Manipulation of states of a degenerate quantum system

We consider the dynamics of an open three-level quantum degenerate system. One of the levels in this system is degenerate. The system interacts with three reservoirs (quantum fields) and a classical external field. We show that nondecaying so-called dark states are generated in this system. Since the interactions of the degenerate level with two different reservoirs are different (correspond to different spaces of dark states), we can describe excitation and manipulations for this kind of states (in particular, observation in spectroscopical experiments). Possible applications of this model in quantum optics, quantum computations, and quantum photosynthesis are discussed.     

A Poisson random field model for intermittent phenomena with application to laminar-turbulent transition and material microstructure

The alternation of a physical system between two phases or states is referred to as intermittency. Examples of intermittent phenomena abound in applications and include the transition from laminar to turbulent flow over a flight vehicle and the presence of imperfections within material microstructure. It is shown that intermittent phenomena of this type can be modeled by two-state random fields with piecewise constant samples; we refer to the states of the random field as “off” and “on” or, equivalently, 0 and 1. These random fields can be calibrated to the available information, which consists of: (1) the marginal probability that the state of the system is “on”; and (2) the average number of fluctuations between states that occur within a bounded region. The proposed model is defined by a sequence of pulses of prescribed shape and unit magnitude, located at random (Poisson) points within a bounded domain. Properties of the model are discussed, and simple algorithms to generate samples of the random field are provided. Various applications are considered, including voids within material microstructure and the random vibration of a flight vehicle subjected to a transition from laminar to turbulent flow over its surface.     

On the Optimal Design of Wall-to-Wall Heat Transport

We consider the problem of optimizing heat transport through an incompressible fluid layer. Modeling passive scalar transport by advection-diffusion, we maximize the mean rate of total transport by a divergence-free velocity field. Subject to various boundary conditions and intensity constraints, we prove that the maximal rate of transport scales linearly in the r.m.s. kinetic energy and, up to possible logarithmic corrections, as the one-third power of the mean enstrophy in the advective regime. This makes rigorous a previous prediction on the near optimality of convection rolls for energy-constrained transport. On the other hand, optimal designs for enstrophy-constrained transport are significantly more difficult to describe: we introduce a “branching” flow design with an unbounded number of degrees of freedom and prove it achieves nearly optimal transport. The main technical tool behind these results is a variational principle for evaluating the transport of candidate designs. The principle admits dual formulations for bounding transport from above and below. While the upper bound is closely related to the “background method,” the lower bound reveals a connection between the optimal design problems considered herein and other apparently related model problems from mathematical materials science. These connections serve to motivate designs. © 2019 Wiley Periodicals, Inc.     

The flow of a supersonic ideal gas with “weak” and “strong” shocks over a wedge

The problem of the flow of a uniform supersonic ideal (inviscid and non-heat-conducting) gas over a wedge is considered. If the turning angle of the flow, which is equal to the angle of inclination of the generatrix of the wedge, is less than the maximum value, the problem has two solutions. In the solution with an oblique low-intensity (“weak”) shock, the uniform flow between the shock and the wedge is almost always supersonic. One exception is a small vicinity of the maximum turning angle. For an ideal gas this vicinity does not exceed a fraction of a degree at all Mach numbers. Behind a high-intensity (“strong”) shock, the flow of an ideal gas is always subsonic. “Weak” shocks are observed in all experiments with finite wedges. Some researchers attribute this preference to the “downstream” boundary conditions (“on the right at infinity” for a flow incident on the wedge from the left), and others attribute it to the instability (“Lyapunov” instability) of a flow with a strong shock when it flows over the wedge and to the stability of flow with a weak shock. The results presented below from calculations of the flows that occur for finite wedges within the two-dimensional unsteady Euler equations, when the parameters behind the strong shock are specified on the right-hand boundary, i.e., on the arc of a circle between the wedge and the shock, demonstrate the correctness of the conclusion of the first group of researchers and the incorrectness of the conclusion of the other group. In these calculations, after both small and fairly large perturbations, the flows investigated (which are, in fact, Lyapunov unstable!) return to the solution with a strong shock. In addition, the problem of steady flow over a wedge was regarded as the limit of the two-dimensional non-steady problems at infinite time. Simplification of one of them leads to the problem of the submerged over-expanded supersonic steady outflow. In the ideal gas model this problem is equivalent to flow over a wedge with both weak and strong shocks. All the solutions considered are stable.     

Clusters die hard: Time-correlated excitation in the Hamiltonian mean field model

The Hamiltonian mean field (HMF) model has a low-energy phase where N particles are trapped inside a cluster. Here, we investigate some properties of the trapping/untrapping mechanism of a single particle into/outside the cluster. Since the single particle dynamics of the HMF model resembles the one of a simple pendulum, each particle can be identified as a high-energy particle (HEP) or a low-energy particle (LEP), depending on whether its energy is above or below the separatrix energy. We then define the trapping ratio as the ratio of the number of LEP to the total number of particles and the “fully-clustered” and “excited” dynamical states as having either no HEP or at least one HEP. We analytically compute the phase-space average of the trapping ratio by using the Boltzmann–Gibbs stable stationary solution of the Vlasov equation associated with the N → ∞ limit of the HMF model. The same quantity, obtained numerically as a time average, is shown to be in very good agreement with the analytical calculation. Another important feature of the dynamical behavior of the system is that the dynamical state changes transitionally: the “fully-clustered” and “excited” states appear in turn. We find that the distribution of the lifetime of the “fully-clustered” state obeys a power law. This means that clusters die hard, and that the excitation of a particle from the cluster is not a Poisson process and might be controlled by some type of collective motion with long memory. Such behavior should not be specific of the HMF model and appear also in systems where itinerancy among different “quasi-stationary” states has been observed. It is also possible that it could mimick the behavior of transient motion in molecular clusters or some observed deterministic features of chemical reactions.     

On the Probabilistic Nature of Quantum Mechanics and the Notion of Closed Systems

The notion of “closed systems” in Quantum Mechanics is discussed. For this purpose, we study two models of a quantum mechanical system P spatially far separated from the “rest of the universe” Q. Under reasonable assumptions on the interaction between P and Q, we show that the system P behaves as a closed system if the initial state of P ∨ Q belongs to a large class of states, including ones exhibiting entanglement between P and Q. We use our results to illustrate the non-deterministic nature of quantum mechanics. Studying a specific example, we show that assigning an initial state and a unitary time evolution to a quantum system is generally not sufficient to predict the results of a measurement with certainty.     

Loop quantum mechanics and the fractal structure of quantum spacetime

We discuss the relation between string quantization based on the Schild path integral and the Nambu-Goto path integral. The equivalence between the two approaches at the classical level is extended to the quantum level by a saddle-point evaluation of the corresponding path integrals. A possible relationship between M-Theory and the quantum mechanics of string loops is pointed out. Then, within the framework of “loop quantum mechanics”, we confront the difficult question as to what exactly gives rise to the structure of spacetime. We argue that the large scale properties of the string condensate are responsible for the effective Riemannian geometry of classical spacetime. On the other hand, near the Planck scale the condensate “evaporates”, and what is left behind is a “vacuum” characterized by an effective fractal geometry.     

SISTA: Learning Optimal Transport Costs under Sparsity Constraints

In this paper, we describe a novel iterative procedure called SISTA to learn the underlying cost in optimal transport problems. SISTA is a hybrid between two classical methods, coordinate descent (“S”-inkhorn) and proximal gradient descent (“ISTA”). It alternates between a phase of exact minimization over the transport potentials and a phase of proximal gradient descent over the parameters of the transport cost. We prove that this method converges linearly, and we illustrate on simulated examples that it is significantly faster than both coordinate descent and ISTA. We apply it to estimating a model of migration, which predicts the flow of migrants using country-specific characteristics and pairwise measures of dissimilarity between countries. This application demonstrates the effectiveness of machine learning in quantitative social sciences. © 2022 Wiley Periodicals LLC.     

ISOMETRIC AND ALLOMETRIC RELATIONSHIPS BETWEEN LARGE AND SMALL‐SCALE TREE SPACING STUDIES

Abstract Basic differences in the relationship between diameter and height have been observed in small and large trees. Small trees (less than 5 m) have little danger of buckling under their own weight, and diameter is proportional to height. Large trees (greater than 5 m) are at risk of buckling under their own weight and are subject to damage from ice and wind. For large trees, diameter cubed is proportional to height squared. This relationship is suggested by the physics of limits to height of cylinders before they buckle under their own weight and has been shown to hold for large trees. Data from large‐scale spacing studies are compared with data from one‐sixteenth scale small spacing studies to determine the validity of this theory. The impact of scaled spacing on scaled diameters at equivalent scaled heights is examined. Results suggest that trees grown at small scales can be “scaled up” to reflect isometric and allometric relationships of trees grown at large scales.     

Uniqueness of quantum Markov chains associated with an XY-model on a cayley tree of order 2

We propose the construction of a quantum Markov chain that corresponds to a “forward” quantum Markov chain. In the given construction, the quantum Markov chain is defined as the limit of finite-dimensional states depending on the boundary conditions. A similar construction is widely used in the definition of Gibbs states in classical statistical mechanics. Using this construction, we study the quantum Markov chain associated with an XY-model on a Cayley tree. For this model, within the framework of the given construction, we prove the uniqueness of the quantum Markov chain, i.e., we show that the state is independent of the boundary conditions.     

A new solution of the equilibrium equations of a system of bodies with elastic joints

Different ways of representing the elastic moments are proposed, which can be used for the finite-dimensional modelling of rod systems using a system of n axisymmetric solids, connected by elastic spherical joints. Using the example of a closed plane rod, possible states of equilibrium of the finite-dimensional model of the rod are analysed for different methods of specifying the elastic torques at the joints. The case when the rod axis has the form of a “figure of eight”, which is modelled by a system of six axisymmetric solids with a relative torsion angle that depends on the bending, is investigated in detail.     

Generation of nonlinear modulated waves in a modified Noguchi electrical transmission network

In the present work, we study the generation of nonlinear modulated waves in a modified version of Noguchi electrical tr ansmission network. In the continuum limit, we have considered the semi discrete approximation and showed that wave modulations in the network are governed by a generalized Chen-Lee-Liu (G-CLL) equation whose “self steepening” parameter is free from line parameters. We have investigated the effects of the “self steepening” parameter of the equation on the dynamics of modulated waves propagating in the network and shown that it can be adequately used either to enhance or to soften the instability of the nonlinear Schrödinger (NLS) equation. Our investigations showed that the introduction of the “self steepening” term in the NLS equation of the network allows bright and dark solitonlike waves to coexist in the same modulational stable and unstable frequency regions of the NLS. Our analytical studies of the G-CLL equation of the network showed that the amplitude of the solitonlike waves propagating in the network decreases as the “self steepening” parameter of the G-CLL equation increases.     

Extensions of the Tensor Algebra and Their Applications

This article presents a natural extension of the tensor algebra. In addition to “left multiplications” by vectors, we can consider “derivations” by covectors as basic operators on this extended algebra. These two types of operators satisfy an analogue of the canonical commutation relations. This algebra and these operators have the following applications: (i) applications to invariant theory related to tensor products and (ii) applications to immanants. The latter includes a new method to study the quantum immanants in the universal enveloping algebras of the general linear Lie algebras and their Capelli type identities (the higher Capelli identities).     

Chaos in M(atrix) theory

We consider the classical and quantum dynamics in M(atrix) theory. Using a simple ansatz we show that a classical trajectory exhibits a chaotic motion. We argue that the holographic feature of M(atrix) theory is related with the repulsive feature of energy eigenvalues in quantum chaotic system. Chaotic dynamics in N = 2 supersymmetric Yang—Mills theory is also discussed. We demonstrate that after the separation of “slow” and “fast” modes there is a singular contribution from the “slow” modes to the Hamiltonian of the “fast” modes.     

On the Dimension of the Space of Dark States in the Tavis–Cummings Model

Mathematical Notes - The space of minimal energy of a qubit system is the dark subspace of quantum states of a system of two-level atoms in the finite-dimensional Tavis–Cummings (TC) model of...     

Role determination in an aerial dogfight

A coplanar aerial dogfight is analyzed by assuming constant, not necessarily identical, speeds and individual maximum turning rates and lethal ranges. A combatant (A) is assumed to be victorious when his opponent (B) has been maneuvered into a relative position within A's lethal range and in the direction of A's velocity. Three variables are required to define the instantaneous “state” of the game, namely the relative position (2) and the angle (1) between their velocities. A computer program has been constructed to divide the 3-dimensional region of possible initial (and subsequent) states into regions corresponding to victory by one or the other combatant, and, if the faster combatant has the smaller lethal range, a “no contest” region corresponding to escape by the faster combatant. The critical separating surface (or surfaces) is composed of a number of pieces corresponding to initial conditions leading either to simultaneous kill or to “near miss” situations of one type or another. Optimal play is defined in the immediate neighborhood of the entire separating surface, guaranteeing victory (or escape) to one combatant or the other, depending on location on one side or the other of the separating surface.     

Least-squares orthogonalization using semidefinite programming

We consider the problem of constructing an optimal set of orthogonal vectors from a given set of vectors in a real Hilbert space. The vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors. We show that the design of the optimal vectors, referred to as the least-squares (LS) orthogonal vectors, can be formulated as a semidefinite programming (SDP) problem. Using the many well-known algorithms for solving SDPs, which are guaranteed to converge to the global optimum, the LS vectors can be computed very efficiently in polynomial time within any desired accuracy.By exploiting the connection between our problem and a quantum detection problem we derive a closed form analytical expression for the LS orthogonal vectors, for vector sets with a broad class of symmetry properties. Specifically, we consider geometrically uniform (GU) sets with a possibly non-abelian generating group, and compound GU sets which consist of subsets that are GU.     

Binary Representation of Coordinate and Momentum in Quantum Mechanics

To simulate a quantum system with continuous degrees of freedom on a quantum computer based on qubits, it is necessary to reduce continuous observables (primarily coordinates and momenta) to binary observables. We consider this problem based on expanding quantum observables in series in powers of two, analogous to the binary representation of real numbers. The coefficients of the series (“digits”) are therefore orthogonal projectors. We investigate the corresponding quantum mechanical operators and the relations between them and show that the binary expansion of quantum observables automatically leads to renormalization of some divergent integrals and series (giving them finite values).     

Slow convergence of sequences of linear operators II: Arbitrarily slow convergence

We study the rate of convergence of a sequence of linear operators that converges pointwise to a linear operator. Our main interest is in characterizing the slowest type of pointwise convergence possible. This is a continuation of the paper Deutsch and Hundal (2010) [14]. The main result is a “lethargy” theorem (Theorem 3.3) which gives useful conditions that guarantee arbitrarily slow convergence. In the particular case when the sequence of linear operators is generated by the powers of a single linear operator, we obtain a “dichotomy” theorem, which states the surprising result that either there is linear (fast) convergence or arbitrarily slow convergence; no other type of convergence is possible. The dichotomy theorem is applied to generalize and sharpen: (1) the von Neumann–Halperin cyclic projections theorem, (2) the rate of convergence for intermittently (i.e., “almost” randomly) ordered projections, and (3) a theorem of Xu and Zikatanov.