Probability Current Research Articles

Boundary conditions at a thin membrane that generate non-Markovian normal diffusion.

We show that some boundary conditions assumed at a thin membrane may result in normal diffusion not being the stochastic Markov process. We consider boundary conditions defined in terms of the Laplace transform in which there is a linear combination of probabilities and probability fluxes defined on both membrane surfaces. The coefficients of the combination may depend on the Laplace transform parameter. Such boundary conditions are most commonly used when considering diffusion in a membrane system unless collective or nonlocal processes in particles diffusion occur. We find Bachelier-Smoluchowski-Chapmann-Kolmogorov (BSCK) equation in terms of the Laplace transform and we derive the criterion to check whether the boundary conditions lead to fundamental solutions of diffusion equation satisfying this equation. If the BSCK equation is not met, then the Markov property is broken. When a probability flux is continuous at the membrane, the general forms of the boundary conditions for which the fundamental solutions meet the BSCK equation are derived. A measure of broken of semi-group property is also proposed. The relation of this measure to the non-Markovian property measure is discussed.

Predicting properties of the stationary probability currents for two-species reaction systems without solving the Fokker-Planck equation.

We derive methods for estimating the topology of the stationary probability current j[over ⃗]_{s} of the two-species Fokker-Planck equation (FPE) without the need to solve the FPE. These methods are chosen such that they become exact in certain limits, such as infinite system size or vanishing coupling between species in the diffusion matrix. The methods make predictions about the fixed points of j[over ⃗]_{s} and their relation to extrema of the stationary probability distribution and to fixed points of the convective field, which is related to the deterministic drift of the system. Furthermore, they predict the rotation sense of j[over ⃗]_{s} around extrema of the stationary probability distribution. Even though these methods cannot be proven to be valid away from extrema, the boundary lines between regions with different rotation senses are obtained with surprising accuracy. We illustrate and test these methods, using simple reaction systems with only one coupling term between the two species as well as a few generic reaction networks taken from the literature. We use it also to investigate the shape of nonphysical probability currents occurring in reaction systems with detailed balance due to the approximations involved in deriving the Fokker-Planck equation.

Capturing nonexponential dynamics in the presence of two decay channels

The most unstable quantum states and elementary particles possess more than a single decay channel. At the same time, it is well known that typically the decay law is not simply exponential. Therefore, it is natural to ask how to spot the non-exponential decay when (at least) two decay channels are opened. In this work, we study the tunneling phenomenon of an initially localized particle in two spatially opposite directions through two different barriers, mimicking two decay channels. In this framework, through a specific quantum mechanical examples which can be accurately solved, we study general properties of a two-channel decay that apply for various unstable quantum states (among which also for unstable particles). Apart from small deviations at early times, the survival probability and the partial tunneling probability along the chosen direction are very well described by the exponential-decay model. In contrast, the ratios of the decay probabilities and probability currents are evidently not a simple constant (as they would be in the exponential limit) but display time-persisting oscillations. Hence, these ratios are optimal witnesses of deviations from the exponential decay law.

Atomic photoionization dynamics in ultrashort cycloidal laser fields

We present numerical simulations of ultrafast multiphoton ionization dynamics in a two-dimensional atomic model driven by co- and counterrotating circularly polarized single-color and bichromatic carrier envelope phase (CEP) stable ultrashort laser pulse sequences. Taking into account phase variations due to CEP fluctuations and the Gouy phase, our results accurately reproduce recently measured photoelectron momentum distributions [Pengel et al., Phys. Rev. Lett. 118, 053003 (2017), Phys. Rev. A 96, 043426 (2017); Kerbstadt et al., Nat. Comm. 10, 685 (2019), Adv. Phys. X 4, 1672583 (2019)]. The time evolution of the complex-valued electron wave function in coordinate and momentum space is calculated to study the bound state- and the vortex formation dynamics. The non-vanishing azimuthal probability current density proves the vortex nature of electron wave packets with odd-numbered rotational symmetry. Their angular momentum expectation value assumes half-integer values of 3.5 (corotating) and 0.5 (counterrotating). Knowledge of the wave function allows us to analyze the photoionization dynamics and to validate the physical pictures proposed in previous experimental studies. As an outlook, we investigate how electron vortices develop from the multiphoton- to the tunnel regime with increasing laser intensity.

Gain-of-function GABRB3 variants identified in vigabatrin-hypersensitive epileptic encephalopathies.

Variants in the GABRB3 gene encoding the β3-subunit of the γ-aminobutyric acid type A ( receptor are associated with various developmental and epileptic encephalopathies. Typically, these variants cause a loss-of-function molecular phenotype whereby γ-aminobutyric acid has reduced inhibitory effectiveness leading to seizures. Drugs that potentiate inhibitory GABAergic activity, such as nitrazepam, phenobarbital or vigabatrin, are expected to compensate for this and thereby reduce seizure frequency. However, vigabatrin, a drug that inhibits γ-aminobutyric acid transaminase to increase tonic γ-aminobutyric acid currents, has mixed success in treating seizures in patients with GABRB3 variants: some patients experience seizure cessation, but there is hypersensitivity in some patients associated with hypotonia, sedation and respiratory suppression. A GABRB3 variant that responds well to vigabatrin involves a truncation variant (p.Arg194*) resulting in a clear loss-of-function. We hypothesized that patients with a hypersensitive response to vigabatrin may exhibit a different γ-aminobutyric acid A receptor phenotype. To test this hypothesis, we evaluated the phenotype of de novo variants in GABRB3 (p.Glu77Lys and p.Thr287Ile) associated with patients who are clinically hypersensitive to vigabatrin. We introduced the GABRB3 p.Glu77Lys and p.Thr287Ile variants into a concatenated synaptic and extrasynaptic γ-aminobutyric acid A receptor construct, to resemble the γ-aminobutyric acid A receptor expression by a patient heterozygous for the GABRB3 variant. The mRNA of these constructs was injected into Xenopus oocytes and activation properties of each receptor measured by two-electrode voltage clamp electrophysiology. Results showed an atypical gain-of-function molecular phenotype in the GABRB3 p.Glu77Lys and p.Thr287Ile variants characterized by increased potency of γ-aminobutyric acid A without change to the estimated maximum open channel probability, deactivation kinetics or absolute currents. Modelling of the activation properties of the receptors indicated that either variant caused increased chloride flux in response to low concentrations of γ-aminobutyric acid that mediate tonic currents. We therefore propose that the hypersensitivity reaction to vigabatrin is a result of GABRB3 variants that exacerbate GABAergic tonic currents and caution is required when prescribing vigabatrin. In contrast, drug strategies increasing tonic currents in loss-of-function variants are likely to be a safe and effective therapy. This study demonstrates that functional genomics can explain beneficial and adverse anti-epileptic drug effects, and propose that vigabatrin should be considered in patients with clear loss-of-function GABRB3 variants.

Continuity Relations, Probability Acceleration Current Sources and Internal Communications in Interacting Fragments

Classical issues of local continuities and density partition in molecular quantum mechanics are reexamined. An effective velocity of the probability current is identified as the current-per-particle and its properties are explored. The local probability acceleration and the associated force concept are introduced. They are shown to identically vanish in the stationary electronic states. This acceleration measure also determines the associated productions of physical currents, e.g., the local source of the resultant content of electronic gradient information. The probability partitioning between reactants is revisited and illustrated using the stockholder division rule of Hirshfeld. A simple orbital model is used to describe the polarized (disentangled) and equilibrium (entangled) molecular fragments containing the distinguishable and indistinguishable groups of electrons, respectively, and their mixed quantum character is emphasized. The fragment density matrix is shown to determine the subsystem internal electron communications.

An infinite square-well potential as a limiting case of a square-well potential in a minimal-length scenario

One of the most widely problem studied in quantum mechanics is of an infinite square-well potential. In a minimal-length scenario its study requires additional care because the boundary conditions at the walls of the well are not well fixed. In order to avoid this we solve the finite square-well potential whose the boundary conditions are well fixed, even in a minimal-length scenario, and then we take the limit of the potential going to infinity to find the eigenfunctions and the energy equation for the infinite square-well potential. Although the first correction for the energy eigenvalues is the same as one found in the literature, our result shows that the eigenfunctions have the first derivative continuous at the square-well walls what is in disagreement with those previous work. That is because in the literature the authors have neglected the hyperbolic solutions and have assumed the discontinuity of the first derivative of the eigenfunctions at the walls of the infinite square-well which is not correct. As we show, the continuity of the first derivative of the eigenfunctions at the square-well walls guarantees the continuity of the probability current density and the unitarity of the time evolution operator.

Dynamics of wave packet transport in branched nanowires

Abstract We have applied wave packet formalism to investigate the dynamics of electron transport in semiconductor nanodevices shaped in branched nanowires. The nanowires are formed by quantum channels with a width of 10 nm, in which a two-dimensional electron gas is scattered. The theoretical model is based on the numerical solution of the time-dependent Schrodinger equation, within the effective-mass and envelope wave function formalism. The split-operator technique is applied and it allows us to calculate the transmission probability, the total probability current, the conductance, and the wave function scattering between the energy subbands. Moreover, the transmission can be tuned in the output channels by applying an external electric field. Hence, the conductivity in the output device can be controlled by an external parameter. We also show how the geometry of the nanowires acts to scatter the electron gas and how the conductance is affected.

Generalized Markov stability of network communities

We address the problem of community detection in networks by introducing a general definition of Markov stability, based on the difference between the probability fluxes of a Markov chain on the network at different timescales. The specific implementation of the quality function and the resulting optimal community structure thus become dependent both on the type of Markov process and on the specific Markov times considered. For instance, if we use a natural Markov chain dynamics and discount its stationary distribution (that is, we take as reference process the dynamics at infinite time) we obtain the standard formulation of the Markov stability. Notably, the possibility to use finite-time transition probabilities to define the reference process naturally allows detecting communities at different resolutions, without the need to consider a continuous-time Markov chain in the small time limit. The main advantage of our general formulation of Markov stability based on dynamical flows is that we work with lumped Markov chains on network partitions, having the same stationary distribution of the original process. In this way the form of the quality function becomes invariant under partitioning, leading to a self-consistent definition of community structures at different aggregation scales.

Beyond brane-Higgs regularization: Clarifying the method and model

The class of higher-dimensional scenarios, based on a brane-localised Higgs boson coupled to bulk fermions, can address both the flavour puzzle and gauge hierarchy problem. A key question arises due to the possibility of fermion profile discontinuities at the Higgs boundary: how to calculate rigorously the fermion mass spectrum and effective four-dimensional (4D) Yukawa couplings? We show that the proper treatment, leading to physically consistent solutions, does not rely on any Higgs peak regularisation but requires the presence of certain Bilinear Brane Terms (BBT). In particular, no profile jump should appear and the Higgs regularisations turn out to suffer from mathematical discrepancies reflected in two non-commutativities of calculation steps debated in the literature. The introduction of BBT can by replaced by vanishing conditions for probability currents at the considered flat interval boundaries. Both contribute to the definition of the field geometrical configuration of the model, even in the free case. The BBT could allow to elaborate an ultra-violet origin of the chiral nature of the Standard Model and of its chirality distribution among quarks/leptons. The current conditions are implemented via essential boundary conditions to be contrasted with the natural boundary conditions derived from the action variation. All these theoretical conclusions are confirmed in particular by the converging exact results of the 4D versus 5D approaches. The analysis is completed by a description of the appropriate energy cut-off procedure. The new calculation methods presented, implying the independence of excited fermion masses and 4D Yukawa couplings on the 'wrong-chirality' Yukawa terms, have impacts on phenomenological results like the relaxing of previously obtained strong bounds on Kaluza-Klein masses induced by flavour changing reactions generated through exchanges of the Higgs field.

Subgrid effects on the filtered velocity gradient dynamics in compressible turbulence

The subgrid effects on the dynamics of the filtered velocity gradient tensor (VGT) in compressible turbulence are studied by means of statistical analysis of the invariants of the filtered VGT in compressible mixing layers. The evolution of the filtered VGT is determined by the interaction among the invariants, the pressure effects, the viscous effects and the subgrid effects. Based on the probability fluxes in the plane of the second ( ) and the third ( ) invariants of the filtered VGT, it is found that the flux for the subgrid effect term changes most with the dilatation compared to the other terms. Further, a Schur decomposition of the filtered VGT into its normal part and non-normal part, which represent the local effect and the non-local effect of the flow dynamics, respectively, is used to deal with their effects of the velocity gradient. It is revealed that the compressibility is mainly related to the normal effect while the behaviour of the subgrid-scale (SGS) energy dissipation is mainly associated with the non-normal effect. A backscattering region of the SGS energy dissipation in the – plane is identified in the locally expanded regions, which is determined by the non-normal effect. Further, an SGS model with the non-local effect is proposed to give a better prediction of the SGS energy dissipation.

The negative flow of probability

We derive an inequality that the parameters of a 1D free-particle Gaussian wave packet with a positive group velocity, approaching a given region x > q, must satisfy such that a negative probability current J exists on q. Local probability conservation implies the counter-intuitive result that the particle detection probability in the region x > q is actually decreasing. The condition J < 0 requires the negative correlation of the position and momentum observables of the state, but the time scales for the negative current and anti-correlation regimes are not identical. Using a probability current operator, we obtain an integral representation of J in momentum space for any free particle wave packet. We use this integral representation to distinguish the separate contributions to J by the positive and negative momentum components, and we identify a third contribution to J composed of cross-terms of both momenta. For the specific case of a Gaussian wave packet with a negative correlation between its position and momentum, the positive momentum component can contribute a negative value to the probability current.

The Pauli and Lévy-Leblond equations, and the spin current density

We review the literature on the Pauli equation and its current density, discussing the progression from the original phenomenological version of Pauli to its derivation by Lévy-Leblond from a linearization of the Schrödinger equation. It was established conclusively by Lévy-Leblond’s work that the spin of a spin-1/2 particle such as an electron is non-relativistic in nature, contrary to what was often stated following Dirac’s derivation of a relativistic wave equation, and his subsequent demonstration that Pauli’s spin interaction term appeared in the non-relativistic limit. In this limit, the Gordon decomposition of the associated probability current density was found to contain a spin-dependent term. Such a term does not follow, however, from the usual derivation of the current density from the Pauli equation, although various physically motivated but otherwise ad hoc explanations were put forward to account for it. We comment on the only exception to these of which we are aware implying the spin term in the current was in fact non-relativistic in nature. However, the earlier work of Lévy-Leblond had already shown, with no additional assumptions, that this term was a prominent feature of the current density derived from his equation. Hence, just as with the spin itself, the spin current was non-relativistic, claims to the contrary notwithstanding. We present a somewhat simplified derivation of the Lévy-Leblond equation and its current density, commenting on possibilities for experimental work that might indicate measurable consequences of the spin term in the current density. The presentation is at a level of quantum mechanics, electromagnetic theory, and mathematical methods typical of courses taken by first-year graduate students in physics, assuming some familiarity with Pauli spin matrices and the Dirac equation. Calculations are presented in detail, making the material suitable for use by teachers as supplementary problems for assignments, and development of theoretical concepts peculiar to the Pauli equation and its probability current density.

Uncovering the Underlying Mechanisms of Cancer Metabolism through the Landscapes and Probability Flux Quantifications.

SummaryCancer metabolism is critical for understanding the mechanism of tumorigenesis, yet the understanding is still challenging. We studied gene-metabolism regulatory interactions and quantified the global driving forces for cancer-metabolism dynamics as the underlying landscape and probability flux. We uncovered four steady-state attractors: a normal state attractor, a cancer OXPHOS state attractor, a cancer glycolysis state attractor, and an intermediate cancer state attractor. We identified the key regulatory interactions through global sensitivity analysis based on the landscape topography. Different landscape topographies of glycolysis switch between normal cells and cancer cells were identified. We uncovered that the normal state to cancer state transformation is associated with the peaks of the probability flux and the thermodynamic dissipation, giving dynamical and thermodynamic origin of cancer formation. We found that cancer metabolism oscillations consume more energy to support cancer malignancy. This study provides a quantitative understanding of cancer metabolism and suggests a metabolic therapeutic strategy.

The edge states properties of the topological insulator with fermion path nonanalyticity points located at the edge

The edge states properties of a finite-size 2D topological insulator (TI) with the open boundary conditions in both directions are studied. It is shown that fermion path nonanalyticity points appearing on the edge of finite-size TI cause a nonuniform edge state distribution along the TI edge. The character of this distribution depends substantially on the edge state’s energy position in the bulk band gap. The edge state with the energy in the middle of the gap tends to be located in the corners between two nonparallel edges of the system, while the edge states with the energy close to the bulk band edge avoid the corners. The monotonic transition from one space distribution of the edge state wave function to another along the edge state’s band is observed. The mentioned edge state’s structure leads to the nonlinear current-voltage characteristic of the square-shaped TI with contacts connected with the corners of the device. It opens the possibility to control the current with the gate voltage and is useful for construction the nano-size devices. For the edge states with energy in the middle of a bulk gap, the vortex structure of the probability flux located near corners is found.

Floquet boundary states in AB -stacked graphite

We report on the effect of laser illumination with circularly polarized light on the electronic structure of AB-stacked graphite samples. By using Floquet theory in combination with Green's function techniques, we find that the polarized light induces band-gap openings at the Floquet zone edge $\hbar\Omega/2$, bridged by chiral boundary states. These states propagate mainly along the borders of the constituting layers as evidenced by the time-averaged local density of states and the probability current density in several geometries. Semianalytic calculations of the Chern number suggest that these states are of topological nature, similar to those found in illuminated 2D samples like monolayer and bilayer graphene. These states are promising candidates for the realization of a three-dimensional version of the quantum Hall effect for Floquet systems.

Quantum tunneling through a rectangular barrier in multi-Weyl semimetals

Multi-Weyl semimetals are new types of topological semimetals whose topological charge is equal to the value of the winding number $J$. Here, we investigate the single-particle ballistic scattering on a rectangular barrier in multi-Weyl semimetals. Because this system has a crystallographic anisotropy, the scattering properties depend on the mutual orientation of the crystalline axis and the barrier. For different $J$, the vertical component of the wave vector ${k}_{\ensuremath{\perp}}$ and the corresponding probability current density ${j}_{\ensuremath{\perp}}$ satisfies ${j}_{\ensuremath{\perp}}\ensuremath{\propto}{k}_{\ensuremath{\perp}}^{2J\ensuremath{-}1}$. In the case of a barrier perpendicular to the $z$-axis, it is found that the reflectionless incident angles are determined by geometrical resonances between the barrier width and the de Broglie length of the scattered electrons in the barrier region. In the $z$-axis direction, the local minimum conductance ${G}_{\mathrm{min}}$ occurs when the chemical potential equals the barrier height and ${G}_{\mathrm{min}}\ensuremath{\propto}1/{L}^{2/J}$, where $L$ is the width of the barrier. Differently, in the case of a barrier perpendicular to the $x$-axis, the angular distribution of the transmission probability is no longer rotation invariant. For the double-Weyl semimetals ($J=2$), the transmission probability decreases rapidly to 0 as the barrier width $L$ increases for a normal incidence, which is similar to conventional nonrelativistic electrons. It is interesting that perfect transmission is again found for normally incident Weyl fermions for the triple-Weyl semimetals ($J=3$). In this case, the tunneling indicates a property similar to that in the case of $J=1$.

Cell renormalized FPK equation for stochastic non-linear systems

The Fokker–Planck–Kolmogorov (FPK) equation, as a well investigated partial differential equation, is of great significance to stochastic dynamics due to its theoretical rigor and exactness. However, practical difficulties with the FPK method are encountered when analysis of multi-degree-of-freedom (MDOF) systems with arbitrary nonlinearity is required. In the present paper, a cell renormalized method (CRM) which is based on a numerical determination of response statistical moments of the FPK equation is developed. Specifically, by invoking the concept of equivalence of probability flux, a cell renormalization procedure and a reconstruction scheme of derivative moments are introduced to divide the continuous state space into a discretized region of cells so that numerical derivative moments is derived. Subsequently, the Cell Renormalized FPK (CR-FPK) equation can be solved by a finite difference algorithm. Two numerical examples are included, and the effectiveness of the proposed method is assessed.

Probability Landscape of Coupled Epigenetic and Genetic Network with Eddy-Like Probability Currents

Gene expression is regulated by multiple processes like transcription factor (TF) binding/unbinding and epigenetic modifications in eukaryotes. These processes are interdependent, though separated in time scale. We investigate the effects of these coupled processes of histone modifications via acetylation and methylation, and TF binding/unbinding in a single landscape. Starting from a model involving discrete state changes of histones and TF binding status, rates, and feedbacks associated with them, we write a master equation and transform it to a set of Langevin equations using a path-integral formalism; the Langevin equations of continuous order parameters representing protein copy numbers, histone states, and TF binding states are obtained.

Investigating Multiwavelength Lognormality with Simulations—Case of Mrk 421

Blazars are highly variable and display complex characteristics. A key characteristic is the flux probability distribution function or flux PDF whose shape depends upon the form of the underlying physical process driving variability. The BL Lacertae Mrk 421 is one of the brightest and most variable blazars across the electromagnetic spectrum. It has been reported to show hints of lognormality across the spectrum from radio to gamma-ray histograms of observed fluxes. This would imply that the underlying mechanisms may not conform to the “standard” additive, multi-zone picture, but could potentially have multiplicative processes. This is investigated by testing the observed lightcurves at different wavelengths with time-series simulations. We find that the simulations reveal a more complex scenario, than a single lognormal distribution explaining the multiwavelength lightcurves of Mrk 421.