Stochastic Formalism Research Articles

Loop correction and resummation of vertex functions for a self interacting scalar field in the de Sitter spacetime

We consider a massless and minimally coupled self interacting quantum scalar field theory in the inflationary de Sitter background of dimension four. The self interaction potential is taken to be either quartic, λϕ4/4!, or quartic plus cubic, λϕ4/4!+βϕ3/3! (λ>0). We compute the four and three point vertex functions up to two loop. The purely local or partly local part of these renormalised loop corrected vertex functions grow unboundedly after sufficient number of de Sitter e-foldings, due to the appearances of secular logarithms. We focus on the purely local part of the vertex functions and attempt a resummation of them in terms of the dynamically generated mass of the scalar field at late times. Such local logarithms have sub-leading powers compared to the non-local leading ones which can be resummed via the stochastic formalism. The variation of these vertex functions are investigated with respect to the tree level couplings numerically. Since neither the secular effect, nor the dynamical generation of field mass is possible in the Minkowski spacetime, the above phenomenon has no flat spacetime analogue. We have also compared our result with the ones that could be found via the recently proposed renormalisation group techniques. All these results suggest that at late times the value of the non-perturbative vertex function should be less than the tree level coupling.

Emergent particles of de Sitter: thermal interpretation of the stochastic formalism and beyond

A thermal interpretation of the stochastic formalism of a slow-rolling scalar field in de Sitter (dS) is given. We construct a correspondence between Hubble patches of dS and particles living in another space called an abstract space. By assuming a dual description of scalar fields and classical mechanics in the abstract space, we show that the stochastic evolution of the infrared part of the field is equivalent to the Brownian motion in the abstract space filled with a heat bath of massless particles. The 1st slow-roll condition and the Hubble expansion are also reinterpreted in the abstract space as the speed of light and a transfer of conserved energy, respectively. Inspired by this, we sketch quantum emergent particles, which may realize the Hubble expansion by an exponential particle production. This gives another meaning of dS entropy as entropy per Hubble volume.

Stern–Gerlach, EPRB and Bell Inequalities: An Analysis Using the Quantum Hamilton Equations of Stochastic Mechanics

The discussion of the recently derived quantum Hamilton equations for a spinning particle is extended to spin measurement in a Stern–Gerlach experiment. We show that this theory predicts a continuously changing orientation of the particles magnetic moment over the course of its motion across the Stern–Gerlach apparatus. The final measurement results agree with experiment and with predictions of the Pauli equation. Furthermore, the Einstein–Podolsky–Rosen–Bohm thought experiment is investigated, and the violation of Bells’s inequalities is reproduced within this stochastic mechanics approach. The origin of the violation of Bell’s inequalities is traced to the the non-local nature of the velocity fields for an entangled state in the stochastic formalism, which is a result of a non-separable probability distribution of the considered particles.

Monomial warm inflation revisited

Abstract We revisit the idea that the inflaton may have dissipated part of its energy into a thermal bath during inflation, considering monomial inflationary potentials and three different forms of dissipation rate. Using a numerical Fokker-Planck approach to describe the stochastic dynamics of inflationary fluctuations, we confront this scenario with current bounds on the spectrum of curvature fluctuations and primordial gravitational waves. We also obtain purely analytical approximations that improve over previously used ones in the small dissipation regime for the amplitude of the spectrum and its tilt. We show that only our numerical Fokker-Planck method is accurate, fast and precise enough to test these models against current data. We advocate its use in future studies of warm inflation. We also apply the stochastic inflation formalism to this scenario, finding that the resulting spectrum is the same as the one obtained with standard perturbation theory. We discuss the origin and convenience of using a commonly implemented large thermal correction to the primordial spectrum and the implications of such a term for a specific scenario. Improved bounds on the scalar spectral index will further constrain warm inflation in the near future.

A jump-diffusion stochastic formalism for muscle contraction models at multiple timescales

Muscle contraction at the macrolevel is a physiological process that is ultimately due to the interaction between myosin and actin proteins at the microlevel. The actin–myosin interaction involves slow attachment and detachment responses and a rapid temporal change in protein conformation called power-stroke. Jump-diffusion models that combine jump processes between attachment and detachment with a mechanical description of the power-stroke have been proposed in the literature. However, the current formulations of these models are not fully compatible with the principles of thermodynamics. To solve the problem of coupling continuous mechanisms with discrete chemical transitions, we rely on the mathematical formalism of Poisson random measures. First, we design an efficient stochastic formulation for existing muscle contraction partial differential equation models. Then, we write a new jump-diffusion model for actin–myosin interaction. This new model describes both the behavior of muscle contraction on multiple time scales and its compatibility with thermodynamic principles. Finally, following a classical calibration procedure, we demonstrate the ability of the model to reproduce experimental data characterizing muscle behavior on fast and slow time scales.

Is Drug Delivery System a Deterministic or Probabilistic Approach? A Theoretical Model Based on the Sequence: Electrodynamics–Diffusion–Bayes

Commonly, it is accepted that oncology treatment would yield outcomes with a certain determinism without any quantitative support or mathematical model that establishes such determinations. Nowadays, with the advent of nanomedicine, the targeting drug delivery scheme has emerged, whose central objective is the uptake of nanoparticles by tumors. Once they are injected into the bloodstream, it is unclear as to which process governs the directing of nanoparticles towards the desired target, deterministic or stochastic. In any scenario, an optimal outcome, small toxicity and minimal dispersion of drugs is expected. Commonly, it is expected that an important fraction of them can be internalized into tumor. In this manner, due to the fraction of nanoparticles that have failed to uptake, the success of the drug delivery scheme might be at risk. In this paper, a theory based on the sequence electrodynamics–diffusion–Bayes theorem is presented. The Bayesian probability that emerges at the end of the sequence might be telling us that dynamical processes based on the injection of electrically charged nanoparticles might be dictated by stochastic formalism. Thus, rather than expecting a deterministic process, the chain of events would convert the drug delivery scheme to be dependent on a sequence of conditional probabilities.

Stochastic thermodynamic engines under time-varying temperature profile

A recently developed stochastic control formalism (Stochastic Thermodynamics) has opened for the first time the possibility to quantify energy exchange and entropy production in finite-time thermodynamic transitions, based on Langevin models for mesoscopic thermodynamic systems. Within this framework we quantify power output and efficiency of overdamped stochastic thermodynamic engines that are powered by a heat bath with temperature that varies periodically with time. Our setting is in contrast to most of the existing literature that considers the Carnot paradigm, alternating contact with heat baths having different fixed temperatures, hot and cold. Specifically, we consider a periodic and bounded but otherwise arbitrary temperature profile and derive explicit bounds on the power and efficiency achievable by a suitably controlling potential that couples the thermodynamic engine to the external world – the time-varying potential represents the control input to the system. A standing assumption in our analysis is that the norm of the gradient of the potentials is bounded – in the absence of any such constraint on the control input, the physically questionable conclusion of arbitrarily large power can be drawn.

Primordial black holes and stochastic inflation beyond slow roll. Part I. Noise matrix elements

Primordial Black Holes (PBHs) may form in the early Universe, from the gravitational collapse of large density perturbations, generated by large quantum fluctuations during inflation. Since PBHs form from rare over-densities, their abundance is sensitive to the tail of the primordial probability distribution function (PDF) of the perturbations. It is therefore important to calculate the full PDF of the perturbations, which can be done non-perturbatively using the `stochastic inflation' framework. In single field inflation models generating large enough perturbations to produce an interesting abundance of PBHs requires violation of slow roll. It is therefore necessary to extend the stochastic inflation formalism beyond slow roll. A crucial ingredient for this are the stochastic noise matrix elements of the inflaton potential. We carry out analytical and numerical calculations of these matrix elements for a potential with a feature which violates slow roll and produces large, potentially PBH generating, perturbations. We find that the transition to an ultra slow-roll phase results in the momentum induced noise terms becoming larger than the field noise whilst each of them falls exponentially for a few e-folds. The noise terms then start rising with their original order restored, before approaching constant values which depend on the nature of the slow roll parameters in the post transition epoch. This will significantly impact the quantum diffusion of the coarse-grained inflaton field, and hence the PDF of the perturbations and the PBH mass fraction.

Explaining large electromagnetic logarithms from loops of inflationary gravitons

Recent progress on nonlinear sigma models on de Sitter background has permitted the resummation of large inflationary logarithms by combining a variant of Starobinsky’s stochastic formalism with a variant of the renormalization group. We reconsider single graviton loop corrections to the photon wave function, and to the Coulomb potential, in light of these developments. Neither of the two 1-loop results have a stochastic explanation, however, the flow of a curvature-dependent field strength renormalization explains their factors of ln(a). We speculate that the factor of ln(Hr) in the Coulomb potential should not be considered as a leading logarithm effect.

Stochastic thermodynamics: dissipativity, accumulativity, energy storage and entropy production.

In this paper, we develop an energy-based dynamical system model driven by a Markov input process to present a unified framework for stochastic thermodynamics predicated on a stochastic dynamical systems formalism. Specifically, using a stochastic dissipativity, losslessness and accumulativity theory, we develop a nonlinear stochastic port-Hamiltonian system model characterized by energy conservation and entropy non-conservation laws that are consistent with statistical thermodynamic principles. In particular, we show that the difference between the average stored system energy and the average supplied system energy for our stochastic thermodynamic model is a martingale with respect to the system filtration, whereas the difference between average system entropy production and the average system entropy consumption is a submartingale with respect to the system filtration. This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 2)'.

Unfinished business in a nonlinear sigma model on de Sitter background

Nonlinear sigma models on de Sitter background possess the same kind of derivative interactions as gravity, and show the same sorts of large spacetime logarithms in correlation functions and solutions to the effective field equations. It was recently demonstrated that these logarithms can be resummed by combining a variant of Starobinsky’s stochastic formalism with a variant of the renormalization group. This work considers one of these models and completes two pieces of analysis which were left unfinished: the evolution of the background at two loop order and the one loop beta function.

The solid effect of dynamic nuclear polarization in liquids.

The solid-state effect of dynamic nuclear polarization (DNP) is operative also in viscous liquids where the dipolar interaction between the electronic and nuclear spins is partially averaged. The proper way to quantify the degree of averaging, and thus calculate the efficiency of the effect, should be based on the time-correlation function of the dipolar interaction. Here we use the stochastic Liouville equation formalism to develop a general theoretical description of the solid effect in liquids. The derived expressions can be used with different dipolar correlations functions depending on the assumed motional model. At high magnetic fields, the theory predicts DNP enhancements at small offsets, far from the classical solid-effect positions that are displaced by one nuclear Larmor frequency from the electronic resonance. The predictions are in quantitative agreement with such enhancement peaks observed at 9.4 T . These non-canonical peaks are not due to thermal mixing or the cross effect but exactly follow the dispersive component of the EPR line.

Optimized attenuated interaction: Enabling stochastic Bethe-Salpeter spectra for large systems.

We develop an improved stochastic formalism for the Bethe-Salpeter equation (BSE), based on an exact separation of the effective-interaction W into two parts, W = (W - vW) + vW, where the latter is formally any translationally invariant interaction, vW(r - r'). When optimizing the fit of the exchange kernel vW to W, using a stochastic sampling W, the difference W - vW becomes quite small. Then, in the main BSE routine, this small difference is stochastically sampled. The number of stochastic samples needed for an accurate spectrum is then largely independent of system size. While the method is formally cubic in scaling, the scaling prefactor is small due to the constant number of stochastic orbitals needed for sampling W.

Finite-temperature ferromagnetic transition in coherently coupled Bose gases

A paramagnetic-ferromagnetic quantum phase transition is known to occur at zero temperature in a two-dimensional coherently coupled Bose mixture of dilute ultracold atomic gases provided the interspecies interaction strength is large enough. Here we study the fate of such a transition at finite temperature by performing numerical simulations with the stochastic (projected) Gross-Pitaevskii formalism, which includes both thermal and beyond mean-field effects. By extracting the average magnetization, the magnetic fluctuations and characteristic relaxation frequency (or critical slowing down), we identify a finite-temperature critical line for the transition. We find that the critical point shifts linearly with temperature and, in addition, the three quantities used to probe the transition exhibit a temperature power-law scaling. The scaling of the critical slowing down is found to be consistent with thermal critical exponents and is very well approximated by the square of the spin excitation gap at zero temperature.

Non-perturbative analysis for a massless minimal quantum scalar with V(ϕ) = λϕ 4/4! + βϕ 3/3! in the inflationary de Sitter spacetime

We consider a massless, minimally coupled quantum scalar field theory with an asymmetric self interaction, V(ϕ) = λϕ 4/4! + βϕ 3/3! (λ > 0) in the inflationary de Sitter spacetime. The potential is bounded from below. While the β=0 case has been much well studied, the motivation behind taking such a hybrid potential corresponds to the fact that it might generate finite negative vacuum expectation values of V(ϕ) as well of ϕ, leading to some dynamical screening of the inflationary cosmological constant Λ, at late times, with the initial conditions, 〈ϕ〉 = 0 = 〈V(ϕ)〉. In this work we first compute the vacuum expectation values of ϕ, ϕ 2 and V(ϕ), using the late time, non-perturbative and infrared effective stochastic formalism. The backreactions to the inflationary Λ are estimated. We also compute the dynamically generated mass of the scalar field using 〈ϕ 2〉. We next compute 〈ϕ 2〉 using quantum field theory with respect to the initial Bunch-Davies vacuum at one and two loops, using the Schwinger-Keldysh formalism. These results show non-perturbative secular logarithms, growing with the cosmological time. Using next a recently proposed renormalisation group inspired formalism, we attempt to find out a resummed 〈ϕ 2〉. We have been able to resum some part of the same which contains contributions only from the local self energy. The corresponding dynamically generated mass is computed. Comparison of the stochastic and the quantum field theory results shows that they differ numerically, although they have similar qualitative behaviour. Possible reasons for such quantitative mismatch is discussed. The manifestation of strong non-classical effects in the results found via both the formalisms has been emphasised.

Qubit control using quantum Zeno effect: Action principle approach

We employ the stochastic path-integral formalism and action principle for continuous quantum measurements – the Chantasri–Dressel–Jordan (CDJ) action formalism Chantasri et al., (2013; 2015) – to understand the stages in which quantum Zeno effect helps control the states of a simple quantum system. The detailed dynamics of a driven two-level system subjected to repeated measurements unravels a myriad of phases, so to say. When the detection frequency is smaller than the Rabi frequency, the oscillations slow down, eventually coming to a halt at an interesting resonance when measurements are spaced exactly by the time of transition between the two states. On the other hand, in the limit of large number of repeated measurements, the dynamics organizes itself in a rather interesting way about two hyperbolic points in phase space whose stable and unstable directions are reversed. Thus, the phase space flow occurs from one hyperbolic point to another, in different ways organized around the separatrices. We believe that the systematic treatment presented here paves the way for a better and clearer understanding of quantum Zeno effect in the context of quantum error correction.

Modeling and Simulation of Shared Electric Automated and Connected Mobility Systems with Autonomous Repositioning: Performance Evaluation and Deployment

The boom seen in artificial intelligence in recent years has led to a revolution in the automotive industry. Numerous automakers around the world, such as Tesla, Toyota, Honda, and BMW, have achieved giant strides in the development of e-autonomous vehicles. Consequently, shared electric automated vehicle mobility (SEAVM) systems, which are a crucial part of future innovative transportation solutions, have attracted significant attention from the research community, particularly from a design perspective. However, the flexibility of shared automated mobility systems may lead to a self-operating technology issue (unequal distribution of vehicles), since users in these systems can pick up and drop off electric vehicles wherever they like. With this in mind, this paper addressed the issues of autonomous repositioning and the assignment of shared autonomous electric vehicle systems to balance a system’s network and fulfill its demand. Modeling, analysis and assessment of the system’s performance were carried out using stochastic Petri nets formalism, which included determining the average time areas were empty/congested and the number of unserved consumers, and estimating the redistribution service launch moment. Furthermore, many simulation scenarios were analyzed, including repositioning and without repositioning scenarios, in order to evaluate the efficiency of the model and to show the potential of using Petri nets as a probabilistic formalism approach for the modeling of e-automated mobility systems.

Spatial correlations of dark energy from quantum fluctuations during inflation

This paper contains a detailed study of the properties of a simple model attempting to explain dark energy as originated from quantum fluctuations of a light spectator scalar field in inflation. In [1] we recently outlined how Starobinsky's stochastic formalism can be used to study the spatial correlations imprinted on dark energy by its quantum origin in this model and we studied their possible role in relieving the Hubble tension. Here we provide a more comprehensive derivation of the results in [1] and we refine some of our estimates, comparing to the approximate results obtained previously. Among the main results, we analyze the non-coincident correlators predicted by a full field theoretical treatment and their relation with those computed within the stochastic formalism. We find that in the region where stochastic theory predicts significant sub-Hubble correlators it is in disagreement with field theoretical predictions. However, agreement can be restored by introducing a reduced speed of sound for the scalar field. We also discuss an alternative approach to the problem of studying correlators within the stochastic formalism based directly on the evolution of probability distributions. We find that the two approaches give the same answer for 2-point functions of the field, but not for 4-point functions relevant to density correlators and we discuss the behaviour of the two methods with respect to Wick's theorem.

Stochastic formalism for U(1) gauge fields in axion inflation

We develop the stochastic formalism for U(1) gauge fields that has the Chern-Simons coupling to a rolling pseudo-scalar field during inflation.The Langevin equations for the physical electromagnetic fields are derived and the analytic solutions are studied. Using numerical simulation we demonstrate that the electromagnetic fields averaged over the Hubble scale continuously change their direction and their amplitudes fluctuate around the analytically obtained expectation values.Though the isotropy is spontaneously broken by picking up a particular local Hubble patch, each Hubble patch is understood independent and the isotropy is conserved globally byaveraging all the Hubble patches.

DNS Cache Poisoning/Honeypot Analysis Based on Data Exfiltration Using Stochastic Petri Nets Method to Enhance Cyber Security Hygiene

According to some experts, organizations make investments of huge sum of dollars concerning firewalls, encryption, and other tightly closed access devices, yet it is all naught, considering that none regarding these solutions tackle the weakest link within the safety chain. That speech perfectly captures the present day stress experienced by skilled network protection specialists. Researchers’ threat detection techniques can pick up a lot of false positive alarms throughout the detection process; false positive alarms are accountable for company’s unneeded system lock down. A stochastic model is necessary to represent a communication system since the nature of the traffic between them is unpredictable. SPN was utilized in this study to build statistical model for networks with security chains. By permitting the formalization of both real-time and non-Markovian behavior, the new stochastic Petri net formalism improves model fidelity. This allowed us to see special structures within the stochastic processes produced by SPN models. We have applied this principle by proposing an effective simulation method that supports deadlock detection and easy-to-compute point estimates and confidence intervals. The method is novel because it can automatically detect hidden regenerative structures that do not conform to different simple conditions, and can be easily determined by analytical methods.