A Note on Causation versus Correlation in an Extreme Situation

Recently, it has been shown that the information flow and causality between two time series can be inferred in a rigorous and quantitative sense, and, besides, the resulting causality can be normalized. A corollary that follows is, in the linear limit, causation implies correlation, while correlation does not imply causation. Now suppose there is an event A taking a harmonic form (sine/cosine), and it generates through some process another event B so that B always lags A by a phase of π/2. Here the causality is obviously seen, while by computation the correlation is, however, zero. This apparent contradiction is rooted in the fact that a harmonic system always leaves a single point on the Poincaré section; it does not add information. That is to say, though the absolute information flow from A to B is zero, i.e., TA→B=0, the total information increase of B is also zero, so the normalized TA→B, denoted as τA→B, takes the form of 00. By slightly perturbing the system with some noise, solving a stochastic differential equation, and letting the perturbation go to zero, it can be shown that τA→B approaches 100%, just as one would have expected.

4-point function from conformally coupled scalar in AdS6

We explore conformally coupled scalar theory in AdS6 extensively and their classical solutions by employing power expansion order by order in its self-interaction coupling λ. We describe how we get the classical solutions by diagrammatic ways which show general rules constructing the classical solutions. We study holographic correlation functions of scalar operator deformations to a certain 5-dimensional conformal field theory where the operators share the same scaling dimension ∆ = 3, from the classical solutions. We do not assume any specific form of the micro Lagrangian density of the 5-dimensional conformal field theory. For our solutions, we choose a scheme where we remove co-linear divergences of momenta along the AdS boundary directions which frequently appear in the classical solutions. This shows clearly that the holographic correlation functions are free from the co-linear divergences. It turns out that this theory provides correct conformal 2- and 3- point functions of the ∆ = 3 scalar operators as expected in previous literature. It makes sense since 2- and 3- point functions are determined by global conformal symmetry not being dependent on the details of the conformal theory. We also get 4-point function from this holographic model. In fact, it turns out that the 4-point correlation function is not conformal because it does not satisfy the special conformal Ward identity although it does dilation Ward identity and respect SO(5) rotation symmetry. However, in the co-linear limit that all the external momenta are in a same direction, the 4-point function is conformal which means that it satisfy the special conformal Ward identity. We inspect holographic n-point functions of this theory which can be obtained by employing a certain Feynman-like rule. This rule is a construction of n-point function by connecting l-point functions each other where l < n. In the co-linear limit, these n-point functions reproduce the conformal n-point functions of ∆ = 3 scalar operators in d = 5 Euclidean space addressed in arXiv:2001.05379.

Integral Representation of Electrostatic Interactions inside a Lipid Membrane

Interactions between charges and dipoles inside a lipid membrane are partially screened. The screening arises both from the polarization of water and from the structure of the electric double layer formed by the salt ions outside the membrane. Assuming that the membrane can be represented as a dielectric slab of low dielectric constant sandwiched by an aqueous solution containing mobile ions, a theoretical model is developed to quantify the strength of electrostatic interactions inside a lipid membrane that is valid in the linear limit of Poisson-Boltzmann theory. We determine the electrostatic potential produced by a single point charge that resides inside the slab and from that calculate charge-charge and dipole-dipole interactions as a function of separation. Our approach yields integral representations for these interactions that can easily be evaluated numerically for any choice of parameters and be further simplified in limiting cases.

Extending Complex Conjugate Control to Nonlinear Wave Energy Converters

This paper extends the concept of Complex Conjugate Control (CCC) of linear wave energy converters (WECs) to nonlinear WECs by designing optimal limit cycles with Hamiltonian Surface Shaping and Power Flow Control (HSSPFC). It will be shown that CCC for a regular wave is equivalent to a power factor of one in electrical power networks, equivalent to mechanical resonance in a mass-spring-damper (MSD) system, and equivalent to a linear limit cycle constrained to a Hamiltonian surface defined in HSSPFC. Specifically, the optimal linear limit cycle is defined as a second-order center in the phase plane projection of the constant energy orbit across the Hamiltonian surface. This concept of CCC described by a linear limit cycle constrained to a Hamiltonian surface will be extended to nonlinear limit cycles constrained to a Hamiltonian surface for maximum energy harvesting by the nonlinear WEC. The case studies presented confirm increased energy harvesting which utilizes nonlinear geometry realization for reactive power generation.

Electron pairing in mirror modes: surpassing the quasi-linear limit

The mirror mode evolving in collisionless magnetised high-temperature thermally anisotropic plasmas is shown to develop an interesting macro-state. Starting as a classical zero-frequency ion fluid instability it saturates quasi-linearly at very low magnetic level, while forming elongated magnetic bubbles which trap the electron component to perform an adiabatic bounce motion along the magnetic field. Further evolution of the mirror mode towards a stationary state is determined by the bouncing trapped electrons which interact with the thermal level of ion sound waves and generate attractive wake potentials which give rise to the formation of electron pairs in the lowest-energy singlet state of two combined electrons. Pairing preferentially takes place near the bounce-mirror points where the pairs become spatially locked with all their energy in the gyration. The resulting large anisotropy of pairs enters the mirror growth rate in the quasi-linearly stable mirror mode. It breaks the quasi-linear stability and causes further growth. Pressure balance is either restored by dissipation of the pairs and their anisotropy or inflow of plasma from the environment. In the first case new pairs will continuously form until equilibrium is reached. In the final state the fraction of pairs can be estimated. This process is open to experimental verification. To our knowledge it is the only process in which high-temperature plasma pairing may occur and has an important observable macroscopic effect: breaking the quasi-linear limit and, via pressure balance, generation of localised diamagnetism.

Stability of single and multiple matter-wave dark solitons in collisionally inhomogeneous Bose–Einstein condensates

We examine the spectral properties of single and multiple matter-wave dark solitons in Bose–Einstein condensates confined in parabolic traps, where the scattering length is periodically modulated. In addition to the large density limit picture previously established for homogeneous nonlinearities, we explore a perturbative analysis in the vicinity of the linear limit, which provides good agreement with the observed spectral modes. Between these two analytically tractable limits, we use numerical computations to fill in the relevant intermediate regime. We find that the scattering length modulation can cause a variety of features absent for homogeneous nonlinearities. Among them, we note the potential oscillatory instability even of the single dark soliton, the potential absence of instabilities in the immediate vicinity of the linear limit for two dark solitons, and the existence of an exponential instability associated with the in-phase motion of three dark solitons.

The response of a continuously stratified fluid to an oscillating flow past an obstacle

An oscillating continuously stratified flow past an isolated obstacle is investigated using scaling arguments and two-dimensional non-hydrostatic numerical experiments. A new dynamic scaling is introduced that incorporates the blocking of fluid with insufficient energy to overcome the background stratification and crest the obstacle. This clarifies the distinction between linear and nonlinear flow regimes near the crest of the obstacle. The flow is decomposed into propagating and non-propagating components. In the linear limit, the non-propagating component is related to the unstratified potential flow past the obstacle and the radiating component exhibits narrow wave beams that are tangent to the obstacle at critical points. When the flow is nonlinear, the near crest flow oscillates between states that include asymmetric, crest-controlled flows. Thin, fast, supercritical layers plunge in the lee, separate from the obstacle and undergo shear instability in the fluid interior. These flow features are localized to the neighbourhood of the crest where the flow transitions from subcriticality to supercriticality and are non-propagating. The nonlinear excitation of energetic non-propagating components reduces the efficiency of topographic radiation in comparison with linear dynamics.