Spatial Translation Research Articles

The coherent states of the two-dimensional isotropic harmonic oscillator and the classical limit of the Landau theory of a charged particle in a uniform magnetic field

In this paper we show how the classical result of a charged particle moving in a circle in the xy plane, when a uniform magnetic field is directed along the z-axis, can be derived from the Landau quantum theory using the coherent states of the two-dimensional isotropic harmonic oscillator in the xy plane. The coherent states in this case are the simultaneous eigen vectors of the annihilation operators a+ and a−. We prove that the time-dependent coordinate space wave packets representing the time-dependent coherent states move in a circle with the cyclotron frequency and with a radius given by the classical expression, but given in terms of the quantum mechanical expectation values. The expectation value of the energy of the particle and of the square of the radius of its circular are proportional to the square of the magnitude of the eigen value of a+ in the coherent state, where as the x and y coordinates of the centre of the circle are proportional to the real and the imaginary parts of the eigen value of a−. The phase of the circular motion is the same as the phase of the complex eigen value of a+. So for a given energy of the particle or for a given radius of the circular orbit, there are an infinite number of circles which differ from each other by the x and y coordinates of the centre as well as the phase of the circular motion. The infinite degeneracy of the Landau levels is due to the invariance of the energy eigen values under spatial translations in the xy plane and rotations about the z-axis. We also show that as the magnitude of the eigen value of a+ becomes much larger than one, the relative uncertainty or fluctuation in the energy and in the radius of the circular orbit becomes negligibly small as we expect for a classical state.

Center manifolds without a phase space

We establish center manifold theorems that allow one to study the bifurcation of small solutions from a trivial state in systems of functional equations posed on the real line. The class of equations includes most importantly nonlinear equations with nonlocal coupling through convolution operators as they arise in the description of spatially extended dynamics in neuroscience. These systems possess a natural spatial translation symmetry, but local existence or uniqueness theorems for a spatial evolution associated with this spatial shift or even a well motivated choice of phase space for the induced dynamics do not seem to be available, due to the infinite range forward- and backward-coupling through nonlocal convolution operators. We perform a reduction relying entirely on functional analytic methods. Despite the nonlocal nature of the problem, we do recover a local differential equation describing the dynamics on the set of small bounded solutions, exploiting that the translation invariance of the original problem induces a flow action on the center manifold. We apply our reduction procedure to problems in mathematical neuroscience, illustrating in particular the new type of algebra necessary for the computation of Taylor jets of reduced vector fields.

Cosmological aspects of the Eisenhart\u2013Duval lift

A cosmological extension of the Eisenhart–Duval metric is constructed by incorporating a cosmic scale factor and the energy-momentum tensor into the scheme. The dynamics of the spacetime is governed by the Ermakov–Milne–Pinney equation. Killing isometries include spatial translations and rotations, Newton–Hooke boosts and translation in the null direction. Geodesic motion in Ermakov–Milne–Pinney cosmoi is analyzed. The derivation of the Ermakov–Lewis invariant, the Friedmann equations and the Dmitriev–Zel’dovich equations within the Eisenhart–Duval framework is presented.

Incoherent transport for phases that spontaneously break translations

We consider phases of matter at finite charge density which spontaneously break spatial translations. Without taking a hydrodynamic limit we identify a boost invariant incoherent current operator. We also derive expressions for the small frequency behaviour of the thermoelectric conductivities generalising those that have been derived in a translationally invariant context. Within holographic constructions we show that the DC conductivity for the incoherent current can be obtained from a solution to a Stokes flow for an auxiliary fluid on the black hole horizon combined with specific thermodynamic quantities associated with the equilibrium black hole solutions.

Diffusion for holographic lattices

We consider black hole spacetimes that are holographically dual to strongly coupled field theories in which spatial translations are broken explicitly. We discuss how the quasinormal modes associated with diffusion of heat and charge can be systematically constructed in a long wavelength perturbative expansion. We show that the dispersion relation for these modes is given in terms of the thermoelectric DC conductivity and static susceptibilities of the dual field theory and thus we derive a generalised Einstein relation from Einstein’s equations. A corollary of our results is that thermodynamic instabilities imply specific types of dynamical instabilities of the associated black hole solutions.

Controllable Micro-Particle Rotation and Transportation Using Sound Field Synthesis Technique

Rotation and transportation of micro-particles using ultrasonically-driven devices shows promising applications in the fields of biological engineering, composite material manufacture, and micro-assembly. Current interest in mechanical effects of ultrasonic waves has been stimulated by the achievements in manipulations with phased array. Here, we propose a field synthesizing method using the fewest transducers to control the orientation of a single non-spherical micro-particle as well as its spatial location. A localized acoustic force potential well is established and rotated by using sound field synthesis technique. The resultant acoustic radiation torque on the trapped target determines its equilibrium angular position. A prototype device consisting of nine transducers with 2 MHz center frequency is designed and fabricated. Controllable rotation of a silica rod with 90 μm length and 15 μm diameter is then successfully achieved. There is a good agreement between the measured particle orientation and the theoretical prediction. Within the same device, spatial translation of the silica rod can also be realized conveniently. When compared with the existing acoustic rotation methods, the employed transducers of our method are strongly decreased, meanwhile, device functionality is improved.

Asymptotic behavior of solutions to a class of diffusive predator–prey systems

We study the large time behavior of a class of diffusive predator–prey systems posed on the whole Euclidean space. By studying a family of similar problems with all possible spatial translations, we first prove the asymptotic persistence of the prey for the spatially heterogeneous case under certain assumptions on the coefficients. Then, applying this persistence theorem, we prove the convergence of the solution to the unique positive equilibrium for the spatially homogeneous case, under certain restrictions on the space dimension and the predation coefficient.

Enhancing the convergence of multipole expansions at intermediate frequency

In this work, we examine near-field acoustic wave scattering from media having a spatial variation in compressibility contrast. Typically, the scattered acoustic pressure can be expressed as a convergent Neumann series when the compressibility contrast is relatively small. However as the magnitude of the compressibility contrast increases a resonant scattering condition ensues, yielding a divergent series. It has been shown that Padé Approximants method can be used in these cases, thereby extending the range of utility of the Neumann series solution. The proposed research explores hybrid methods that combine multipole-methods, and Padé Approximant approaches to calculate the near-field scattered pressure. As part of this work, we will examine the low-frequency breakdown in the plane wave, and monopole-monopole expansion approaches as it affects the numerical stability of spatial translation operations.

On the Stability of KMS States in Perturbative Algebraic Quantum Field Theories

We analyze the stability properties shown by KMS states for interacting massive scalar fields propagating over Minkowski spacetime, recently constructed in the framework of perturbative algebraic quantum field theories by Fredenhagen and Lindner \cite{FredenhagenLindner}. In particular, we prove the validity of the return to equilibrium property when the interaction Lagrangian has compact spatial support. Surprisingly, this does not hold anymore, if the adiabatic limit is considered, namely when the interaction Lagrangian is invariant under spatial translations. Consequently, an equilibrium state under the adiabatic limit for a perturbative interacting theory evolved with the free dynamics does not converge anymore to the free equilibrium state. Actually, we show that its ergodic mean converges to a non equilibrium steady state for the free theory.

Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria

The Hamiltonian structure of ideal translationally symmetric extended MHD (XMHD) is obtained by employing a method of Hamiltonian reduction on the three-dimensional noncanonical Poisson bracket of XMHD. The existence of the continuous spatial translation symmetry allows the introduction of Clebsch-like forms for the magnetic and velocity fields. Upon employing the chain rule for functional derivatives, the 3D Poisson bracket is reduced to its symmetric counterpart. The sets of symmetric Hall, Inertial, and extended MHD Casimir invariants are identified, and used to obtain energy-Casimir variational principles for generalized XMHD equilibrium equations with arbitrary macroscopic flows. The obtained set of generalized equations is cast into Grad-Shafranov-Bernoulli (GSB) type, and special cases are investigated: static plasmas, equilibria with longitudinal flows only, and Hall MHD equilibria, where the electron inertia is neglected. The barotropic Hall MHD equilibrium equations are derived as a limiting case of the XMHD GSB system, and a numerically computed equilibrium configuration is presented that shows the separation of ion-flow from electro-magnetic surfaces.

ESTIMATION OF AEOLIAN DUNE MIGRATION OVER MARTIAN SURFACE EMPLOYING HIGH PRECISION PHOTOGRAMMETRIC MEASUREMENTS

Abstract. At the present time, arguments continue regarding the migration speeds of Martian dune fields and their correlation with atmospheric circulation. However, precisely measuring the spatial translation of Martian dunes has been rarely successful due to the technical difficulties to quantitatively observe expected small surface migrations. Therefore, we developed a generic procedure to measure the migration of dune fields employing a high-accuracy photogrammetric processor and sub-pixel image correlator on the 25-cm resolution High Resolution Imaging Science Experiment (HiRISE). The established algorithms have been tested over a few Martian dune fields. Consequently, migrations over well-known crater dune fields appeared to be almost static for the considerable temporal periods and were weakly correlated with wind directions estimated by the Mars Climate Database. Only over some Martian dune fields, such as Kaiser crater, meaningful migration speeds (> 1m/year) considering photogrammetric error residual have been detected. Currently a technically improved processor to compensate error residual using time series observation is under development and expected to produce the long term migration speed over Martian dune fields where constant HiRISE image acquisitions are available.

Boomerang RG flows in M-theory with intermediate scaling

We construct novel RG flows of D=11 supergravity that asymptotically approach AdS4 × S7 in the UV with deformations that break spatial translations in the dual field theory. In the IR the solutions return to exactly the same AdS4 × S7 vacuum, with a renormalisation of relative length scales, and hence we refer to the flows as ‘boomerang RG flows’. For sufficiently large deformations, on the way to the IR the solutions also approach two distinct intermediate scaling regimes, each with hyperscaling violation. The first regime is Lorentz invariant with dynamical exponent z = 1 while the second has z = 5/2. Neither ofthe two intermediatescaling regimesare associatedwith exact hyperscaling violation solutions of D = 11 supergravity. The RG flow solutions are constructed using the four dimensional N = 2 STU gauged supergravity theory with vanishing gauge fields, but non-vanishing scalar and pseudoscalar fields. In the ABJM dual field theory the flows are driven by spatially modulated deformation parameters for scalar and fermion bilinear operators.

Lorentz violation bounds from torsion trace fermion sector and galaxy M 51 data and chiral dynamos

Earlier we have computed a Lorentz violation (LV) bound for torsion terms via galactic dynamos and found bounds similar to the one obtained by Kostelecky et al. (Phys Rev Lett 100:111102, 2008) which is of the order of 10^{-31} GeV. Their result was found making use of the axial torsion vector in terms of Dirac spinors and minimal torsion coupling in flat space-time of fermions. In this paper, a torsion dynamo equation obtained using the variation of the torsion trace and galaxy M51 data of 500 pc are used to place an upper bound of 10^{-26} GeV in LV, which agrees with the one by Kostelecky and his group using an astrophysical framework background. Their lowest bound was obtained in earth laboratory using dual masers. One of the purposes of this paper is to apply the Faraday self-induction magnetic equation, recently extended to torsioned space-time, by the author to show that it lends support to physics in Riemann–Cartan space-time, in several distinct physical backgrounds. Backreaction magnetic effects are used to obtain the LV bounds. Previously Bamba et al. (JCAP 10:058, 2012) have used the torsion trace in their teleparallel investigation of the IGMF, with the argument that the torsion trace leads to less weaker effects than the other irreducible components of the torsion tensor. LV is computed in terms of a chiral-torsion-like current in the new dynamo equation analogous to the Dvornikov and Semikoz dynamo equation with chiral magnetic currents. Making use of the chiral-torsion dynamo equation we estimate the LV bounds in the early universe to be of the order of 10^{-24} GeV, which was the order of the charged-lepton sector. Our main result is that it is possible to obtain more stringent bounds than the ones found in the fermion sector of astrophysics in the new revised 2017 data table for CPT and Lorentz violation by Kostelecky and Mewes. They found in several astrophysical backgrounds, orders of magnitude such as 10^{-24} and 10^{-23} GeV which are not so stringent as the one found here, of 10^{-26} GeV, in the torsion fermionic sector with the help of galaxy M51 data. It is also shown that non-gauge invariance of chiral dynamos and axial anomalies are obtained from the noninvariance of the electric fields under torsion spatial translations.

Correlation functions in theories with Lifshitz scaling

The 2+1 dimensional quantum Lifshitz model can be generalised to a class of higher dimensional free field theories that exhibit Lifshitz scaling. When the dynamical critical exponent equals the number of spatial dimensions, equal time correlation functions of scaling operators in the generalised quantum Lifshitz model are given by a d-dimensional higher-derivative conformal field theory. Autocorrelation functions in the generalised quantum Lifshitz model in any number of dimensions can on the other hand be expressed in terms of autocorrelation functions of a two-dimensional conformal field theory. This also holds for autocorrelation functions in a strongly coupled Lifshitz field theory with a holographic dual of Einstein-Maxwell-dilaton type. The map to a two-dimensional conformal field theory extends to autocorrelation functions in thermal states and out-of-equilbrium states preserving symmetry under spatial translations and rotations in both types of Lifshitz models. Furthermore, the spectrum of quasinormal modes of scalar field perturbations in Lifshitz black hole backgrounds can be obtained analytically at low spatial momenta and exhibits a linear dispersion relation at z = d. At high momentum, the mode spectrum can be obtained in a WKB approximation and displays very different behaviour compared to holographic duals of conformal field theories. This has implications for thermalisation in strongly coupled Lifshitz field theories with z > 1.

ADAR1-mediated 3\u2032 UTR editing and expression control of antiapoptosis genes fine-tunes cellular apoptosis response

Adenosine-to-inosine RNA editing constitutes a crucial component of the cellular transcriptome and critically underpins organism survival and development. While recent high-throughput approaches have provided comprehensive documentation of the RNA editome, its functional output remains mostly unresolved, particularly for events in the non-coding regions. Gene ontology analysis of the known RNA editing targets unveiled a preponderance of genes related to apoptosis regulation, among which proto-oncogenes XIAP and MDM2 encode two the most abundantly edited transcripts. To further decode this potential functional connection, here we showed that the main RNA editor ADAR1 directly targets this 3′ UTR editing of XIAP and MDM2, and further exerts a negative regulation on the expression of their protein products. This post-transcriptional silencing role was mediated via the inverted Alu elements in the 3′ UTR but independent of alteration in transcript stability or miRNA targeting. Rather, we discovered that ADAR1 competes transcript occupancy with the RNA shuttling factor STAU1 to facilitate nuclear retention of the XIAP and MDM2 mRNAs. As a consequence, ADAR1 may acquire functionality in part by conferring spatial distribution and translation efficiency of the target transcripts. Finally, abrogation of ADAR1 expression or catalytic activity elicited a XIAP-dependent suppression of apoptotic response, whereas ectopic expression reversed this protective effect on cell death. Together, our results extended the known functions of ADAR1 and RNA editing to the critical fine-tuning of the intracellular apoptotic signaling and also provided mechanistic explanation for ADAR1’s roles in development and tumorigenesis.

Black holes will break up solitons and white holes may destroy them

We consider a quantum analogue of black holes and white holes using Bose–Einstein condensates. The model is described by the nonlinear Schrödinger equation with a ‘stream flow’ potential, that induces a spatial translation to standing waves. We then mainly consider the dynamics of dark solitons in a black hole or white hole flow analogue and their interactions with the event horizon. A reduced equation describing the position of the dark solitons was obtained using variational method. Through numerical computations and comparisons with the analytical approximation we show that solitons can pass through black hole horizons even though they will break up into several solitons after the collision. In the interaction with a white hole horizon, we show that solitons either pass through the horizon or will be destroyed by it.

Observation of a discrete time crystal.

Spontaneous symmetry breaking is a fundamental concept in many areas of physics, including cosmology, particle physics and condensed matter. An example is the breaking of spatial translational symmetry, which underlies the formation of crystals and the phase transition from liquid to solid. Using the analogy of crystals in space, the breaking of translational symmetry in time and the emergence of a 'time crystal' was recently proposed, but was later shown to be forbidden in thermal equilibrium. However, non-equilibrium Floquet systems, which are subject to a periodic drive, can exhibit persistent time correlations at an emergent subharmonic frequency. This new phase of matter has been dubbed a 'discrete time crystal'. Here we present the experimental observation of a discrete time crystal, in an interacting spin chain of trapped atomic ions. We apply a periodic Hamiltonian to the system under many-body localization conditions, and observe a subharmonic temporal response that is robust to external perturbations. The observation of such a time crystal opens the door to the study of systems with long-range spatio-temporal correlations and novel phases of matter that emerge under intrinsically non-equilibrium conditions.

Singularity analysis of a class of kinematically redundant parallel Schönflies motion generators

This paper provides an analysis of the singularities of a class of kinematically redundant parallel manipulators. The studied mechanisms are capable of spatial translations in addition to rotation around an axis with constant orientation. Such motion is classified as Schönflies motion and is of high importance for industrial applications. The studied mechanisms include a manipulated platform composed of two platform sections, which are connected by a passive joint. The mechanisms include five actuated kinematic chains of a specific type, where three kinematic chains are connected to one platform section and two to the other section. Several novel mechanisms in this class are introduced and the singularities of all introduced variants are analyzed utilizing screw theory. In particular, all configurations leading to a redundant passive motion (RPM) singularity are determined. In such configurations, at least one of the passive joints in the mechanism may have a non-zero velocity while the velocity of the tool and all actuated joints are zero. For the studied mechanisms, such configurations correspond to the system of wrenches acting on the platform section including the tool has full rank while the wrench system acting on the other platform section is rank deficient.

Transient dynamics and their control in time-delay autonomous Boolean ring networks.

Biochemical systems with switch-like interactions, such as gene regulatory networks, are well modeled by autonomous Boolean networks. Specifically, the topology and logic of gene interactions can be described by systems of continuous piecewise-linear differential equations, enabling analytical predictions of the dynamics of specific networks. However, most models do not account for time delays along links associated with spatial transport, mRNA transcription, and translation. To address this issue, we have developed an experimental test bed to realize a time-delay autonomous Boolean network with three inhibitory nodes, known as a repressilator, and use it to study the dynamics that arise as time delays along the links vary. We observe various nearly periodic oscillatory transient patterns with extremely long lifetime, which emerge in small network motifs due to the delay, and which are distinct from the eventual asymptotically stable periodic attractors. For repeated experiments with a given network, we find that stochastic processes give rise to a broad distribution of transient times with an exponential tail. In some cases, the transients are so long that it is doubtful the attractors will ever be approached in a biological system that has a finite lifetime. To counteract the long transients, we show experimentally that small, occasional perturbations applied to the time delays can force the trajectories to rapidly approach the attractors.

Discrete Solitary Waves in Systems with Nonlocal Interactions and the Peierls–Nabarro Barrier

We study a class of discrete focusing nonlinear Schr{\"o}dinger equations (DNLS) with general nonlocal interactions. We prove the existence of onsite and offsite discrete solitary waves, which bifurcate from the trivial solution at the endpoint frequency of the continuous spectrum of linear dispersive waves. We also prove exponential smallness, in the frequency-distance to the bifurcation point, of the Peierls-Nabarro energy barrier (PNB), as measured by the difference in Hamiltonian or mass functionals evaluated on the onsite and offsite states. These results extend those of the authors for the case of nearest neighbor interactions to a large class of nonlocal short-range and long-range interactions. The appearance of distinct onsite and offsite states is a consequence of the breaking of continuous spatial translation invariance. The PNB plays a role in the dynamics of energy transport in such nonlinear Hamiltonian lattice systems. Our class of nonlocal interactions is defined in terms of coupling coefficients, $J_m$, where $m\in\mathbb{Z}$ is the lattice site index, with $J_m\simeq m^{-1-2s}, s\in[1,\infty)$ and $J_m\sim e^{-\gamma|m|},\ s=\infty,\ \gamma>0,$ (Kac-Baker). For $s\ge1$, the bifurcation is seeded by solutions of the (effective / homogenized) cubic focusing nonlinear Schr{\"o}dinger equation (NLS). However, for $1/4<s<1$, the bifurcation is controlled by the fractional nonlinear Schr{\"o}dinger equation, FNLS, with $(-\Delta)^s$ replacing $-\Delta$. The proof is based on a Lyapunov-Schmidt reduction strategy applied to a momentum space formulation. The PN barrier bounds require appropriate uniform decay estimates for the discrete Fourier transform of DNLS discrete solitary waves. A key role is also played by non-degeneracy of the ground state of FNLS, recently proved by Frank, Lenzmann \& Silvestre.