Geometry of Hamiltonian Chaos
L. Horwitz, et al.
PRL 98, 234302 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e234302
This group from Israel has recast classical Hamiltonian dynamics in the language of differential geometry. In doing so, they have discovered a new criterion for chaotic motion: the deviation of parallel geodesics. The geodesics of the system are classical trajectories. If nearby geodesics diverge, it indicates an instability in the system. This is a different measure than the Lyapunov exponent, although it’s possible there is a deeper connection between the two. I would speculate that the Lyapunov exponents could be derived from this geometric theory.
The general idea behind the method is to rewrite the Hamiltonian using a metric in the space of conjugate momenta. From this metric, one can derive the connection and curvature. From the curvature, one can calculate geodesic deviation. However, the authors find that the geodesic equation and the Hamiltonian equations of motion do not agree. It turns out that the geodesics in the dual manifold (defined by dx^{i} = g^{ij} dx_{j}) do indeed agree with the Hamiltonian equations of motion. The authors simply state this fact without discussion. There must be a deeper significance of this result. Or maybe the authors just started their calculation on the wrong manifold. Regardless of the connection, the geodesics on the dual manifold and the Hamiltonian equations of motion are mathematically equivalent; thus, one has a geometric way of thinking about phase space trajectories.
The new connection is anti-symmetric, so there is torsion. I don’t know much about torsion. In my course on general relativity, one of the postulates we used in deriving Einstein’s equations was the “no torsion condition.” It has something to do with twist of the geodesics. It is irrelevant in the derivation of the authors because the antisymmetric terms cancel from the geodesic equation, leaving only a symmetric connection.
In the end, the authors find that the curvature derived from this connection determines the stability of an orbit. If the curvature is negative, geodesics diverge and the system is chaotic. If curvature is positive, geodesics converge, and the system is stable. This seems like a very useful tool for analyzing unstable systems. The authors point out that this approach does not require one to use approximations of the true equations of motion, unlike other methods.
The idea behind the paper is quite interesting. My one complaint with the paper is that the phrase “a complete discussion will be given elsewhere” was used at least four times, but the authors don’t indicate where. It would be interesting to see these discussions, as well as a discussion of the connection between the curvature studied by the authors and the Lyapunov exponents.
Saturday, June 16, 2007
Wednesday, June 13, 2007
Excitons Insensitive to Environment
Screening of Excitons in Single, Suspended Carbon Nanotubes
A.G. Walsh, et al.
Nanoletters 7, 1485--1488 (2007)
URL: http://pubs.acs.org/cgi-bin/abstract.cgi/nalefd/2007/7/i06/abs/nl070193p.html
This group from Boston University has investigated the binding energy of excitons in nanotubes as a function of the dielectric constant of the environment. They repeated photoluminescence measurements on nanotubes in dry air, humid air, and water. In general, they find that the binding energy is not very sensitive to changes in the dielectric environment. The binding energy only changes by a couple tens of meV between dry air and water.
The theoretical model they use is a 1D interaction of the form V / ( |z| + Z ). Surprisingly, they find that the parameter Z scales linearly with the dielectric constant! They claim this model was solved exactly in 1959. I’m going to track down that reference. With the exact solution and scaling relation for Z, they are able to reproduce the scaling of their data as well as a scaling relation reported by Perebeinos a couple years ago.
A.G. Walsh, et al.
Nanoletters 7, 1485--1488 (2007)
URL: http://pubs.acs.org/cgi-bin/abstract.cgi/nalefd/2007/7/i06/abs/nl070193p.html
This group from Boston University has investigated the binding energy of excitons in nanotubes as a function of the dielectric constant of the environment. They repeated photoluminescence measurements on nanotubes in dry air, humid air, and water. In general, they find that the binding energy is not very sensitive to changes in the dielectric environment. The binding energy only changes by a couple tens of meV between dry air and water.
The theoretical model they use is a 1D interaction of the form V / ( |z| + Z ). Surprisingly, they find that the parameter Z scales linearly with the dielectric constant! They claim this model was solved exactly in 1959. I’m going to track down that reference. With the exact solution and scaling relation for Z, they are able to reproduce the scaling of their data as well as a scaling relation reported by Perebeinos a couple years ago.
Tuesday, June 12, 2007
Exchange and Correlation in Graphene
Chirality and Correlations in Graphene
Y. Barlas, T. Pereg-Barnea, M. Polini, R. Asgari, and A.H. MacDonald
PRL 98, 236601 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e236601
This article was a pleasure to read. I've been reading about the use of techniques from field theory in condensed matter physics lately, and I got to see them in action here.
This group from Texas has evaluated the effects of exchange and correlation on the charge and spin susceptibility of graphene. They start from the massless Dirac equation that describes low energy excitations in graphene and calculate the interaction energy. The technique they used was new to me.
First, they write down an expression for the interaction energy that depends on the structure factor of the interacting system. The interaction energy appears to be calculated by adiabatically turning on the coupling between particles. The structure factor is calculated from the density-density response function, a relation based on the fluctuation dissipation theorem. The response function can be calculated in the random phase approximation using methods of quantum field theory. The authors combine a lot of neat tools to obtain the interaction energy. It's all condensed into one paragraph in this paper. I'll have to check out their reference, Giuliani and Vignale's "Quantum Theory of the Electron Liquid."
Though the techniques are well-established, they lead to surprising results in graphene. The authors find that the sign of the correections to the spin and charge susceptibilities is opposite that of the usual electron gas. The similarity of the response functions is also different from an electron gas. The exchange effect decreases the susceptibilities in graphene, while it enhances them in the normal 2DEG. The authors trace the difference in behavior to the renormalized fermi velocity, which increases with the interaction strength.
Once I am more comfortable with the techniques I'm learning from Abrikosov and Mattuck, I'm going to try and reproduce these results.
Y. Barlas, T. Pereg-Barnea, M. Polini, R. Asgari, and A.H. MacDonald
PRL 98, 236601 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e236601
This article was a pleasure to read. I've been reading about the use of techniques from field theory in condensed matter physics lately, and I got to see them in action here.
This group from Texas has evaluated the effects of exchange and correlation on the charge and spin susceptibility of graphene. They start from the massless Dirac equation that describes low energy excitations in graphene and calculate the interaction energy. The technique they used was new to me.
First, they write down an expression for the interaction energy that depends on the structure factor of the interacting system. The interaction energy appears to be calculated by adiabatically turning on the coupling between particles. The structure factor is calculated from the density-density response function, a relation based on the fluctuation dissipation theorem. The response function can be calculated in the random phase approximation using methods of quantum field theory. The authors combine a lot of neat tools to obtain the interaction energy. It's all condensed into one paragraph in this paper. I'll have to check out their reference, Giuliani and Vignale's "Quantum Theory of the Electron Liquid."
Though the techniques are well-established, they lead to surprising results in graphene. The authors find that the sign of the correections to the spin and charge susceptibilities is opposite that of the usual electron gas. The similarity of the response functions is also different from an electron gas. The exchange effect decreases the susceptibilities in graphene, while it enhances them in the normal 2DEG. The authors trace the difference in behavior to the renormalized fermi velocity, which increases with the interaction strength.
Once I am more comfortable with the techniques I'm learning from Abrikosov and Mattuck, I'm going to try and reproduce these results.
Wednesday, June 6, 2007
Ripping a Fluid Apart
Motion of a Viscoelastic Micellar Fluid around a Cylinder: Flow and Fracture
J.R. Gladden and A. Belmonte
PRL 98, 224501 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e224501
This duo from Penn state has experimentally demonstrated both the viscous and elastic regimes of a viscoelastic fluid. As with most fluid dynamics experiments, the photos are beautiful and fascinating.
Fluids flow. Solids deform or fracture. A viscoelastic fluid is a material that exhibits both types of response depending on how it is probed. In this Letter, Gladden and Belmonte have probed both types of response in a very simple way. The take a cylinder of diameter D and pull it through a layer of their fluid at constant velocity V. By varying the diameter and velocity, they observe three different types of response:
1) Flow: the fluid moves smoothly around the cylinder and recombines behind it.
2) Cut: the fluid still flows around the cylinder, but a cavity forms around it , and some air bubbles are trapped in its wake.
3) Tear: the cylinder rips through the fluid, leaving a trail of fin-shaped tears behind it, like a cylinder pulled through a thin plastic sheet.
These three types of response are displayed very effectively in Fig. 1. By plotting response as a function of V and D, the authors create a phase diagram for the fluid. There is a linear boundary between cut and flow, and a hyperbolic boundary between tear and the other two states. There is even a triple point.
The authors give an excellent discussion of their data. The boundary between flow and cut occurs when the time scale of fluid flow around the cylinder, D/V, exceeds the relaxation time of the fluid. The boundary between tear and the other two types of response has D*V constant. The authors use scaling arguments to determine the physical meaning of this constant, which is proportional to the tear strength of the fluid.
One more interesting observation is that when the cylinder is pulled fast enough, a crack forms in front of it. To analyze the stresses on the fluid, the authors used cross polarizers. They found a dipole pattern for the stress due to the moving cylinder, but the most fascinating picture from the paper is Fig. 6, the stress field due to a cubic probe. Amazing.
The pictures alone make this article worth reading. The authors analysis of the data is excellent.
J.R. Gladden and A. Belmonte
PRL 98, 224501 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e224501
This duo from Penn state has experimentally demonstrated both the viscous and elastic regimes of a viscoelastic fluid. As with most fluid dynamics experiments, the photos are beautiful and fascinating.
Fluids flow. Solids deform or fracture. A viscoelastic fluid is a material that exhibits both types of response depending on how it is probed. In this Letter, Gladden and Belmonte have probed both types of response in a very simple way. The take a cylinder of diameter D and pull it through a layer of their fluid at constant velocity V. By varying the diameter and velocity, they observe three different types of response:
1) Flow: the fluid moves smoothly around the cylinder and recombines behind it.
2) Cut: the fluid still flows around the cylinder, but a cavity forms around it , and some air bubbles are trapped in its wake.
3) Tear: the cylinder rips through the fluid, leaving a trail of fin-shaped tears behind it, like a cylinder pulled through a thin plastic sheet.
These three types of response are displayed very effectively in Fig. 1. By plotting response as a function of V and D, the authors create a phase diagram for the fluid. There is a linear boundary between cut and flow, and a hyperbolic boundary between tear and the other two states. There is even a triple point.
The authors give an excellent discussion of their data. The boundary between flow and cut occurs when the time scale of fluid flow around the cylinder, D/V, exceeds the relaxation time of the fluid. The boundary between tear and the other two types of response has D*V constant. The authors use scaling arguments to determine the physical meaning of this constant, which is proportional to the tear strength of the fluid.
One more interesting observation is that when the cylinder is pulled fast enough, a crack forms in front of it. To analyze the stresses on the fluid, the authors used cross polarizers. They found a dipole pattern for the stress due to the moving cylinder, but the most fascinating picture from the paper is Fig. 6, the stress field due to a cubic probe. Amazing.
The pictures alone make this article worth reading. The authors analysis of the data is excellent.
Bubble Dynamics
Role of Dimensionality and Axisymmetry in Fluid Pinch-Off and Coalescence
J.C. Burton and P. Taborek
PRL 98, 224502 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e224502
A very interesting paper. Figure 2 is amazing!
This pair of researchers from UC Irvine has investigated two phenomena that occure at fluid interfaces: pinch off and coalescence. As the authors point out, these are topological transitions which involve a conversion of interfacial energy into kinetic energy of fluid flow.
Pinch off is what happens when a droplet is pulled apart. At some point it breaks up into two or more pieces. Coalescence is the merging of two droplets into one. The authors have used high-speed, high-resolution video to study these two processes for both two-dimensional and three-dimensional alkane droplets.
What Burton and Taborek observed in 2D pinch off blows me away. Watch the video. When a 3D droplet is pulled apart, it transforms into two globules connected by a thin filament just before pinch off. At one of the filament-globule interfaces, the filament separates, leaving a flat surface and a cone whose opening angle is determined by the fluid properties of the droplet. For the 2D droplets in this paper, the filament starts to pinch off at both ends, so there are two large globules on the left and right connected to a small globule in the middle. As these filaments pinch off, they repeat the process on a smaller scale. The authors observe 5 generations of successive pinch offs, with each generation of droplets smaller than its parents by a factor of 3. The end result is that a single droplet has broken up into about 30 smaller droplets, the smallest of which are almost 1000 times smaller than the original droplet. If that's confusing, watch the video. It's amazing!
Burton and Taborek did not observe any striking differences in the coalescence of 2D and 3D droplets. As two spherical droplets start to merge, the radius of the connected region grows linearly in time. At longer times, the radius grows with the square root of time. The authors use the two scaling laws to determine the approximate size of the droplet at the crossover. The crossover length scale is 2 orders of magnitude larger than the natural length scale of the fluid system.
This is a very interesting paper, with beautiful images, clean data, and insightful explanations.
J.C. Burton and P. Taborek
PRL 98, 224502 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e224502
A very interesting paper. Figure 2 is amazing!
This pair of researchers from UC Irvine has investigated two phenomena that occure at fluid interfaces: pinch off and coalescence. As the authors point out, these are topological transitions which involve a conversion of interfacial energy into kinetic energy of fluid flow.
Pinch off is what happens when a droplet is pulled apart. At some point it breaks up into two or more pieces. Coalescence is the merging of two droplets into one. The authors have used high-speed, high-resolution video to study these two processes for both two-dimensional and three-dimensional alkane droplets.
What Burton and Taborek observed in 2D pinch off blows me away. Watch the video. When a 3D droplet is pulled apart, it transforms into two globules connected by a thin filament just before pinch off. At one of the filament-globule interfaces, the filament separates, leaving a flat surface and a cone whose opening angle is determined by the fluid properties of the droplet. For the 2D droplets in this paper, the filament starts to pinch off at both ends, so there are two large globules on the left and right connected to a small globule in the middle. As these filaments pinch off, they repeat the process on a smaller scale. The authors observe 5 generations of successive pinch offs, with each generation of droplets smaller than its parents by a factor of 3. The end result is that a single droplet has broken up into about 30 smaller droplets, the smallest of which are almost 1000 times smaller than the original droplet. If that's confusing, watch the video. It's amazing!
Burton and Taborek did not observe any striking differences in the coalescence of 2D and 3D droplets. As two spherical droplets start to merge, the radius of the connected region grows linearly in time. At longer times, the radius grows with the square root of time. The authors use the two scaling laws to determine the approximate size of the droplet at the crossover. The crossover length scale is 2 orders of magnitude larger than the natural length scale of the fluid system.
This is a very interesting paper, with beautiful images, clean data, and insightful explanations.
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