A soluble random-matrix model for relaxation in quantum systems

We study the relaxation of a degenerate two-level system interacting with a heat bath, assuming a random-matrix model for the system-bath interaction. For times larger than the duration of a collision and smaller than the Poincaré recurrence time, the survival probability of still finding the system at timet in the same state in which it was prepared att=0 is exactly calculated.

On an exact populationary model of genetic algorithms

Genetic algorithms (GAs) are search methods that are being employed in a multitude of applications with extremely large search spaces. Recently, there has been considerable interest among GA researchers in understanding and formalizing the working of GAs. In an earlier paper, we have introduced the notion of binomially distributed populations as the central idea behind an exact ''populationary'' model of the large-population dynamics of the GA operators for objective functions called ''functions of unitation.'' In this paper, we extend this populationary model of GA dynamics to a more general class of objective functions called functions of unitation variables. We generalize the notion of a binomially distributed population to a generalized binomially distributed population (GBDP). We show that the effects of selection, crossover, and mutation can be exactly modelled after decomposing the population into GBDPs. Based on this generalized model, we have implemented a GA simulator for functions of two unitation variables-GASIM 2, and the distributions predicted by GASIM 2 match with those obtained from actual GA runs. The generalized populationary model of GA dynamics not only presents a novel and natural way of interpreting the workings of GAs with large populations, but it also provides for an efficient implementation of the model as a GA simulator. (C) Elsevier Science Inc. 1997.

Absolute Entropy and Energy of Carbon Dioxide Using the Two-Phase Thermodynamic Model

The two-phase thermodynamic (2PT) model is used to determine the absolute entropy and energy of carbon dioxide over a wide range of conditions from molecular dynamics trajectories. The 2PT method determines the thermodynamic properties by applying the proper statistical mechanical partition function to the normal modes of a fluid. The vibrational density of state (DoS), obtained from the Fourier transform of the velocity autocorrelation function, converges quickly, allowing the free energy, entropy, and other thermodynamic properties to be determined from short 20-ps MD trajectories. The anharmonic effects in the vibrations are accounted for by the broadening of the normal modes into bands from sampling the velocities over the trajectory. The low frequency diffusive modes, which lead to finite DoS at zero frequency, are accounted for by considering the DoS as a superposition of gas-phase and solid-phase components (two phases). The analytical decomposition of the DoS allows for an evaluation of properties contributed by different types of molecular motions. We show that this 2PT analysis leads to accurate predictions of entropy and energy of CO2 over a wide range of conditions (from the triple point to the critical point of both the vapor and the liquid phases along the saturation line). This allows the equation of state of CO2 to be determined, which is limited only by the accuracy of the force field. We also validated that the 2PT entropy agrees with that determined from thermodynamic integration, but 2PT requires only a fraction of the time. A complication for CO2 is that its equilibrium configuration is linear, which would have only two rotational modes, but during the dynamics it is never exactly linear, so that there is a third mode from rotational about the axis. In this work, we show how to treat such linear molecules in the 2PT framework.

Model exact low-lying states and spin dynamics in ferric wheels: Fe-6 to Fe-12

Using an efficient numerical scheme that exploits spatial symmetries and spin parity, we have obtained the exact low-lying eigenstates of exchange Hamiltonians for ferric wheels up to Fe-12. The largest calculation involves the Fe-12 ring which spans a Hilbert space dimension of about 145x10(6) for the M-S=0 subspace. Our calculated gaps from the singlet ground state to the excited triplet state agree well with the experimentally measured values. Study of the static structure factor shows that the ground state is spontaneously dimerized for ferric wheels. The spin states of ferric wheels can be viewed as quantized states of a rigid rotor with the gap between the ground and first excited states defining the inverse of the moment of inertia. We have studied the quantum dynamics of Fe-10 as a representative of ferric wheels. We use the low-lying states of Fe-10 to solve exactly the time-dependent Schrodinger equation and find the magnetization of the molecule in the presence of an alternating magnetic field at zero temperature. We observe a nontrivial oscillation of the magnetization which is dependent on the amplitude of the ac field. We have also studied the torque response of Fe-12 as a function of a magnetic field, which clearly shows spin-state crossover.

Phase diagram of a two-species lattice model with a linear instability

We discuss the properties of a one-dimensional lattice model of a driven system with two species of particles in which the mobility of one species depends on the density of the other. This model was introduced by Lahiri and Ramaswamy (Phys. Rev. Lett., 79, 1150 (1997)) in the context of sedimenting colloidal crystals, and its continuum version was shown to exhibit an instability arising from linear gradient couplings. In this paper we review recent progress in understanding the full phase diagram of the model. There are three phases. In the first, the steady state can be determined exactly along a representative locus using the condition of detailed balance. The system shows phase separation of an exceptionally robust sort, termed strong phase separation, which survives at all temperatures. The second phase arises in the threshold case where the first species evolves independently of the second, but the fluctuations of the first influence the evolution of the second, as in the passive scalar problem. The second species then shows phase separation of a delicate sort, in which long-range order coexists with fluctuations which do not damp down in the large-size limit. This fluctuation-dominated phase ordering is associated with power law decays in cluster size distributions and a breakdown of the Porod law. The third phase is one with a uniform overall density, and along a representative locus the steady state is shown to have product measure form. Density fluctuations are transported by two kinematic waves, each involving both species and coupled at the nonlinear level. Their dissipation properties are governed by the symmetries of these couplings, which depend on the overall densities. In the most interesting case,, the dissipation of the two modes is characterized by different critical exponents, despite the nonlinear coupling.

Interfacial growth as a model of tube-width heterogeneities in concentrated solutions of stiff polymers

Recent experimental measurements of the distribution P(w) of transverse chain fluctuations w in concentrated solutions of F-actin filaments B. Wang, J Guan, S. M. Anthony, S. C. Bae, K. S. Schweizer, and S. Granick, Phys. Rev. Lett. 104, 118301 (2010); J. Glaser, D. Chakraborty, K. Kroy, I. Lauter, M. Degawa, N. Kirchgessner, B. Hoffmann, R. Merkel, and M. Giesen, Phys. Rev. Lett. 105, 037801 (2010)] are shown to be well-fit to an expression derived from a model of the conformations of a single harmonically confined weakly bendable rod. The calculation of P(w) is carried out essentially exactly within a path integral approach that was originally applied to the study of one-dimensional randomly growing interfaces. Our results are generally as successful in reproducing experimental trends as earlier approximate results obtained from more elaborate many-chain treatments of the confining tube potential.

Interfacial growth as a model of tube-width heterogeneities in concentrated solutions of stiff polymers

Recent experimental measurements of the distribution P(w) of transverse chain fluctuations w in concentrated solutions of F-actin filaments B. Wang, J Guan, S. M. Anthony, S. C. Bae, K. S. Schweizer, and S. Granick, Phys. Rev. Lett. 104, 118301 (2010); J. Glaser, D. Chakraborty, K. Kroy, I. Lauter, M. Degawa, N. Kirchgessner, B. Hoffmann, R. Merkel, and M. Giesen, Phys. Rev. Lett. 105, 037801 (2010)] are shown to be well-fit to an expression derived from a model of the conformations of a single harmonically confined weakly bendable rod. The calculation of P(w) is carried out essentially exactly within a path integral approach that was originally applied to the study of one-dimensional randomly growing interfaces. Our results are generally as successful in reproducing experimental trends as earlier approximate results obtained from more elaborate many-chain treatments of the confining tube potential. (C) 2013 AIP Publishing LLC.

A Statistical Model of Current Loops and Magnetic Monopoles

We formulate a natural model of loops and isolated vertices for arbitrary planar graphs, which we call the monopole-dimer model. We show that the partition function of this model can be expressed as a determinant. We then extend the method of Kasteleyn and Temperley-Fisher to calculate the partition function exactly in the case of rectangular grids. This partition function turns out to be a square of a polynomial with positive integer coefficients when the grid lengths are even. Finally, we analyse this formula in the infinite volume limit and show that the local monopole density, free energy and entropy can be expressed in terms of well-known elliptic functions. Our technique is a novel determinantal formula for the partition function of a model of isolated vertices and loops for arbitrary graphs.

Three-Dimensional Network Model And Its Parameter Calibration

A material model, whose framework is parallel spring-bundles oriented in 3-D space, is proposed. Based on a discussion of the discrete schemes and optimum discretization of the solid angles, a 3-D network cell consisted of one-dimensional components is developed with its geometrical and physical parameters calibrated. It is proved that the 3-D network model is able to exactly simulate materials with arbitrary Poisson ratio from 0 to 1/2, breaking through the limit that the previous models in the literature are only suitable for materials with Poisson ratio from 0 to 1/3. A simplified model is also proposed to realize high computation accuracy within low computation cost. Examples demonstrate that the 3-D network model has particular superiority in the simulation of short-fiber reinforced composites.

Kinetic model of continuous-wave flow chemical lasers

In this paper an analysis of the kinetic theory of the continuous-wave flow chemical lasers(CWFCL) is presented with emphasis being laid on the effects of inhomogeneous broadeningon CWFCL's performance. The results obtained are applicable to the case where laser fre-quency is either coincident or incoincident with that of the eenter of the line shape. This rela-tion has been,compared with that of the rate model in common use. These two models are almostidentical as the broadening parameter η is larger than 1. The smaller the value of η, thegreater the difference between the results of these two models will be. For fixed η, the dif-ferences between fhe results of the two models increase with the increase of the frequencyshift parameter ξ. When η is about less than 0.2. the kinetic model can predict exactly the in-homogeneous broadening effects,while the rate model cannot.

Mathematical study of complex networks : brain, internet, and power grid

The dissertation is concerned with the mathematical study of various network problems. First, three real-world networks are considered: (i) the human brain network (ii) communication networks, (iii) electric power networks. Although these networks perform very different tasks, they share similar mathematical foundations. The high-level goal is to analyze and/or synthesis each of these systems from a “control and optimization” point of view. After studying these three real-world networks, two abstract network problems are also explored, which are motivated by power systems. The first one is “flow optimization over a flow network” and the second one is “nonlinear optimization over a generalized weighted graph”. The results derived in this dissertation are summarized below.

Brain Networks: Neuroimaging data reveals the coordinated activity of spatially distinct brain regions, which may be represented mathematically as a network of nodes (brain regions) and links (interdependencies). To obtain the brain connectivity network, the graphs associated with the correlation matrix and the inverse covariance matrix—describing marginal and conditional dependencies between brain regions—have been proposed in the literature. A question arises as to whether any of these graphs provides useful information about the brain connectivity. Due to the electrical properties of the brain, this problem will be investigated in the context of electrical circuits. First, we consider an electric circuit model and show that the inverse covariance matrix of the node voltages reveals the topology of the circuit. Second, we study the problem of finding the topology of the circuit based on only measurement. In this case, by assuming that the circuit is hidden inside a black box and only the nodal signals are available for measurement, the aim is to find the topology of the circuit when a limited number of samples are available. For this purpose, we deploy the graphical lasso technique to estimate a sparse inverse covariance matrix. It is shown that the graphical lasso may find most of the circuit topology if the exact covariance matrix is well-conditioned. However, it may fail to work well when this matrix is ill-conditioned. To deal with ill-conditioned matrices, we propose a small modification to the graphical lasso algorithm and demonstrate its performance. Finally, the technique developed in this work will be applied to the resting-state fMRI data of a number of healthy subjects.

Communication Networks: Congestion control techniques aim to adjust the transmission rates of competing users in the Internet in such a way that the network resources are shared efficiently. Despite the progress in the analysis and synthesis of the Internet congestion control, almost all existing fluid models of congestion control assume that every link in the path of a flow observes the original source rate. To address this issue, a more accurate model is derived in this work for the behavior of the network under an arbitrary congestion controller, which takes into account of the effect of buffering (queueing) on data flows. Using this model, it is proved that the well-known Internet congestion control algorithms may no longer be stable for the common pricing schemes, unless a sufficient condition is satisfied. It is also shown that these algorithms are guaranteed to be stable if a new pricing mechanism is used.

Electrical Power Networks: Optimal power flow (OPF) has been one of the most studied problems for power systems since its introduction by Carpentier in 1962. This problem is concerned with finding an optimal operating point of a power network minimizing the total power generation cost subject to network and physical constraints. It is well known that OPF is computationally hard to solve due to the nonlinear interrelation among the optimization variables. The objective is to identify a large class of networks over which every OPF problem can be solved in polynomial time. To this end, a convex relaxation is proposed, which solves the OPF problem exactly for every radial network and every meshed network with a sufficient number of phase shifters, provided power over-delivery is allowed. The concept of “power over-delivery” is equivalent to relaxing the power balance equations to inequality constraints.

Flow Networks: In this part of the dissertation, the minimum-cost flow problem over an arbitrary flow network is considered. In this problem, each node is associated with some possibly unknown injection, each line has two unknown flows at its ends related to each other via a nonlinear function, and all injections and flows need to satisfy certain box constraints. This problem, named generalized network flow (GNF), is highly non-convex due to its nonlinear equality constraints. Under the assumption of monotonicity and convexity of the flow and cost functions, a convex relaxation is proposed, which always finds the optimal injections. A primary application of this work is in the OPF problem. The results of this work on GNF prove that the relaxation on power balance equations (i.e., load over-delivery) is not needed in practice under a very mild angle assumption.

Generalized Weighted Graphs: Motivated by power optimizations, this part aims to find a global optimization technique for a nonlinear optimization defined over a generalized weighted graph. Every edge of this type of graph is associated with a weight set corresponding to the known parameters of the optimization (e.g., the coefficients). The motivation behind this problem is to investigate how the (hidden) structure of a given real/complex valued optimization makes the problem easy to solve, and indeed the generalized weighted graph is introduced to capture the structure of an optimization. Various sufficient conditions are derived, which relate the polynomial-time solvability of different classes of optimization problems to weak properties of the generalized weighted graph such as its topology and the sign definiteness of its weight sets. As an application, it is proved that a broad class of real and complex optimizations over power networks are polynomial-time solvable due to the passivity of transmission lines and transformers.

Baryon and lepton numbers in particle physics beyond the standard model

The works presented in this thesis explore a variety of extensions of the standard model of particle physics which are motivated by baryon number (B) and lepton number (L), or some combination thereof. In the standard model, both baryon number and lepton number are accidental global symmetries violated only by non-perturbative weak effects, though the combination B-L is exactly conserved. Although there is currently no evidence for considering these symmetries as fundamental, there are strong phenomenological bounds restricting the existence of new physics violating B or L. In particular, there are strict limits on the lifetime of the proton whose decay would violate baryon number by one unit and lepton number by an odd number of units.

The first paper included in this thesis explores some of the simplest possible extensions of the standard model in which baryon number is violated, but the proton does not decay as a result. The second paper extends this analysis to explore models in which baryon number is conserved, but lepton flavor violation is present. Special attention is given to the processes of μ to e conversion and μ → eγ which are bound by existing experimental limits and relevant to future experiments.

The final two papers explore extensions of the minimal supersymmetric standard model (MSSM) in which both baryon number and lepton number, or the combination B-L, are elevated to the status of being spontaneously broken local symmetries. These models have a rich phenomenology including new collider signatures, stable dark matter candidates, and alternatives to the discrete R-parity symmetry usually built into the MSSM in order to protect against baryon and lepton number violating processes.

SteelConverter and Caltech VirtualShaker: Rapid Nonlinear Cloud-Based Structural Model Conversion and Analysis

STEEL, the Caltech created nonlinear large displacement analysis software, is currently used by a large number of researchers at Caltech. However, due to its complexity, lack of visualization tools (such as pre- and post-processing capabilities) rapid creation and analysis of models using this software was difficult. SteelConverter was created as a means to facilitate model creation through the use of the industry standard finite element solver ETABS. This software allows users to create models in ETABS and intelligently convert model information such as geometry, loading, releases, fixity, etc., into a format that STEEL understands. Models that would take several days to create and verify now take several hours or less. The productivity of the researcher as well as the level of confidence in the model being analyzed is greatly increased.

It has always been a major goal of Caltech to spread the knowledge created here to other universities. However, due to the complexity of STEEL it was difficult for researchers or engineers from other universities to conduct analyses. While SteelConverter did help researchers at Caltech improve their research, sending SteelConverter and its documentation to other universities was less than ideal. Issues of version control, individual computer requirements, and the difficulty of releasing updates made a more centralized solution preferred. This is where the idea for Caltech VirtualShaker was born. Through the creation of a centralized website where users could log in, submit, analyze, and process models in the cloud, all of the major concerns associated with the utilization of SteelConverter were eliminated. Caltech VirtualShaker allows users to create profiles where defaults associated with their most commonly run models are saved, and allows them to submit multiple jobs to an online virtual server to be analyzed and post-processed. The creation of this website not only allowed for more rapid distribution of this tool, but also created a means for engineers and researchers with no access to powerful computer clusters to run computationally intensive analyses without the excessive cost of building and maintaining a computer cluster.

In order to increase confidence in the use of STEEL as an analysis system, as well as verify the conversion tools, a series of comparisons were done between STEEL and ETABS. Six models of increasing complexity, ranging from a cantilever column to a twenty-story moment frame, were analyzed to determine the ability of STEEL to accurately calculate basic model properties such as elastic stiffness and damping through a free vibration analysis as well as more complex structural properties such as overall structural capacity through a pushover analysis. These analyses showed a very strong agreement between the two softwares on every aspect of each analysis. However, these analyses also showed the ability of the STEEL analysis algorithm to converge at significantly larger drifts than ETABS when using the more computationally expensive and structurally realistic fiber hinges. Following the ETABS analysis, it was decided to repeat the comparisons in a software more capable of conducting highly nonlinear analysis, called Perform. These analyses again showed a very strong agreement between the two softwares in every aspect of each analysis through instability. However, due to some limitations in Perform, free vibration analyses for the three story one bay chevron brace frame, two bay chevron brace frame, and twenty story moment frame could not be conducted. With the current trend towards ultimate capacity analysis, the ability to use fiber based models allows engineers to gain a better understanding of a building’s behavior under these extreme load scenarios.

Following this, a final study was done on Hall’s U20 structure [1] where the structure was analyzed in all three softwares and their results compared. The pushover curves from each software were compared and the differences caused by variations in software implementation explained. From this, conclusions can be drawn on the effectiveness of each analysis tool when attempting to analyze structures through the point of geometric instability. The analyses show that while ETABS was capable of accurately determining the elastic stiffness of the model, following the onset of inelastic behavior the analysis tool failed to converge. However, for the small number of time steps the ETABS analysis was converging, its results exactly matched those of STEEL, leading to the conclusion that ETABS is not an appropriate analysis package for analyzing a structure through the point of collapse when using fiber elements throughout the model. The analyses also showed that while Perform was capable of calculating the response of the structure accurately, restrictions in the material model resulted in a pushover curve that did not match that of STEEL exactly, particularly post collapse. However, such problems could be alleviated by choosing a more simplistic material model.

Relativistic quark models and the current algebra

In this thesis we are concerned with finding representations of the algebra of SU(3) vector and axial-vector charge densities at infinite momentum (the "current algebra") to describe the mesons, idealizing the real continua of multiparticle states as a series of discrete resonances of zero width. Such representations would describe the masses and quantum numbers of the mesons, the shapes of their Regge trajectories, their electromagnetic and weak form factors, and (approximately, through the PCAC hypothesis) pion emission or absorption amplitudes.

We assume that the mesons have internal degrees of freedom equivalent to being made of two quarks (one an antiquark) and look for models in which the mass is SU(3)-independent and the current is a sum of contributions from the individual quarks. Requiring that the current algebra, as well as conditions of relativistic invariance, be satisfied turns out to be very restrictive, and, in fact, no model has been found which satisfies all requirements and gives a reasonable mass spectrum. We show that using more general mass and current operators but keeping the same internal degrees of freedom will not make the problem any more solvable. In particular, in order for any two-quark solution to exist it must be possible to solve the "factorized SU(2) problem," in which the currents are isospin currents and are carried by only one of the component quarks (as in the K meson and its excited states).

In the free-quark model the currents at infinite momentum are found using a manifestly covariant formalism and are shown to satisfy the current algebra, but the mass spectrum is unrealistic. We then consider a pair of quarks bound by a potential, finding the current as a power series in 1/m where m is the quark mass. Here it is found impossible to satisfy the algebra and relativistic invariance with the type of potential tried, because the current contributions from the two quarks do not commute with each other to order 1/m³. However, it may be possible to solve the factorized SU(2) problem with this model.

The factorized problem can be solved exactly in the case where all mesons have the same mass, using a covariant formulation in terms of an internal Lorentz group. For a more realistic, nondegenerate mass there is difficulty in covariantly solving even the factorized problem; one model is described which almost works but appears to require particles of spacelike 4-momentum, which seem unphysical.

Although the search for a completely satisfactory model has been unsuccessful, the techniques used here might eventually reveal a working model. There is also a possibility of satisfying a weaker form of the current algebra with existing models.

Convex approaches to model wavelet sparsity patterns

Statistical dependencies among wavelet coefficients are commonly represented by graphical models such as hidden Markov trees (HMTs). However, in linear inverse problems such as deconvolution, tomography, and compressed sensing, the presence of a sensing or observation matrix produces a linear mixing of the simple Markovian dependency structure. This leads to reconstruction problems that are non-convex optimizations. Past work has dealt with this issue by resorting to greedy or suboptimal iterative reconstruction methods. In this paper, we propose new modeling approaches based on group-sparsity penalties that leads to convex optimizations that can be solved exactly and efficiently. We show that the methods we develop perform significantly better in de-convolution and compressed sensing applications, while being as computationally efficient as standard coefficient-wise approaches such as lasso. © 2011 IEEE.