Why Statistical Physics Should Be A Foundation For Materials Design

Marc Miskin

Author: Marc Miskin

Affiliation:
James Franck Institute and Department of Physics, The University of Chicago, USA.

The fact that major epochs in human history have been named after materials like bronze, steel, silicon, and stone expresses both how important materials are for technology and how long it can take before a new material is discovered. Even today, the timespan to convert materials' discoveries into functioning technologies takes upwards of 20 years. In part this is because creating technologically useful materials requires selecting a wide range of parameters to optimize the material’s performance. While tools like statistical physics are useful for describing a material’s behavior given a set of parameters, it remains unknown how to generally invert these relationships to target desired behavior. This task is called materials design and it is a new concept at the forefront of materials research.

Recently, several methods have emerged across disciplines that draw upon optimization and simulation to create computer programs that tailor material responses to specified behaviors. However, so far the methods developed either involve black-box techniques, in which the optimizer operates without explicit knowledge of the physical laws that underpin the material’s behavior, or require carefully tuned algorithms with applicability limited to a narrow subclass of materials.

Our recent publication titled “Turning Statistical Physics Models into Materials Design Engines” [1] presents a new perspective for material design. In contrast to prior approaches, it is broad enough to be applied without modification to any system that is well described by statistical mechanics and also retains much of the key insight that is at the heart of statistical physics. In short, our formalism allows a user to transform the capacity to predict material behavior into an optimizer that tunes it.

We achieved this by examining the fundamental relationship between microstate configurations and material properties. Statistical physics poses the idea that materials are intrinsically statistical objects: the properties a bulk material has are best calculated by averaging over all the possible configurations for the material's microscopic parts. Our insight was that design programs should focus on tailoring materials at the level of micro-states themselves, rather than simply focusing on the bulk emergent properties.

For instance, suppose that the pressure and temperature affect the stiffness of a given material and the goal is to set these two parameters to make the stiffest material. The black-box approach is to view this as an optimization problem with pressure and temperature as inputs and stiffness as an output. Yet this view completely ignores the micro-states. A better perspective is to treat the control parameters as means to alter the likelihood of micro-states. To design the material, an algorithm should tweak the parameters so that the material is more likely to be in micro-states with the target behavior.

This idea is the kernel of our approach: we built a formalism out of this concept and tackle materials' design at the level of micro-state information. This gave us a program that is broad enough to address the range of materials that are well described by statistical physics, and we achieve this with a boost in efficiency, thanks to the extra information extracted from the microstate configurations.

To test our approach, we constructed test problems that a good materials optimization scheme should be able to address by itself. Material scientists need optimizers that can solve problems where the search landscape has little variation between candidate materials, juggle multiple potentially competing physical effects, operate in high dimensional search spaces, tune the processing conditions that a material is subject to, and operate when on real-world scale optimization problems. We then translated these challenges into physical test problems, and compared our approach against optimization schemes that we have used successfully for materials design in the past. Given the criterion that the best optimizer is the one that has to make the fewest guesses to arrive at the material that performs a target function, our optimizer outperformed all of our old standards.

Probably the two most fascinating solutions presented in the paper [1] are the polymer folding problem and the directed self-assembly problem. In the polymer folding problem, we asked our optimizer to tune the interaction strengths between 6 beads attached to each other along a linear chain (Figure 1). Because the interactions are attractive, when they are strong enough the chain will fold itself up into a compact shape. The goal here was to make the chain fold into a specific shape: an octahedron. Its an interesting problem because it's well known that simply making all the interactions large will not produce an octahedral geometry. Instead, the interactions needed to be developed into three separate families to generate octahedron and it took hard work from the colloidal self-assembly community to show that this worked. So it was very exciting for us when our optimizer not only produced a virtually identical motif, but managed to yield the result in the span of hours.

Figure 1: Given a polymer of 6 beads each of different color, how should the strength of each short-ranged interaction be picked so that the polymer self-folds into an octahedron geometry? Each image shows a typical polymer configuration obtained at each stage in an optimization using our new approach. The optimizer essentially starts from random, chain like geometries and after ~200 cycles transforms them into the target shape.

There is a similar story behind our directed self-assembly problem. In this case, the material is a polymer made of two types of beads. The goal is to pattern a substrate with a thin strip that has an affinity for a particular one of those two beads (Figure 2). By setting the strip width and the strength of affinity just right, it is possible to make the polymer self-assemble into stripes containing only one polymer type followed by a stripe of the other polymer type and so on. This idea holds serious promise as a next-generation manufacturing technology for semiconductors because the sizes of these stripes are on the order of nanometers. By using the polymer stripes as stencils or masks, it is possible to make next generation circuits or hard drive media with features significantly smaller than what current processing techniques allow. What we found was that not only can our optimizer produce solutions to the problem of tailoring interaction strengths for this kind of directed self-assembly, but that it does so between 5-130X faster than approaches we had tried in the past. To put this into context, solving a directed self assembly problem in the past took us roughly 1 week. Now we can solve them in just under 12 hours.

Figure 2: Given melt of polymer chains each made from half a-type (red) and half b-type (blue) beads, how should one tune the interactions between the a and b beads and a substrate so that the polymer melt self-assembles into equally spaced stripes of a and b? On the top is an image of the original polymer melt configuration for randomly seeded interaction parameters. On the bottom shows the structures that result from using our algorithm to elicit self-assembly. Note the substrate has been colored based on the affinity for each type.

Speaking broadly, materials by design is a radical shift in how we transform bulk matter into useful technology. Historically materials have been either discovered by accident or appropriated from nature to perform technological functions. What we're after is the capacity to systematically identify which materials produce a target response. The benefit of this paradigm is that increasingly complex materials could be cooked up automatically to meet specific technological demands. Our algorithm is a small part of this growing field, but the hope is that it will inspire others to consider their expertise on materials within the new context of design.

Our experience in the past has been that it can be difficult to get started on building design engines for a new material. If the problem isn't posed the right way or the optimizer isn't appropriately tailored, it may require a substantial investment of time to construct an optimization scheme that actually works. What we tried to show in this paper is that our formalism works broadly over a range of very different physical problems without any need for additional modifications. It works out of the box, so to speak, for designing any material that can be simulated using a statistical physics approach. Our hope is that this robustness will translate into a reduction in the time researchers need to spend building design algorithms, and free them up to focus on the task of making exotic materials.

Some of our recent work along these lines can be found here:
[1] Marc Z. Miskin, Gurdaman Khaira, Juan J. de Pablo, Heinrich M. Jaeger, "Turning statistical physics models into materials design engines", Proceedings of the National Academy of Sciences, 113, 34-39 (2016). Full Text.
[2] Marc Z. Miskin, Heinrich M. Jaeger, "Adapting granular materials through artificial evolution", Nature Materials, 12, 326-331 (2013). Abstract.
[3] Marc Z. Miskin, Heinrich M. Jaeger, "Evolving design rules for the inverse granular packing problem", Soft Matter, 10, 3708-3715 (2014). Abstract.
[4] Jian Qin, Gurdaman S. Khaira, Yongrui Su, Grant P. Garner, Marc Miskin, Heinrich M. Jaeger, Juan J. de Pablo, "Evolutionary pattern design for copolymer directed self-assembly", Soft Matter, 9, 11467–11472 (2013). Abstract.

Demon-like Algorithmic Quantum Cooling

Man-Hong Yung

Author: Man-Hong Yung

Affiliation: Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China

Efficient cooling methods are indispensable for studying fundamental physical phenomena at low temperatures, leading to important discoveries such as superconductivity, Bose-Einstein condensation and the quantum Hall effect in the laboratory. As numerical tools, cooling algorithms are crucial for studying ground-state problems in computational physics, and for solving complex optimization problems in computer science. Recent advances in quantum information science open new venues for scientific discovery in physics and chemistry. However, simulating low-temperature properties of many-body systems remains one of the major challenges for both theory and experiment. In a recent work published in Nature Photonics [1], we developed a novel cooling method, called demon-like algorithmic quantum cooling (DLAC), that provides a solution to the problem of cooling quantum systems in a way similar to that of Maxwell’s demon. The method of DLAC share some features of heat-bath algorithmic cooling [2,3,4] and an approach of cooling that we call the dissipative open-system approach [5,6].

Heat-bath algorithmic cooling [2,3,4] is one way of cooling classical spin systems, which would be useful for initializing qubits in the ‘0’ state. The idea of heat-bath algorithmic cooling is to first re-distribute the entropy within a group of spins. Then, the spin (or the group of spins) containing relatively high entropy is put into contact with a heat reservoir to pump away the excess entropy. This procedure causes the overall entropy of the system to decrease.

The dissipative open-system approach [5,6] aims to drive a quantum state to the ground state of a certain kind of Hamiltonians, called frustrated Hamiltonians. This approach is based on the application of appropriate quantum jump operators to the subspace of the quantum state that would give an energy penalty.

Our DLAC approach is similar to the heat-bath algorithmic cooling, in the sense that the entropy of the system is re-distributed, but in a superposition of states, instead of different spins. DLAC is also similar to the dissipative open-system approach, in the sense that the superposition is collapsed once through a projective measurement of the ancilla qubit. The resulting operation is equivalent to applying a jump operator that causes the entropy to increase or decrease, depending on the measurement outcome. A more detailed analysis is shown in Fig. 1.

Fig 1. (View higher resolution by clicking on the figure) Basic principle of the cooling method. (a) Logic diagram of the feedback cooling system. The cooling module produces two outcomes, correlated with heating and cooling. The measurement result can be mapped into the position x of a 1D random walker; when the walker goes beyond its starting position x = 0 to the negative position x = −1, the particle is either rejected or recycled. (b) The quantum circuit diagram of the cooling module. It includes a controlled evolution for time t and an energy bias parameter γ. (c) The relative change in the population of the output state post-conditioned with a “cooling” measurement result depends on the eigenenergy. The lower the energy, the higher the gain.

The DLAC approach can be implemented in many systems such as trapped ions, quantum dots, and superconducting qubits. We chose to implement it in a quantum-optical system operated by the optics group Key Laboratory of Quantum Information of the University of Science and Technology of China (USTC). The optics group is lead by Prof. Chuan-Feng Li and the experimental setup (see Fig. 2) is mainly designed and performed by Prof. Jin-Shi Xu. The quantum system to be cooled is the polarization of a single photon.

Fig 2. (View higher resolution by clicking on the figure) Experimental setup. (a) The input photon state is prepared by a polarization beam splitter (PBS), a half-wave plate (HWP) and a quarter-wave plate (QWP). The cooling module is a polarization-dependent Sagnac interferometer. Split by the PBS, the horizontal and vertical components of a polarized photon propagate in opposite paths within the interferometer. Two HWPs operate on the corresponding polarization components. The quartz plate compensator (PC) is used to compensate for the relative phase in the interferometer. These elements simulate the system Hamiltonian. Both paths are then recombined on the same PBS. The photons at output ports 0 and 1 represent the cases of cooling and heating, respectively. The photon proceeds to the next cooling module depending on the value of the position x of the random walker. The final quantum state is reconstructed using quantum state tomography, with the measurement bases defined by a QWP, HWP and PBS. The photon is detected by a single photon detector (SPD) equipped with a 3 nm interference filter (IF). (b) Sketch of the random walker evolution.

In the experiment, we were able to achieve three steps of cooling (see Fig. 3) with two different strategies, namely evaporative and recycling. These two strategies determine what we do when we have a resulting state that has an entropy higher than the initial state. The evaporative strategy basically throws away those states and recycling strategy replace them with the original state.

Fig 3. (View higher resolution by clicking on the figure) a, Mean energies of the evaporative and recycling pseudo cooling strategies. b, The corresponding theory and experiment for the number n of copies obtained out of N initial copies at consecutive steps. The total probability of the recycling algorithm is set to 1, correcting for experimental losses. c, Mean energies for 3 pseudo cooling steps of the evaporative (non-recycling) strategy for different θ. Different maps correspond to theory and experiment for Hamiltonians Pauli Z and Pauli X, and different initial states (step 0). Error bars represent the statistical distribution of experimental measurements.

In conclusion, this optical experiment has adequately verified the feasibility of the our DLAC approach of cooling quantum systems. To extend the implementation of the cooling method to a larger quantum system, the key consideration is to choose a system where both projective measurement and feedback control can be efficiently incorporated. These technologies are currently not widely available, but they should become mature in the near future. For an extended discussion of this work in the context of Maxwell’s demon, see the associated News and View [7] by Seth Lloyd.

References:
[1] Jin-Shi Xu, Man-Hong Yung, Xiao-Ye Xu, Sergio Boixo, Zheng-Wei Zhou, Chuan-Feng Li, Alán Aspuru-Guzik, Guang-Can Guo, "Demon-like algorithmic quantum cooling and its realization with quantum optics". Nature Photonics, 8, 113-118 (2014). Abstract.
[2] P. Oscar Boykin, Tal Mor, Vwani Roychowdhury, Farrokh Vatan, Rutger Vrijen, "Algorithmic cooling and scalable NMR quantum computers". Proceedings of the National Academy of Scienses USA, 99, 3388-3393 (2002). Full Article.
[3] J. Baugh, O. Moussa, C.A. Ryan, A. Nayak, R. Laflamme, "Experimental implementation of heat-bath algorithmic cooling using solid-state nuclear magnetic resonance". Nature, 438, 470-473 (2005). Abstract.
[4] José M. Fernandez, Seth Lloyd, Tal Mor, Vwani Roychowdhury, "Algorithmic cooling of spins: a practicable method for increasing polarization". International Journal of Quantum Information, 2, 461–477 (2004). Abstract.
[5] B. Kraus, H. P. Büchler, S. Diehl, A. Kantian, A. Micheli, P. Zoller, "Preparation of entangled states by quantum Markov processes". Physical Review A, 78, 042307 (2008). Abstract.
[6] Frank Verstraete, Michael M. Wolf, J. Ignacio Cirac, "Quantum computation and quantum-state engineering driven by dissipation". Nature Physics, 5, 633-636 (2009). Abstract.
[7] Seth Lloyd, "Quantum optics: Cool computation, hot bits". Nature Photonics, 8, 90–91 (2014). Abstract.

The Formation of Two Supermassive Black Holes from A Single Collapsing Supermassive Star

From left to right: (top row) Christian Reisswig, Christian D. Ott, Ernazar Abdikamalov; (bottom row) Roland Haas, Philipp Mösta, Erik Schnetter

Authors: Christian Reisswig^1,*, Christian D. Ott^1,2,+, Ernazar Abdikamalov¹, Roland Haas¹, Philipp Moesta¹, Erik Schnetter^3,4,5

Affiliation:
¹TAPIR, California Institute of Technology, Pasadena, CA, USA
²Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), The University of Tokyo, Kashiwa, Japan
³Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
⁴Department of Physics, University of Guelph, Guelph, ON, Canada
⁵Center for Computation & Technology, Louisiana State University, Baton Rouge, LA, USA

^*NASA Einstein Fellow
⁺Alfred P. Sloan Research Fellow

The existence of supermassive black holes with masses a billion times the mass of our sun at high redshifts z>7 [1] is one of the mysteries in our understanding of the early history of the universe. At redshift z=7, the universe was less then one billion years old. This leads to a serious problem: how is it possible for black holes to acquire this tremendous amount of mass over a short timescale of just one billion years? A common theory of black hole growth assumes as a starting point the collapse of the very first stars, so called Population III stars. Population III stars may have had masses around 100 times the mass of our sun. The collapse of such a star can leave behind a black hole of similar mass that then grows via subsequent accretion of material from its surroundings. This process can yield quite massive black holes, but in order to reach supermassive scales within only one billion years, the accretion process must be rather extremely rapid to enable fast black hole growth. These high required accretion rates, however, seem to be difficult to be maintained due to, e.g. strong outgoing radiation that can blow away the surrounding gas that otherwise would be accreted onto the black hole [2]. The model, therefore, has difficulties of explaining the existence of very massive black holes in the early universe.

Another model which has recently regained attention is supermassive star collapse. Supermassive stars have originally been proposed by Hoyle & Fowler in the 1960s as a model for strong distant radio sources [3]. Such stars have masses up to a million times the mass of our sun and potentially formed in the monolithic collapse of primordial gas clouds that existed in the early universe [4, 5]. Unlike ordinary stars, which are mainly powered by nuclear burning, supermassive stars are mainly stabilized against gravity by their own photon radiation field that originates from the very high interior temperatures generated by gravitational contraction. During their short lives, they slowly cool due to the emitted photon radiation that keeps the stars in hydrostatic equilibrium. The colder stellar gas can be more easily compressed by the inward gravitational pull, and as a consequence, the stars slowly contract and become more compact. This process continues for a few million years until the stars reach sufficient compactness for gravitationally instability to set in. This general relativistic instability inevitably leads to gravitational collapse. One possible outcome of the collapse is a massive black hole containing most of the original mass of the star. Since the 'seed' mass of the nascent black hole is already pretty large, subsequent growth via accretion from the surroundings can easily push the black hole to supermassive scales within the available time without the need of extreme accretion rates and thus without any strong photon radiation that may blow away the surrounding accreting gas.

In our recent article published in Physical Review Letters [6], we study non-axisymmetric effects in the collapse of supermassive stars. The starting point of our models are supermassive stars which are at the onset of gravitational collapse. We use general relativistic hydrodynamic supercomputer simulations with fully dynamical non-linear space-time evolution to investigate the behavior and dynamics of collapsing supermassive stars. Such computer models have been considered in previous studies [7,8,9,10], however, mostly in axisymmetry.

 In an axisymmetric configuration, a supermassive star maintains a spherical shape during its collapse, which is possibly flattened due to rotation. In these previous studies, it has been shown that the possible outcome is either a single massive rotating black hole, or, alternatively, a powerful supernova explosion which completely disrupts the star. In our case, we select an initial stellar model which is rapidly rotating and leads to black hole formation. In fact, it is so rapidly rotating that the shape of our star is no longer spheroidal, but rather resembles the shape of a 'quasi'-torus where the maximum density is off-center and thus forming a central high-density ring (see upper left panel of Figure 1).

Figure 1: (To view higher resolution click on the image) The various stages encountered during the collapse of a supermassive star with an initial m=2 standing density wave perturbation. Each panel shows the density distribution in the equatorial plane.

Such a configuration is unstable to tiny density perturbations that may be present at the onset of collapse [10]. This instability is particularly strong for perturbations in the form of standing poloidal density waves with one (m=1) or two (m=2) maxima. Due to this instability, these perturbations grow exponentially during the collapse, and can lead to significant deformations away from axisymmetry. The nature of the instability typically leads to the formation of orbiting high-density clumps of matter inside the collapsing star (see upper right panel of Figure 1). Since the m=1 and m=2 perturbations grow fastest, either one or two high-density clumps will form, depending on the initial perturbation of the stellar density. These high density fragments continue to grow rapidly during the collapse, thus becoming denser and hotter. 

Once temperatures of more than one billion Kelvins are reached, a process sets, which is called electron-positron pair creation. The creation of particle pairs is possible because there is enough energy available in the surrounding gas to spontaneously create a particle and its anti-particle, in this case electrons and positrons. The pair creation process has the effect of taking out energy from the gas fragments, thus dramatically reducing their local pressure. The reduction in pressure support leads to a rapid increase in the central density within each fragment up to the point at which the fragments become so dense that event horizons appear around each of them (center left panel of Figure 1). In the case of an initial m=2 density perturbation, two black holes form that orbit each other. Since two black holes in close orbit emit very powerful gravitational radiation - ripples of space-time that travel at the speed of light - , the associated loss of energy causes the black hole orbits to shrink, leading to an inspiralling motion (red lines in the center left panel of Figure 1). The leading order mode of the corresponding emitted gravitational wave signal is shown in the lower panel Figure 2.

Figure 2: (To view higher resolution click on the image) The upper panel shows the time evolution of the density maximum until black hole formation. The center panel shows the mass and spin evolution of the black holes. The lower panel shows the emitted leading order gravitational wave signal.

It resembles the typical quasi-sinusoidal oscillatory signal expected from binary black hole mergers: as the orbit shrinks, the emitted radiation becomes higher in frequency. The inspiral continues until a common event horizon appears, marking the merger of the two black holes (lower left panel of Figure 1). The black hole merger remnant is initially deformed into a peanut shape, which quickly relaxes into a spherical shape by emitting exponentially decaying gravitational ring-down radiation. This is shown in the lower panel of Figure 2. The peak amplitude of the waveform corresponds to the black hole merger. From there, the signal quickly decays due to black hole ring-down. By the end of our simulation, the remnant black hole is rapidly rotating and is surrounded by a massive accretion disk (lower right panel of Figure 1).

The formation of two merging black holes requires a particular choice of initial stellar model parameters at the onset of collapse: (i) we require rapid rotation and (ii) a poloidal m=2 standing density wave perturbation must be present. This naturally leads to the question of the likelihood of our model. Recent cosmological simulations of collapsing primordial gas clouds - the potential birth sites for supermassive stars - indicate that rapid rotation is very likely [4]. Curiously, the same simulations also show that an m=2 deformation arises at the center of the clouds where the supermassive star will eventually form. Unfortunately, these simulations currently do not offer sufficient spatial resolution to investigate the formation of supermassive stars in the collapse of primordial gas clouds in detail. Further research will be necessary to self-consistently model the formation of supermassive stars that may inform us about the stellar conditions at the onset of collapse.


The new and exciting prediction that two black holes can form in the collapse of a single star gives rise to very efficient gravitational wave emission compared to models where only one black hole forms. The emitted gravitational radiation in our model configuration is so powerful that future space-borne gravitational wave observatories might see the signal from the edge of our universe. This has important implications for cosmology. If detected, the signal will inform us about the formation processes of supermassive stars and supermassive black holes in the early universe and will allow us to test the validity of the supermassive star collapse pathway to supermassive black hole formation.

Acknowledgements: This research is partially supported by NSF grant nos. PHY-1151197, AST-1212170, PHY-1212460, and OCI-0905046, by the Alfred P. Sloan Foundation, and by the Sherman Fairchild Foundation. CR acknowledges support by NASA through Einstein Postdoctoral Fellowship grant number PF2-130099 awarded by the Chandra X-ray center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. RH acknowledges support by the Natural Sciences and Engineering Council of Canada. The simulations were performed on the Caltech compute cluster Zwicky (NSF MRI award No. PHY-0960291), on supercomputers of the NSF XSEDE network under computer time allocation TG-PHY100033, on machines of the Louisiana Optical Network Initiative under grant loni_numrel08, and at the National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the US Department of Energy under contract DE-AC02-05CH11231.

References:
[1] Daniel J. Mortlock, Stephen J. Warren, Bram P. Venemans, Mitesh Patel, Paul C. Hewett, Richard G. McMahon, Chris Simpson, Tom Theuns, Eduardo A. Gonzáles-Solares, Andy Adamson, Simon Dye, Nigel C. Hambly, Paul Hirst, Mike J. Irwin, Ernst Kuiper, Andy Lawrence, Huub J. A. Röttgering, "A luminous quasar at a redshift of z = 7.085", Nature, 474, 616 (2011). Abstract.
[2] Marcelo A. Alavarez, John H. Wise, Tom Abel, "Accretion onto the first stellar-mass black holes", Astrophysical Journal Letters, 701:L133 (2009). Abstract.
[3] F. Hoyle, William A. Fowler, "Nature of strong radio sources", Nature, 197, 533 (1963). Abstract.
[4] Jun-Hwan Choi, Isaac Shlosman, Mitchell C. Begelman, "Supermassive black hole formation at high redshifts via direct collapse: physical processes in the early stage", Astrophysical Journal, 774:149, 18 (2013). Abstract.
[5] M. A. Latif, D. R. G. Schleicher, W. Schmidt, J. Niemeyer, "Black hole formation in the early Universe", Monthly Notices of the Royal Astronomical Society, 433, 1607-1618 (2013). Abstract.
[6] Christian Reisswig, Christian D. Ott, Ernazar Abdikamalov, Roland Haas, Philipp Moesta, "Formation and Coalescence of Cosmological Supermassive-Black-Hole Binaries in Supermassive-Star Collapse", Physical Review Letters, 111, 15, 151101 (2013). Abstract.
[7] Pedro J. Montero, Hans-Thomas Janka, Ewald Mueller, "Relativistic collapse and explosion of rotating supermassive stars with thermonuclear effects", Astrophysical Journal, 749:37, 14 (2012). Article.
[8] Motoyuki Sajio, Ian Hawke, "Collapse of differentially rotating supermassive stars: post black hole formation", Physical Review D, 80, 064001 (2009). Abstract.
[9] Masaru Sibata, Stuart L. Shapiro, "Collapse of a rotating supermassive star to a supermassive black hole: fully relativistic simulations", Astrophysical Journal, 572:L39 (2002). Article.
[10] Burkhard Zink, Nikolas Stergioulas, Ian Hawke, Christian D. Ott, Erik Schnetter, Ewald Mueller, "Nonaxisymmetric instability and fragmentation of general relativistic quasitoroidal stars", Physical Review D, 76, 024019 (2007). Abstract.

Three-Body Force in Nucleus

David Dean (left) and Hai Ah Nam (right)

The nucleus of an atom, like most everything else, is more complicated than we first thought. Just how much more complicated is the subject of a Petascale Early Science project led by Oak Ridge National Laboratory's David Dean.

According to findings outlined by Dean and his colleagues in a recent paper in the journal Physical Review Letters [1], researchers who want to understand how and why a nucleus hangs together as it does and disintegrates when and how it does have a very tough job ahead of them. Scientists from five institutes have contributed to this project: (from USA) Department of Physics and Astronomy, Iowa State University; Lawrence Livermore National Laboratory; Michigan State University; Oak Ridge National Laboratory; (from Canada) TRIUMF, Vancouver.

Specifically, they must take into account the complex nuclear interactions known as the three-body force.

Nuclear theory to this point has assumed that the two-body force is sufficient to explain the workings of a nucleus. In other words, the half-life or decay path of an unstable nucleus was to be understood through the combined interactions of pairs of protons and neutrons within.

An accurate picture of the carbon-14 nucleus must consider the interactions among protons and neutrons both in pairs (known as the two-body force, left) and in threes (known as the three-body force, right).

Dean's team, however, determined that the two-body force is not enough; researchers must also tackle the far more difficult challenge of calculating combinations of three particles at a time (three protons, three neutrons, or two of one and one of the other). This approach yields results that are both different from and more accurate than those of the two-body force.

Nuclei are held together by the strong force, one of four basic forces that govern the universe (The other three are gravity, which holds planets, solar systems, and galaxies together and pins us to the ground, the electromagnetic force, which holds matter together and keeps us from, for instance, falling through the ground, and the weak force, which drives nuclear decay). The strong force acts primarily to combine elementary particles known as quarks into protons and neutrons through the exchange of force carriers known as gluons. Each proton or neutron has three quarks. The strong force also holds neighboring protons and neutrons together into a nucleus.

It does so imperfectly, however. Many nuclei are unstable and will eventually decay, emitting one or more particles and becoming a smaller nucleus. While we cannot say specifically when an individual nucleus will decay, we can determine the likelihood it will do so within a certain time. Thus an isotope's half-life is the time it takes half the nuclei in a sample to decay. Known half-lives range from an absurdly small fraction of a second for beryllium-8 to more than 2 trillion trillion years for tellurium-128.

One job of nuclear theory, then, is to determine why nuclei have different half-lives and predict what those half-lives are.

"For a long time, nuclear theory assumed that two-body forces were the most important and that higher-body forces were negligible," noted team member and ORNL computational physicist Hai Ah Nam. "You have to start with an assumption: How to capture the physics best with the least complexity?"

Two factors complicate the choice of approaches. First, two-body interactions do accurately describe some nuclei. Second, accurate calculations including three-body forces are very difficult and demand state-of-the-art supercomputers such as ORNL's Jaguar, the most powerful system in the United States. With the ability to churn through as many as 2.33 thousand trillion calculations each second, or 2.33 petaflops, Jaguar gave the team the computing muscle it needed to analyze the carbon-14 nucleus using the three-body force.

Carbon-14, with six protons and eight neutrons, is the isotope behind carbon dating, allowing researchers to determine the age of plant- or animal-based relics going back as far as 60,000 years. It was an ideal choice for this project because studies using only two-body forces dramatically underestimate the isotope's half-life, which is around 5,700 years.

"With Jaguar we are able to do ab initio calculations, using three-body forces, of the half-life for carbon-14," Nam said. "It's an observable that is sensitive to the three-body force. This is the first time that we've demonstrated at this large scale how the three-body force contributes."

The three-body force does not replace the two-body force in these calculations, she noted; rather, the two approaches are combined to present a more refined picture of the structure of the nucleus. In the carbon-14 calculation, the three-body force serves to correct a serious underestimation of the isotope's half-life produced by the two-body force alone.

Dean and his colleagues used an application known as Many Fermion Dynamics, nuclear, or MFDn, which was created by team member James Vary of Iowa State University. With it, they tackled the carbon-14 nucleus using an approach known as the nuclear shell model and performing ab initio calculations—or calculations based on the fundamental forces between protons and neutrons.

Analogous to the atomic shell model that explains how many electrons can be found at any given orbit, the nuclear shell model describes the number of protons and neutrons that can be found at a given energy level. Generally speaking, the nucleons gather at the lowest available energy level until the addition of any more would violate the Pauli exclusion principle, which states that no two particles can be in the same quantum state. At that point, some nucleons bump up to the next higher energy level, and so on. The force between nucleons complicates this picture and creates an enormous computational problem to solve.

The carbon-14 calculation, for instance, involved a billion-by-billion matrix containing a quintillion values. Fortunately, most of those values are zero, leaving about 30 trillion nonzero values to then be multiplied by a billion vector values. As Nam noted, just keeping the problem straight is a phenomenally complex task, even before the calculation is performed; those 30 trillion matrix elements take up 240 terabytes of memory.

"Jaguar is the only system in the world with the capability to store that much information for a single calculation," Nam said. "This is a huge, memory-intensive calculation."

The job is even more daunting with larger nuclei, and researchers will have a long wait for supercomputers powerful enough to compute the nature of the largest nuclei using the three-body force. Even so, if the three-body force gives more accurate results than the two-body force, should researchers be looking at four, five, or more nucleons at a time?

"Higher-body forces are still under investigation, but it will require more computational resources than we currently have available," Nam said.

Reference
[1] P. Maris, J. P. Vary, P. Navráti, W. E. Ormand, H. Nam, and D. J. Dean, "Origin of the Anomalous Long Lifetime of ¹⁴C", Physical Review Letters, 106, 202502 (2011). Abstract.

[The text is written by Leo Williams of Oak Ridge National Laboratory]

Solving the Superconductor Puzzle

Thomas A. Maier [photo courtesy: Oak Ridge National Laboratory]

Superconducting materials, which transmit power resistance-free, are found to perform optimally when high- and low-charge density varies on the nanoscale level, according to research performed at the US Department of Energy's Oak Ridge National Laboratory (ORNL) and Institut für Theoretische Physik, Zürich, Switzerland.

In research toward better understanding the dynamics behind high-temperature superconductivity, the ORNL scientists rewrote computational code for the numerical Hubbard model that previously assumed copper-compound superconducting materials known as cuprates to be homogenous — the same electron density — from atom to atom. The paper is published in Physical Review Letters [1].

Lead author Thomas Maier and colleagues Gonzalo Alvarez, Michael Summers and Thomas Schulthess received the Association for Computing Machinery Gordon Bell Prize two years ago for their high-performance computing application. The application has now been used to examine the nanoscale inhomogeneities in superconductors that had long been noticed but left unexplained.













Researchers have found that atom clusters with inhomogenous stripes of lower density (shown in red) raise critical temperature needed to reach superconductor state [Courtsey: ORNL]

"Cuprates and other chemical compounds used as superconductors require very cold temperatures, nearing absolute zero, to transition from a phase of resistance to no resistance," said Jack Wells, director of the Office of Institutional Planning and a former Computational Materials Sciences group leader.

Liquid nitrogen is used to cool superconductors into phase transition. The colder the conductive material has to get to reach the resistance-free superconductor phase, the less efficient and more costly are superconductor power infrastructures. Such infrastructures include those used on magnetic levitation trains, hospital Magnetic Resonance Imaging, particle accelerators and some city power utilities.

In angle-resolved photoemission experiments and transport studies on a cuprate material that exhibits striped electronic inhomogeneity, scientists for years observed that superconductivity is heavily affected by the nanoscale features and in some respect even optimized.

"The goal following the Gordon Bell Prize was to take that supercomputing application and learn whether these inhomogenous stripes increased or decreased the temperature required to reach transition," Wells said. "By discovering that striping leads to a strong increase in critical temperature, we can now ask the question: is there an optimal inhomogeneity?"

In an ideal world, a material could become superconductive at an easily achieved and maintained low temperature, eliminating much of the accompanying cost of the cooling infrastructure.

"The next step in our progress is a hard problem," Wells said. "But from our lab's point of view, all of the major tools suited for studying this phenomenon — the computational codes we've written, the neutron scattering experiments that allow us to examine nanoscale properties — are available to us here."

Reference
[1] T. A. Maier, G. Alvarez, M. Summers, T. C. Schulthess, "Dynamic Cluster Quantum Monte Carlo Simulations of a Two-Dimensional Hubbard Model with Stripelike Charge-Density-Wave Modulations: Interplay between Inhomogeneities and the Superconducting State", Phys. Rev. Lett. 104, 247001 (2010). Abstract.

[We thank Oak Ridge National Laboratory for materials used in this posting]