Wednesdays at 4 PM in Eckhart 202 unless otherwise
noted.
We show that, in the hydrodynamic limit, the Heisenberg dynamics of the energy, momentum, and particle densities for fermions with short-range pair interactions converges to the compressible Euler equations with the pressure function given by quantum statistical mechanics. Our derivation is based on a quantum version of the entropy method and a suitable quantum virial theorem. We require a number of technical conditions that will be discussed in the talk (joint work with H.T. Yau).
We discuss recent results of (semi)classical limits of (weakly) nonlinear Schroedinger equations, including Hartree type nonlinearities and/or strong nonlinearities of cubic NLS type. Without entering technicalities we explain the problems and the methods and compare results using Wigner measure techniques and WKB techniques. Results based on Wigner measures for the 1-d "pure state" case, global in time, for the limit of Schroedinger-Poisson to Vlasov-Poisson encounter the notorious problem of nonuniqueness of the V-P system for measure valued data (as dealt with e.g. by A. Majda and Y. Zheng) Also we present 3-d results for (semi)classical limits of Hartree type equations, based on WKB methods, where we discuss the different scalings of the nonlinearity with respect to the semiclassical parameter. We motivate our analysis by models in quantum semiconductor modelling and illustrate it by numerical simulations. References : [ZZM2002] "The limit from the Schrodinger-Poisson to Vlasov-Poisson equation with general data in one dimension", P. Zhang, Y. Zheng and N.J. Mauser, Comm. Pure and Appl. Math. 55 (5) (2002) 582- 632 [M2002] "(Semi)classical limits of Schrodinger-Poisson systems via Wigner transforms", N.J. Mauser, J. EDP, Univ. Nantes, (2002) 1- 12 [CMS2004] "(Semi)classical limit of the Hartree equation with harmonic potential", R. Carles, N.J. Mauser and H.P. Stimming, preprint (2004) [SMM2003] "Multivalued geometrical optics: Wigner transforms vs. WKB-methods", C. Sparber, P. Markowich and N.J. Mauser, Asymptotic Analysis 33 (2) (2003) 153-187 [BMS2004] "Effective one particle quantum dynamics of electrons : the Schrodinger-Poisson-Xalpha model", W. Bao, N.J. Mauser and H.P. Stimming, Comm. Math. Sciences 1 (4) (2004) 809-828
We investigate two-dimensional pattern formation in a non-local neural model of Wilson-Cowan type. The first step in the analysis is to consider the scalar case and derive an equivalent partial differential equation. We then show how instability of axially symmetric solutions leads to the formation of multibump solutions. In the second part of the talk we extend the model to a system by including a recovery variable. In the system the formation of spiral waves has led Jian-Young Wu(Georgetown U.) to search for spirals in tangential slices of rat cortex. Videos will be shown which illustrate spirals in the model and also the experimentally observed spiral waves recently found by Wu's group. Finally, we will illustrate the formation of ring shaped waves which result from a stimulus applied to the center of the medium. Tangential slices of rat cortex have recently been shown to exhibit similar ring waves and these will also be illustrated.
The convective Cahn-Hilliard equation, u_t+(u_xx+u-u^3)_xx-Duu_x=0, has been suggested recently for the description of several physical phenomena, including spinodal decomposition of phase separating systems in an external field, step instability on a crystal surface, and faceting of thermodynamically unstable surfaces. This equation provides a "bridge" between the standard Cahn-Hilliard equation and the Kuramoto-Sivashinsky equation. Depending on the value of the parameter D, either domain coarsening (I) or the formation of regular or irregular patterns takes place. In the case (I), we investigate the dynamics of kinks that governs the domain coarsening. The important role of kink-antikink pairs and kink triplets is revealed. In the case (II), we investigate the stability of stationary and traveling-wave solutions.
Many friction dominated models from the sciences come in form of a gradient flow. The gradient flow structure separates the driving energetics, modeled by the energy functional $E$, from the limiting dissipation mechanisms, encoded in a metric tensor $g$. In many applications, this metric structure is genuinely Riemannian. We introduce such a (formal) Riemannian structure on the space of densities. This is for instance the geometry which makes the well--studied porous medium equation a gradient flow with respect to a generalized entropy functional $E$. We argue that $E$ is convex in this geometry, which is an infinitesimal reformulation of McCann's displacement convexity. We employ this convexity and Riemannian calculus to deduce convergence to the self--similar profile at optimal rate.
The above Riemannian structure is also natural in probability theory: It endows the space of probability measures on a given smooth manifold with a Riemannian structure. We argue that its induced distance is the Wasserstein distance. A Fokker--Planck equation is the gradient flow w. r. t. to a relative entropy. The Bakry--Emery condition is just the condition for intrinsic convexity of the relative entropy. We also show that the logarithmic Sobolev inequality and the Talagrand inequality can be naturally understood and derived within this framework.
This is partially joint work with C. Villani.
For a very strong external field $h_{ext}$ in direction of, say, $(1,0,0)$, the minimal magnetization $m^*=m^*(h_{ext})$ is very close to $(1,0,0)$. As one reduces the strength and possibly reverses the sign of the external field $h_{ext}$, the magnetization along this saturation branch becomes unstable. This nucleation is at the onset of switching. Mathematically, it is characterized by a degenerated Hessian Hess$E(m^*(h_{ext}))$ of the micromagnetic energy functional $E$. The elements of the degenerate subspace describe the nucleation mode. The value of the external field at which nucleation happens is the critical field $h_{crit}$.
Nucleation has been studied analytically for special sample geometries $\Omega$, like ellipsoids and infinite prisms of arbitrary cross--sections. For these samples, $m^*=(1,0,0)$ is an exact stationary point for all field strengths and the critical field $h_{crit}$ can be characterized variationally by a Rayleigh quotient. Based on a series of theoretical papers in the physics literature, Aharoni claims that there are at most three different types of nucleation modes: coherent rotation, buckling and curling. Which of these regimes occurs depends on the ratio of the sample dimension to the intrinsic exchange length $d$.
We rigorously show that for an infinite prism which mimics a thin--film elements, i. e. $\Omega=\mathbb{R}\times(0,\ell)\times(0,t)$ with $t\ll\ell$, there are exactly four different scaling regimes of $h_{crit}$ in the two non--dimensional parameters $\ell/d\gg t/d$. The new feature of the fourth regime is that it displays an oscillation in the infinite direction on a length scale $w$ which depends on the microscopic length $d$. We relate $w$ to the length scale in experimentally observed concertina pattern. The different scaling regimes are identified by establishing lower and upper bounds on the Rayleigh quotient which match in terms of scaling. The analytically interesting part are the Ansatz--free lower bounds, which rely on appropriate interpolation inequalities.
This is joint work with R. Cantero--Alvarez.
We consider capillarity--driven spreading of a thin droplet of a viscous liquid on a solid. The spreading is determined by the quasi--static balance between surface tension and viscous forces. On the level of the lubrication approximation, this leads to a degenerate fourth--order parabolic equation for the film height $h$. However, as a consequence of the no--slip boundary condition at the liquid--solid interface, a logarithmic divergence occurs at the moving contact line.
This well--known singularity is for instance removed by relaxing the no--slip condition, thereby introducing a microscopic lengthscale $b$. Matched asymptotics yields a relationship between the speed of the contact line and an ``apparent'' contact angle. The microscopic length scale $b$ only enters logarithmically. Hence also the macroscopic rate of spreading only depends logarithmically on the microscopic length $b$. Surprisingly, a simple heuristic argument by de Gennes, only based on energy and energy dissipation, yields the same macroscopic law.
We show that this robust argument by de Gennes can be made rigorous by PDE techniques. We introduce a notion of ``apparent support'' $S$ and derive three relations between $S$, the capillary energy $E$ and their rates of change $\dot S, \dot E$. The proof of these relations relies on interpolation estimates for critical exponents, thus involving logarithms.
This is joint work with L. Giacomelli.
All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. The speaker will recount some recent history of universality ideas in physics starting with Wigner's model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting.
We consider here some models of thermo-diffusive lean spray flames where the gaseous reactant is provided by vaporizing liquid fuel droplets. First, we prove the existence of travelling waves for general combustion and vaporization laws. The proof classically relies on the application of the Leray Topological Degree, after suitable a priori estimates are found. The most important estimate is that of the velocity of the travelling wave. The proof of the bound of the velocity distinguishes the case where the droplets are completely vaporized before the reaction zone, and the opposite case. This alternative is emphasized in the high energy activation limit for the system, where one assumes that the combustion is the fastest phenomenon in the problem. Here, explicit expressions for the limiting profiles are obtained, and we can prove the existence of two different regimes of deflagration. On the one hand, when the droplets are completely vaporized before the reaction zone (i.e. when the droplets are small enough), the velocity of the flame is that of the equivalent gas flame. On the other hand, when the size of the droplets is larger than a critical value, then the droplets interact with the combustion zone, and one obtains a "vaporization-limited" combustion regime leading to a decrease of the velocity of the flame. In the latter case, the structure of the reaction zone is more complex than in the classical gaseous case and contains both a singular and a regular part. This is in agreement with previous numerical studies. Also, at contrast with the gaseous case, more general high energy activation asymptotics are possible and will be discussed. Finally, we present a general framework for the study of the nonlinear stability of the system, and show its applicability in situations where analytical calculations are possible, for example when the reaction or combustion rates have simple expressions, or in the high energy activation limit.
We discuss a class of dispersive nonlinear nonlocal partial differential equations (PDEs) modeling mechanics and neural functions of the inner ear, then show analytical and numerical properties of wave like solutions under single and multiple frequency sound inputs. An intriguing nonlinear phenomenon is masking, namely, an audible sound becomes inaudible in the presence of another sound. We present a PDE based two level psychoacoustic model on masking of banded noise, and relate it to perceptual coding in digital music (MP3).
Computing the entropy solution to conservation laws with nonconvex fluxes or to systems of conservation laws with nonconvex equations of state is a challenging problem. Different numerical methods are available, but most of them are computationally expensive and employ characteristic decomposition and Riemann problem solvers.
In the last decade, Godunov-type central schemes emerge as simple, reliable, efficient, high-resolution methods for solving time-dependent PDE's with wide applications. We illustrate that second order central-upwind schemes are an attractive tool for solving ``nonconvex problems'', where the nature of the problem requires at every time step a careful selection of the approximate solution. Various test examples, such as the Euler equations of gas dynamics and others, are presented.
We investigate stability of the initial modulational perturbations of stationary solutions to the two-dimensional Navier-Stokes equations with a time-independent periodic rapidly oscillating forcing. The stationary solutions are cellular flows and they are determined by the stream function $\phi = \sin x_1/\epsilon \sin x_2/\epsilon + \delta \cos x_1/\epsilon \cos x_2/\epsilon$, $0 \leq \delta \leq 1$. The perturbations satisfy the modulation equation. For small Reynolds number we determine the stability for the fully nonlinear modulation equation. For any Reynolds number we determine the stability of the linearized modulation equation.
Given a time dependent orbit of probability densities $\rho(x,t)$ (a cloud), the velocity field v(x,t) which transports this cloud is the one which solves the continuity equation $\rho_t + div(\rho v)=0$. In some cases such a field is not uniquely defined (or not defined at all). The right approach to this problem is via the optimal mass transportation known as the Monge-Kantorovich problem. I'll review the Monge Kantorovich and discuss some of its recent applications to circle maps and physical optics.
In his celebrated paper on Brownian motion Einstein established a linear relation between the mobility of a charged particle moving under the influence of an external electric field in an environment that is in thermal equilibrium and the diffusivity of an unperturbed Brownian particle. In this talk we present a rigorous argument showing that such a relation holds for certain models of systems appearing in statistical mechanics. We start showing a quite general perturbative argument that leads to Einstein relation for systems whose dynamics possesses a spectral gap. This shall be followed by a result establishing such a relation for a certain model of passive tracer motion, namely a random walk among random conductivities. In that case the environment is static, thus no spectral gap estimate is available and the proof uses a non-perturbative argument.
We present a new semi-discrete Godunov-type scheme for approximating solutions of Hamilton-Jacobi equations on unstructured grids. Similarly to previous works with structured grids, a semi-discrete formulation of central schemes is made possible due to estimates of the local speeds of propagation. The consistency of the method is obtained following Abgrall's calculations for the consistency of an upwind Lax-Friedrichs-type scheme on unstructured grids. This is a joint work with S. Nayak. Along the way, we plan to review the recent developments in Godunov-type schemes for multidimensional Hamilton-Jacobi equations. In particular, we will discuss the fully-discrete and semi-discrete central-upwind schemes we recently developed with S. Bryson. In all our schemes, high-order accuracy is obtained by using a weighted essentially non-oscillatory reconstruction of the derivatives. The derivation is done in an arbitrary number of space dimensions and we prove the monotonicity of the fluxes we use.
A common table top demonstration of river braiding and meandering is produced by a jet of fluid running down a partially wetting inclined plane. However, the physics is quite different from that causing river morphology, since no erosion is present. We show that the meandering of the jet is caused by the disturbances in the inlet and can be completely eliminated in a careful experiment. When the meandering is suppressed, we observe a beautiful flow pattern when the thickness of the jet running down is oscillating (braiding). We develop a complete theory of braiding, based on the interplay of surface tension, inertia, and contact angle. The theory shows an excellent agreement with the experiment. We also study the transition from braiding to no-braiding regime (rivulets) and construct the corresponding bifurcation diagram.
For questions, contact Eduard Kirr at ekirr@math.uchicago.edu
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