Probability Current Research Articles

Stochastic line integrals and stream functions as metrics of irreversibility and heat transfer

Stochastic line integrals are presented as a useful metric for quantitatively characterizing irreversibility and detailed balance violation in noise-driven dynamical systems. A particular realization is the stochastic area, recently studied in coupled electrical circuits. Here we provide a general framework for understanding properties of stochastic line integrals and clarify their implementation for experiments and simulations. For two-dimensional systems, stochastic line integrals can be expressed in terms of a stream function, the sign of which determines the orientation of nonequilibrium steady-state probability currents. Theoretical results are supported by numerical studies of an overdamped two-dimensional mass-spring system driven out of equilibrium. Additionally, the stream function permits analytical understanding of the scaling dependence of stochastic area growth rate on key parameters such as the noise strength for both linear and nonlinear springs.

Electromagnetic Signatures of Possible Charge Anomalies in Tunneling

We reconsider some well-known tunneling processes from the point of view of Aharonov-Bohm electrodynamics, a unique extension of Maxwell’s theory which admits charge-current sources that are not locally conserved. In particular we are interested into tunneling phenomena having relatively long range (otherwise the non-Maxwellian effects become irrelevant, especially at high frequency) and involving macroscopic wavefunctions and coherent matter, for which it makes sense to evaluate the classical e.m. field generated by the tunneling particles. For some condensed-matter systems, admitting discontinuities in the probability current is a possible way of formulating phenomenological models. In such cases, the Aharonov-Bohm theory offers a logically consistent approach and allows to derive observable consequences. Typical e.m. signatures of the failure of local conservation are at high frequency the generation of a longitudinal electric radiation field, and at low frequency a small effect of “missing” magnetic field. Possible causes of this failure are instant tunneling and phase slips in superconductors. For macroscopic quantum systems in which the phase-number uncertainty relation ΔNΔφ∼1 applies, the expectation value of the anomalous source I=∂tρ+∇·j has quantum fluctuations, thus becoming a random source of weak non-Maxwellian fields.

Quantum Hall effect in q-formalism based on Fermi gas model

To the best of our knowledge, for the first time, the quantum Hall effect (QHE) is considered in thermal non-equilibrium conditions (TNEC) via [Formula: see text]-algebra. The [Formula: see text]-algebra is a topological tool that provides a controlling [Formula: see text]-parameter to approximate theoretical results to the laboratory ones. We obtain the QHE spectrum and [Formula: see text]-deformed Hall potential (or Hall potential in TNEC) via the density probability current. Because of the Pauli Exclusion Principle, we model the QHE using relativistic Fermi electron gas and the [Formula: see text]-deformed Fermi–Dirac distribution. We find the dependence of deformed Hall potential to temperature and magnetic field for different geometric [Formula: see text]-regions. Further, we geometrically prove that the magnitude of the Hall potentials depends on the [Formula: see text]-regions (i.e. [Formula: see text]). We find that for some [Formula: see text]-amounts, the Landau energy gap on graphene-based materials can be explained in the room-temperature quantum Hall effect (RTQHE). Our method may be applied to analyze new electronic devices based on graphene in TNEQ.

Spectral analysis of bilateral birth–death processes: some new explicit examples

AbstractWe consider the spectral analysis of several examples of bilateral birth–death processes and compute explicitly the spectral matrix and the corresponding orthogonal polynomials. We also use the spectral representation to study some probabilistic properties of the processes, such as recurrence, the invariant distribution (if it exists), and the probability current.

The effect of large-scale forcing on small-scale dynamics of incompressible turbulence

The subtle but crucial effects of large-scale forcing on the small-scale velocity-gradient (VG) dynamics is examined using direct numerical simulation (DNS) data of incompressible turbulence. The interplay among large-scale forcing, inertia, pressure and viscous effects is characterised as a function of local streamline geometry and VG magnitude (Frobenius norm). When conditioned on local topology, forcing: (i) counteracts inertial and viscous action in the strain-dominated nodal topologies; and (ii) balances pressure action in the rotation-dominated unstable focal topologies. Unexpectedly, forcing acts to reduce VG magnitudes in regions of strong dissipation. In these regions, forcing balances the non-local pressure effects whereas viscous action offsets the nonlinear inertial effects. In regions of very low dissipation, forcing combines with inertia and pressure effects to offset viscous action. With regard to the probability distribution of the normalised VG invariants, the primary role of forcing is to nullify certain features (dilatational probability currents) of inertia, pressure and viscous action. This results in the emergence of universal statistical features (solenoidal probability currents) that are determined largely by inertia–pressure–viscous interactions. These findings serve to enhance our understanding of small-scale processes in turbulence and guide VG model development.

State-space renormalization group theory of nonequilibrium reaction networks: Exact solutions for hypercubic lattices in arbitrary dimensions.

Nonequilibrium reaction networks (NRNs) underlie most biological functions. Despite their diverse dynamic properties, NRNs share the signature characteristics of persistent probability fluxes and continuous energy dissipation even in the steady state. Dynamics of NRNs can be described at different coarse-grained levels. Our previous work showed that the apparent energy dissipation rate at a coarse-grained level follows an inverse power-law dependence on the scale of coarse-graining. The scaling exponent is determined by the network structure and correlation of stationary probability fluxes. However, it remains unclear whether and how the (renormalized) flux correlation varies with coarse-graining. Following Kadanoff's real space renormalization group (RG) approach for critical phenomena, we address this question by developing a state-space renormalization group theory for NRNs, which leads to an iterative RG equationfor the flux correlation function. In square and hypercubic lattices, we solve the RG equationexactly and find two types of fixed point solutions. There is a family of nontrivial fixed points where the correlation exhibits power-law decay, characterized by a power exponent that can take any value within a continuous range. There is also a trivial fixed point where the correlation vanishes beyond the nearest neighbors. The power-law fixed point is stable if and only if the power exponent is less than the lattice dimension n. Consequently, the correlation function converges to the power-law fixed point only when the correlation in the fine-grained network decays slower than r^{-n} and to the trivial fixed point otherwise. If the flux correlation in the fine-grained network contains multiple stable solutions with different exponents, the RG iteration dynamics select the fixed point solution with the smallest exponent. The analytical results are supported by numerical simulations. We also discuss a possible connection between the RG flows of flux correlation with those of the Kosterlitz-Thouless transition.

Spatially-heterogeneous embedded stochastic SEIR models for the 2014–2016 Ebola outbreak in West Africa

The dynamics of human infectious diseases are challenging to understand, particularly when a pathogen spreads spatially over a large region. We present a stochastic, spatially-heterogeneous model framework derived from the foundational SEIR compartmental model. These models utilize a graph structure of spatial locations, facilitating mobility via random walks while progressing through disease states, parameterized by the net probability flux between locations. The analysis is bolstered by Approximate Bayesian Computation, by which epidemiological and mobility parameter distributions are estimated, including an empirically adjusted reproductive number, while model structure proposals are compared using Bayes Factors. The utility of this novel class of models is demonstrated through application to the 2014–2016 Ebola outbreak in West Africa. The flexibility of such models, whose complexity may be adjusted as desired, and complementary methods of analysis enable the exploration of various spatial divisions and mobility schema, while maintaining the essential spatiotemporal disease dynamics.

Generalized phase-space description of nonlinear Hamiltonian systems and Harper-like dynamics

Phase-space features of the Wigner flow for generic one-dimensional systems with a Hamiltonian, $H^{W}(q,\,p)$, constrained by the $\partial ^2 H^{W} / \partial q \partial p = 0$ condition are analytically obtained in terms of Wigner functions and Wigner currents. Liouvillian and stationary profiles are identified for thermodynamic (TD) and Gaussian quantum ensembles to account for exact corrections due to quantum modifications over a classical phase-space pattern. General results are then specialized to the Harper Hamiltonian system which, besides working as a feasible test platform for the framework here introduced, admits a statistical description in terms of TD and Gaussian ensembles, where the Wigner flow properties are all obtained through analytical tools. Quantum fluctuations over the classical regime are therefore quantified through probability and information fluxes whenever the classical Hamiltonian background is provided. Besides allowing for a broad range of theoretical applications, our results suggest that such a generalized Wigner approach works as a probe for quantumness and classicality of Harper-like systems, in a framework which can be extended to any quantum system described by Hamiltonians in the form of $H^{W}(q,\,p) = K(p) + V(q)$.

Negative rectification and anomalous diffusion in nonlinear substrate potentials: Dynamical relaxation and information entropy.

We numerically investigate the rectification of the probability flux and dynamical relaxation of particles moving in a system with and without noise. The system, driven by two external forces, consists of two substrate potentials that have identical shapes and different potential barriers with different friction coefficients. The deterministic model exhibits the perfect rectification of the probability flux, ratchet effect, and the dependence of the unpredictability of the dynamics on basin of attraction. In contrast, the stochastic model displays that the rectification is sensitive to the temperature and an external bias. They can induce kinetic phase transitions between no transport and a finite net transport. These transitions lead to an unexpected phenomenon, called negative rectification. The results are analyzed through the corresponding time-dependent diffusion coefficient, information entropy (IE), etc. At a low temperature, anomalous diffusions occur in system. For the occurrence of the flux in certain parameter regimes, the larger the diffusion is, the smaller the corresponding IE is, and vice versa. We also present the selected parameter regimes for the emergence of the rectification and negative rectification. Additionally, we study the rectification of the interacting particles in the system and find that the flux may depend on the coupling strength and the number of the interacting particles, and that collective motions occur for the forward flux. Our work provides not only a way of the rectification for the transport of various particles (e.g., ions, electrons, photons, phonons, molecules, DNA chains, nanoswimmers, dust particles, etc.) in physics, chemistry, biology, and material science, but also a design of various circuits.

Outbreak Size Distribution in Stochastic Epidemic Models.

Motivated by recent epidemic outbreaks, including those of COVID-19, we solve the canonical problem of calculating the dynamics and likelihood of extensive outbreaks in a population within a large class of stochastic epidemic models with demographic noise, including the susceptible-infected-recovered (SIR) model and its general extensions. In the limit of large populations, we compute the probability distribution for all extensive outbreaks, including those that entail unusually large or small (extreme) proportions of the population infected. Our approach reveals that, unlike other well-known examples of rare events occurring in discrete-state stochastic systems, the statistics of extreme outbreaks emanate from a full continuum of Hamiltonian paths, each satisfying unique boundary conditions with a conserved probability flux.

Theory of laser-induced photoemission from a metal surface with nanoscale dielectric coating

This paper presents an analytical quantum model for photoemission from metal surfaces coated with an ultrathin dielectric, by solving the 1D time-dependent Schrödinger equation subject to an oscillating double-triangular potential barrier. The model is valid for an arbitrary combination of metal (of any work function and Fermi level), dielectric (of any thickness, relative permittivity, and electron affinity), laser field (strength and wavelength), and dc field. The effects of dielectric properties on photoemission are systematically investigated. It is found that a flat metal surface with dielectric coating can photoemit a larger current density than the uncoated case when the dielectric has smaller relative permittivity and larger electron affinity. Resonant peaks in the photoemission probability and emission current are observed as a function of dielectric thickness or electron affinity due to the quantum interference of electron waves inside the dielectric. Our model is compared with the effective single-barrier quantum model and modified Fowler–Nordheim equation, for both 1D flat cathodes and pyramid-shaped nanoemitters. While the three models show quantitatively good agreement in the optical field tunneling regime, the present model may be used to give a more accurate evaluation of photoemission from coated emitters in the multiphoton absorption regime.

The onset of dissipative chaos driven by nonequilibrium conditions.

Dissipative chaos appears widely in various nonequilibrium systems; however, it is not clear how dissipative chaos originates from nonequilibrium. We discuss a framework based on the potential-flux approach to study chaos from the perspective of nonequilibrium dynamics. In this framework, chaotic systems possess a wide basinon the potential landscape, in which the rotational flux dominates the system dynamics, and chaos occurs with the appearance of this basin. In contrast, the probability flux is particularly associated with the detailed balance-breaking in nonequilibrium systems. This implies that the appearance of dissipative chaos is driven by nonequilibrium conditions.

Midisuperspace foam and the cosmological constant

Wheeler’s conjectured ‘spacetime foam’—large quantum fluctuations of spacetime at the Planck scale—could have important implications for quantum gravity, perhaps even explaining why the cosmological constant seems so small. Here I explore this problem in a midisuperspace model consisting of metrics with local spherical symmetry. Classically, an infinite class of ‘foamy’ initial data can be constructed, in which cancellations between expanding and contracting regions lead to a small average expansion even if Λ is large. Quantum mechanically, the model admits corresponding stationary states, for which the probability current is also nearly zero. These states appear to describe a self-reproducing spacetime foam with very small average expansion, effectively hiding the cosmological constant.

Quantum backflow across a black hole horizon in a toy model approach

Quantum backflow, discovered quite a few years back, is a generic purely quantum phenomenon, in which the probability of finding a particle in a direction is nonzero (and increasing for a certain period of time) even when the particle has with certainty a velocity in the opposite direction. In this paper, we study quantum backflow of a quantum particle across the event horizon of a Schwarzschild black hole. In a toy model approach, we consider a superposition of two ingoing solutions and observe the probability density and probability current. We explicitly demonstrate a nonvanishing quantum back flow in a small region around the event horizon. This is in contrast to the classical black hole picture, that once an excitation crosses the horizon, it is lost forever from the outside world. Deeper implications of this phenomenon are speculated. We also study quantum backflow for another spacetime with horizon, the Rindler spacetime, where the phenomenon can be studied only within the Rindler wedge.

Steady state of overdamped particles in the non-conservative force field of a simple non-linear model of optical trap

Optically trapped particles are often subject to a non-conservative scattering force arising from radiation pressure. In this paper, we present an exact solution for the steady state statistics of an overdamped Brownian particle subjected to a commonly used force field model for an optical trap. The model is the simplest of its kind that takes into account non-conservative forces. In particular, we present the exact results for certain marginals of the full three-dimensional steady state probability distribution, in addition to results for the toroidal probability currents that are present in the steady state, as well as for the circulation of these currents. Our analytical results are confirmed by numerical solution of the steady state Fokker–Planck equation.

Seed priming with proline improved photosystem II efficiency and growth of wheat (Triticum aestivum L.)

BackgroundProline can promote growth of plants by increasing photosynthetic activity under both non-stress and abiotic stress conditions. However, its role in non-stressed conditions is least studied. An experiment was conducted to assess as to whether increase in growth of wheat due to seed priming with proline under non-stress condition was associated with proline-induced changes in photosystem II (PSII) activity. Seeds of four wheat varieties (S-24, Sehar-06, Galaxy-13, and Pasban-90) were primed with different concentrations of proline (0, 5, 15 and 25 mM) for 12 h and allowed to grow under normal conditions. Biomass accumulation and photosynthetic performance, being two most sensitive features/indicators of plant growth, were selected to monitor proline modulated changes.ResultsSeed priming with proline increased the fresh and dry weights of shoots and roots, and plant height of all four wheat varieties. Maximum increase in growth attributes was observed in all four wheat varieties at 15 mM proline. Maximum growth improvement due to proline was found in var. Galaxy-13, whereas the reverse was true for S-24. Moreover, proline treatment changed the Fo, Fm, Fv/Fo, PIABS, PITot in wheat varieties indicating changes in PSII activity. Proline induced changes in energy fluxes for absorption, trapping, electron transport and heat dissipation per reaction center indicated that var. Galaxy-13 had better ability to process absorbed light energy through photosynthetic machinery. Moreover, lesser PSII efficiency in var. S-24 was due to lower energy flux for electron transport and greater energy flux for heat dissipation. This was further supported by the fact that var. S-24 had disturbance at acceptor side of PSI as reflected by reduction in ΔVIP, probability and energy flux for electron transport at the PSI end electron acceptors.ConclusionSeed priming with proline improved the growth of wheat varieties, which depends on type of variety and concentration of proline applied. Seed priming with proline significantly changed the PSII activity in wheat varieties, however, its translation in growth improvement depends on potential of processing of absorbed light energy by electron acceptors of electron transport chain, particularly those present at PSI end.

Induced Pluripotent Stem Cell-Derived Cardiomyocytes with SCN5A R1623Q Mutation Associated with Severe Long QT Syndrome in Fetuses and Neonates Recapitulates Pathophysiological Phenotypes.

Simple SummaryIn this study, the induced pluripotent stem cell-derived cardiomyocyte model from a patient with long QT syndrome harboring a heterozygous Nav1.5 R1623Q mutation exhibited prolonged field potential duration corrected by Fridericia’s formula (FPDcF, analogous to QTcF). FPDcF was shortened with mexiletine treatment and increased the frequency of arrhythmia-like EAD events following E4031, an Ikr blocker, administration. These characteristics partly reflect the patient phenotypes. As the R1623Q mutation is related to severe congenital LQT syndrome in fetuses and neonates, the effect of the neonatal variants on the electrophysiological properties of the R1623Q mutant was examined using an automated patch-clamp system. Our results demonstrated that both R1623Q and neonatal R1623Q delayed inactivation of INa and increased late Na current. We speculated that neonatal Nav1.5 ameliorates QTc prolongation. Developmental switching of neonatal/adult Nav1.5 isoforms might play a role in the mechanisms underlying severe long QT syndrome in fetuses and neonates.The SCN5A R1623Q mutation is one of the most common genetic variants associated with severe congenital long QT syndrome 3 (LQT3) in fetal and neonatal patients. To investigate the properties of the R1623Q mutation, we established an induced pluripotent stem cell (iPSC) cardiomyocyte (CM) model from a patient with LQTS harboring a heterozygous R1623Q mutation. The properties and pharmacological responses of iPSC-CMs were characterized using a multi-electrode array system. The biophysical characteristic analysis revealed that R1623Q increased open probability and persistent currents of sodium channel, indicating a gain-of-function mutation. In the pharmacological study, mexiletine shortened FPDcF in R1623Q-iPSC-CMs, which exhibited prolonged field potential duration corrected by Fridericia’s formula (FPDcF, analogous to QTcF). Meanwhile, E4031, a specific inhibitor of human ether-a-go-go-related gene (hERG) channel, significantly increased the frequency of arrhythmia-like early after depolarization (EAD) events. These characteristics partly reflect the patient phenotypes. To further analyze the effect of neonatal isoform, which is predominantly expressed in the fetal period, on the R1623Q mutant properties, we transfected adult form and neonatal isoform SCN5A of control and R1623Q mutant SCN5A genes to 293T cells. Whole-cell automated patch-clamp recordings revealed that R1623Q increased persistent Na+ currents, indicating a gain-of-function mutation. Our findings demonstrate the utility of LQT3-associated R1623Q mutation-harboring iPSC-CMs for assessing pharmacological responses to therapeutic drugs and improving treatment efficacy. Furthermore, developmental switching of neonatal/adult Nav1.5 isoforms may be involved in the pathological mechanisms underlying severe long QT syndrome in fetuses and neonates.

Stochastic Hopf bifurcations in vacuum optical tweezers

The forces acting on an isotropic microsphere in optical tweezers are effectively conservative. However, reductions in the symmetry of the particle or trapping field can break this condition. Here we theoretically analyze the motion of a particle in a linearly nonconservative optical vacuum trap, concentrating on the case where symmetry is broken by optical birefringence, causing nonconservative coupling between rotational and translational degrees of freedom. Neglecting thermal fluctuations, we first show that the underlying deterministic motion can exhibit a Hopf bifurcation in which the trapping point destabilizes and limit cycles emerge whose amplitude grows with decreasing viscosity. When fluctuations are included, the bifurcation of the underlying deterministic system is expressed as a transition in the statistical description of the motion. For high viscosities, the probability distribution is normal, with a kurtosis of three, and persistent probability currents swirl around the stable trapping point. As the bifurcation is approached, the distribution and currents spread out in phase space. Following the bifurcation, the probability distribution function hollows out, reflecting the underlying limit cycle, and the kurtosis halves abruptly. The system is seen to be a noisy self-sustained oscillator featuring a highly uneven limit cycle. A variety of applications, from autonomous stochastic resonance to synchronization, is discussed.

Research on Total Ionizing Dose Effect and Reinforcement of SOI-TFET.

Since the oxide/source overlap structure can improve the tunneling probability and on-state current of tunneling field effect transistor (TFET) devices, and the silicon-on-insulator (SOI) structure has the effect of resisting the single event effect, SOI-TFET with the oxide/source overlap structure is a device with development potential. The total ionizing dose (TID) effect on SOI-TFET was studied by discussing the switching ratio, band–band tunneling rate, threshold voltage, sub-threshold swing and bipolar effect of the device under different doses of irradiation. At the same time, simulations prove that selecting the proper thickness of the buried oxide (BOX) layer can effectively reduce the influence of the TID effect. This provides a way of direction and method for research on the irradiation effects on the device in the future.

Energy harvesting from anisotropic fluctuations.

We consider a rudimentary model for a heat engine, known as the Brownian gyrator, that consists of an overdamped system with two degrees of freedom in an anisotropic temperature field. Whereas the hallmark of the gyrator is a nonequilibrium steady-state curl-carrying probability current that can generate torque, we explore the coupling of this natural gyrating motion with a periodic actuation potential for the purpose of extracting work. We show that path lengths traversed in the manifold of thermodynamic states, measured in a suitable Riemannian metric, represent dissipative losses, while area integrals of a work density quantify work being extracted. Thus, the maximal amount of work that can be extracted relates to an isoperimetric problem, trading off area against length of an encircling path. We derive an isoperimetric inequality that provides a universal bound on the efficiency of all cyclic operating protocols, and a bound on how fast a closed path can be traversed before it becomes impossible to extract positive work. The analysis presented provides guiding principles for building autonomous engines that extract work from anisotropic fluctuations.