Math. Phys. 43 (2002), no. 10, 4668-4680. (with R. Littlejohn) [link to journal]

  Abstract: The Darboux–Christoffel formula is a closed-form expression for the kernel of the operator that projects onto the first N of a system of one-dimensional polynomials, orthonormal with respect to some weighting function. It is a key element in the theory of Gaussian integration and in the theory of discrete variable representation or Lagrangian mesh methods for diagonalizing quantum Hamiltonians of a few degrees of freedom. The one-dimensional Darboux–Christoffel formula turns out to have a generalization that is valid in a semiclassical or asymptotic sense for a wider class of orthonormal functions than orthonormal polynomials. This class consists of the bound eigenfunctions of one-dimensional Hamiltonians with time-reversal invariance, such as kinetic-plus-potential Hamiltonians. It also has certain generalizations involving the unbound eigenfunctions of such Hamiltonians.

I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible.
--Carl Friedrich Gauss, quoted in G. Simmon's Calculus Gems (New York 1992).

The purpose of models is not to fit the data but to sharpen the questions.
--Samuel Karlin, 11th R A Fisher Memorial Lecture, Royal Society April 1983.