Reversible Diffusion Processes Research Articles

Stability of the Poincaré constant

We study stability of the sharp Poincaré constant of the invariant probability measure of a reversible diffusion process satisfying some natural conditions. The proof is based on the spectral interpretation of Poincaré inequalities and Stein’s method. In particular, these results are applied to gamma distributions, to the Brownian motion on spheres and to the Brascamp-Lieb inequality for one-dimensional log-concave measures.

How Much Is Enough? A Study on Diffusion Times in Score-Based Generative Models

Score-based diffusion models are a class of generative models whose dynamics is described by stochastic differential equations that map noise into data. While recent works have started to lay down a theoretical foundation for these models, a detailed understanding of the role of the diffusion time T is still lacking. Current best practice advocates for a large T to ensure that the forward dynamics brings the diffusion sufficiently close to a known and simple noise distribution; however, a smaller value of T should be preferred for a better approximation of the score-matching objective and higher computational efficiency. Starting from a variational interpretation of diffusion models, in this work we quantify this trade-off and suggest a new method to improve quality and efficiency of both training and sampling, by adopting smaller diffusion times. Indeed, we show how an auxiliary model can be used to bridge the gap between the ideal and the simulated forward dynamics, followed by a standard reverse diffusion process. Empirical results support our analysis; for image data, our method is competitive with regard to the state of the art, according to standard sample quality metrics and log-likelihood.

Dif-Fusion: Towards High Color Fidelity in Infrared and Visible Image Fusion with Diffusion Models.

Color plays an important role in human visual perception, reflecting the spectrum of objects. However, the existing infrared and visible image fusion methods rarely explore how to handle multi-spectral/channel data directly and achieve high color fidelity. This paper addresses the above issue by proposing a novel method with diffusion models, termed as Dif-Fusion, to generate the distribution of the multi-channel input data, which increases the ability of multi-source information aggregation and the fidelity of colors. In specific, instead of converting multi-channel images into single-channel data in existing fusion methods, we create the multi-channel data distribution with a denoising network in a latent space with forward and reverse diffusion process. Then, we use the the denoising network to extract the multi-channel diffusion features with both visible and infrared information. Finally, we feed the multi-channel diffusion features to the multi-channel fusion module to directly generate the three-channel fused image. To retain the texture and intensity information, we propose multi-channel gradient loss and intensity loss. Along with the current evaluation metrics for measuring texture and intensity fidelity, we introduce Delta E as a new evaluation metric to quantify color fidelity. Extensive experiments indicate that our method is more effective than other state-of-the-art image fusion methods, especially in color fidelity. The source code is available at https://github.com/GeoVectorMatrix/Dif-Fusion.

Adaptive invariant density estimation for continuous-time mixing Markov processes under sup-norm risk

Up to now, the nonparametric analysis of multidimensional continuous-time Markov processes has focussed strongly on specific model choices, mostly related to symmetry of the semigroup. While this approach allows to study the performance of estimators for the characteristics of the process in the minimax sense, it restricts the applicability of results to a rather constrained set of stochastic processes and in particular hardly allows incorporating jump structures. As a consequence, for many models of applied and theoretical interest, no statement can be made about the robustness of typical statistical procedures beyond the beautiful, but limited framework available in the literature. To close this gap, we identify $\beta$-mixing of the process and heat kernel bounds on the transition density as a suitable combination to obtain $\sup$-norm and $L^2$ kernel invariant density estimation rates matching the case of reversible multidimenisonal diffusion processes and outperforming density estimation based on discrete i.i.d. or weakly dependent data. Moreover, we demonstrate how up to $\log$-terms, optimal $\sup$-norm adaptive invariant density estimation can be achieved within our general framework based on tight uniform moment bounds and deviation inequalities for empirical processes associated to additive functionals of Markov processes. The underlying assumptions are verifiable with classical tools from stability theory of continuous time Markov processes and PDE techniques, which opens the door to evaluate statistical performance for a vast amount of Markov models. We highlight this point by showing how multidimensional jump SDEs with L\'evy driven jump part under different coefficient assumptions can be seamlessly integrated into our framework, thus establishing novel adaptive $\sup$-norm estimation rates for this class of processes.

Stability of higher order eigenvalues in dimension one

We study stability of the eigenvalues of the generator of a one dimensional reversible diffusion process satisfying some natural conditions. The proof is based on Stein’s method. In particular, these results are applied to the Normal distribution (via the Ornstein–Uhlenbeck process), to Gamma distributions (via the Laguerre process) and to Beta distributions (via the Jacobi process).

Variational Principles for Asymptotic Variance of General Markov Processes

A variational formula for the asymptotic variance of general Markov processes is obtained. As application, we get an upper bound of the mean exit time of reversible Markov processes, and some comparison theorems between the reversible and non-reversible diffusion processes.

Non-reversible Metastable Diffusions with Gibbs Invariant Measure II: Markov Chain Convergence

This article considers a class of metastable non-reversible diffusion processes whose invariant measure is a Gibbs measure associated with a Morse potential. In a companion paper (Lee and Seo in Probab Theory Relat Fields 182:849–903, 2022), we proved the Eyring–Kramers formula for the corresponding class of metastable diffusion processes. In this article, we further develop this result by proving that a suitably time-rescaled metastable diffusion process converges to a Markov chain on the deepest metastable valleys. This article is also an extension of (Rezakhanlou and Seo in https://arxiv.org/abs/1812.02069, 2018), which considered the same problem for metastable reversible diffusion processes. Our proof is based on the recently developed resolvent approach to metastability.

One Dimensional Critical Kinetic Fokker–Planck Equations, Bessel and Stable Processes

We consider a particle moving in one dimension, its velocity being a reversible diffusion process, with constant diffusion coefficient, of which the invariant measure behaves like $$(1+|v|)^{-\beta }$$ for some $$\beta >0$$ . We prove that, under a suitable rescaling, the position process resembles a Brownian motion if $$\beta \ge 5$$ , a stable process if $$\beta \in [1,5)$$ and an integrated symmetric Bessel process if $$\beta \in (0,1)$$ . The critical cases $$\beta =1$$ and $$\beta =5$$ require special rescalings. We recover some results of Nasreddine and Puel (ESAIM Math Model Numer Anal 49:1–17, 2015), Cattiaux et al. (Kinet Relat Models, to appear), Lebeau and Puel (Commun Math Phys, to appear. arXiv:1711.03060 ) and Barkai et al. (Phys Rev X 4:021036, 2014) on the kinetic Fokker–Planck equation, with an alternative approach.

Spectral thresholding for the estimation of Markov chain transition operators

We consider nonparametric estimation of the transition operator $P$ of a Markov chain and its transition density $p$ where the singular values of $P$ are assumed to decay exponentially fast. This is for instance the case for periodised, reversible multi-dimensional diffusion processes observed in low frequency. We investigate the performance of a spectral hard thresholded Galerkin-type estimator for $P$ and ${p}$, discarding most of the estimated singular triples. The construction is based on smooth basis functions such as wavelets or B-splines. We show its statistical optimality by establishing matching minimax upper and lower bounds in $L^2$-loss. Particularly, the effect of the dimensionality $d$ of the state space on the nonparametric rate improves from $2d$ to $d$ compared to the case without singular value decay.

Reversible Al Propagation in Si x Ge1-x Nanowires: Implications for Electrical Contact Formation.

While reversibility is a fundamental concept in thermodynamics, most reactions are not readily reversible, especially in solid-state physics. For example, thermal diffusion is a widely known concept, used among others to inject dopants into the substitutional positions in the matrix and improve device properties. Typically, such a diffusion process will create a concentration gradient extending over increasingly large regions, without possibility to reverse this effect. On the other hand, while the bottom-up growth of semiconducting nanowires is interesting, it can still be difficult to fabricate axial heterostructures with high control. In this paper, we report a thermally assisted partially reversible thermal diffusion process occurring in the solid-state reaction between an Al metal pad and a SixGe1–x alloy nanowire observed by in situ transmission electron microscopy. The thermally assisted reaction results in the creation of a Si-rich region sandwiched between the reacted Al and unreacted SixGe1–x part, forming an axial Al/Si/SixGe1–x heterostructure. Upon heating or (slow) cooling, the Al metal can repeatably move in and out of the SixGe1–x alloy nanowire while maintaining the rodlike geometry and crystallinity, allowing to fabricate and contact nanowire heterostructures in a reversible way in a single process step, compatible with current Si-based technology. This interesting system is promising for various applications, such as phase change memories in an all crystalline system with integrated contacts as well as Si/SixGe1–x/Si heterostructures for near-infrared sensing applications.

Anomalous diffusion for multi-dimensional critical kinetic Fokker–Planck equations

We consider a particle moving in $d\geq2$ dimensions, its velocity being a reversible diffusion process, with identity diffusion coefficient, of which the invariant measure behaves, roughly, like $(1+|v|)^{-\beta}$ as $|v|\to\infty$, for some constant $\beta>0$. We prove that for large times, after a suitable rescaling, the position process resembles a Brownian motion if $\beta\geq4+d$, a stable process if $\beta\in[d,4+d)$ and an integrated multi-dimensional generalization of a Bessel process if $\beta\in(d-2,d)$. The critical cases $\beta=d$, $\beta=1+d$ and $\beta=4+d$ require special rescalings.

The influence of the pack decarburizing process with Pinctada maxima shell powder agent on the properties of high carbon steel

In the present study, ductility enhancement of high carbon steel AISI 420 was conducted by pack decarburizing method to improve mechanical properties of this steel. This specimen was placed in a rectangular box containing pinctada maxima shell powder (PMSP) mixed with the carburizing agent with different percentage variations and heat treated in an oxygen atmosphere at different temperatures and soaking times. Phase analysis results indicated that the pack decarburizing process at a temperature of 900 °C, for soaking time 3 hours and an additional 30 % PMSP in the carburizing agent causing the martensit microstructure, the surface hardness number and thickness of carbon layer decreased but the impact energy of high carbon steel AISI 420 increased. The surface hardness number, carbon layer thickness each respectively decreased by 63 % and 60 %, but impact energy or impact strength increased by 33 %. This phenomenon indicates that the pack decarburizing treatment causes carbon diffusion from the surface of the specimens to the carburizing agent or reverse carbon diffusion occurs, because the concentration of carbon in the carburizing agent is higher than the surface of the specimen. The addition of PMSP in the carburizing agent increases the occurrence of carbon diffusion from the surface of specimens to the carburizing agent or reverse carbon diffusion occurs, because differences in concentration and influence of PMSP contains elements of Ca which function as catalysts or energizers. The results showthat the pack decarburizing process with an additional PMSP in the carburizing agent accelerates the diffusion of carbon atoms out the surface of the specimens (reverse carbon diffusion process), thus decreasing the thickness of the surface carbon layer, surface hardness number and increasing the impact energy

In-situ characterization of moisture absorption and hygroscopic swelling of silicone/phosphor composite film and epoxy mold compound in LED packaging

In this paper, moisture absorption/desorption and hygroscopic swelling of silicone/phosphor composite film and epoxy mold compound (EMC) used in chip-on-board (COB) LED packaging is studied with in-situ characterization. Material properties, including diffusivity, activation energy, saturated moisture concentration and the coefficient of hygroscopic swelling, are extracted from the experimental data. For silicone/phosphor film, the moisture diffusion can be described by the conventional Fickian diffusion model. However, the saturated moisture concentration increases with temperatures under the same relative humidity condition. The dependence of saturated moisture concentration on temperature is characterized by an Arrhenius-like equation, and the net absorption heat is found to be 8–11 kJ/mol. Furthermore, the silicone/phosphor composite film shows reversible diffusion process during absorption and desorption cycle, which confirms the Fickian behavior of moisture absorption. Moisture diffusion in EMC, on the other hand, exhibits non-Fickian diffusion, and thus, a dual stage diffusion model is adopted to fit experimental data. Moreover, EMC shows partially non-reversible moisture absorption during absorption and desorption cycle, which is consistent with the dual-stage model description. In hygroscopic swelling testing, silicone/phosphor film shows a permanent swelling during moisture absorption despite that the moisture absorption is completely reversible. EMC shows partially reversible swelling after moisture is removed, which indicates the residual moisture in material. Further study is needed to understand the mechanism of permanent hygroscopic swelling for silicone/phosphor composite during moisture absorption and desorption.

Generalisation of the Eyring–Kramers Transition Rate Formula to Irreversible Diffusion Processes

In the small noise regime, the average transition time between metastable states of a reversible diffusion process is described at the logarithmic scale by Arrhenius' law. The Eyring-Kramers formula classically provides a subexponential prefactor to this large deviation estimate. For irreversible diffusion processes, the equivalent of Arrhenius' law is given by the Freidlin-Wentzell theory. In this paper, we compute the associated prefactor and thereby generalise the Eyring-Kramers formula to irreversible diffusion processes. In our formula, the role of the potential is played by Freidlin-Wentzell's quasipotential, and a correction depending on the non-Gibbsianness of the system along the instanton is highlighted. Our analysis relies on a WKB analysis of the quasistationary distribution of the process in metastable regions, and on a probabilistic study of the process in the neighbourhood of saddle-points of the quasipotential.

A gradient flow approach to large deviations for diffusion processes

In this work, we investigate links between the formulation of the flow of marginals of reversible diffusion processes as gradient flows in the space of probability measures and path wise large deviation principles for sequences of such processes. An equivalence between the LDP principle and Gamma-convergence for a sequence of functionals appearing in the gradient flow formulation is proved. As an application, we study large deviations from the hydrodynamic limit for two variants of the Ginzburg–Landau model endowed with Kawasaki dynamics.

Characterization of Silicon Accumulation in Maize Cell Suspension Cultures

Silicon (Si) is an abundant element in the earth’s crust and is available to plants as silicic acid. Silicon uptake by plants is correlated with increased tolerance to various biotic and abiotic stresses. However, cellular mechanisms responsible for its beneficial effects are still unknown. Even its cellular import mechanisms are not well understood. We thus aimed to characterize silicon localization within minimally differentiated Zea mays (Black Mexican Sweet) cells in suspension. Cells were grown in a medium containing silicon, and the mRNA levels of silicon transporters were measured by real-time PCR. Cells were separated into an insoluble (mainly walls and starch) and a cytoplasmic fraction. Soluble and total silicon was measured by inductively-coupled-plasma – atomic-emission-spectroscopy. Silicon distribution was assessed by transmission electron microscopy. The cell walls were analyzed chemically, and by Raman micro-spectroscopy and thermal gravimetric analysis. Silicon treatment reduced the levels of silicon transporters transcripts, without affecting cell proliferation. About 70 % of the silicon was localized in the cytoplasm, mostly in vesicles. We found indications that silicon affected the secondary structure of proteins and thermally stabilized starch. Silicon was loosely bound, and diffused out of the cells within 24 hours. Our results show that silicon binds spontaneously to cell walls/starch and accumulates in cytoplasm vesicles. These processes allow the cells to accumulate silicon against its concentration gradient in solution. However, cellular intake acts against reversible diffusion processes, probably through the aquaporin silicon channels (Lsi1, Lsi6) that exchange the cellular silicon with the surrounding medium.

Electrochemical Redox of Benzoquinone in Ionic Liquids

The reduction of 1,4-benzoquinone(BQ) on Pt electrode was investigated in various ionic liquids by cyclic voltammetry(CV). Over a wide range of scan rates,two couples of redox peaks were clearly observed in cyclic voltammogram,showing a quasi reversible diffusion process of BQ with the two-step electrochemical reduction. Diffusion coefficient and activation energy were obtained from cyclic voltammogram, which displayed the increase of diffusion coefficient accompanying temperature rise from 298. 15 K to 343. 15 K. The techniques of cyclic voltammetry,in situ FTIR spectroelectrochemistry(FTIR),cyclic voltabsorptometry(CVA) and derivative cyclic voltabsorptometry(DCVA) were used to investigate the electrochemical behavior of BQ in ionic liquids,BMIMBF4and BMIMPF6. The results support that BQ undergoes two successive one-electron reductions,corresponding to the formation of a radical anion intermediate and then the dianion,in ionic liquids,while presenting a different shape of cyclic voltammogram compared to anhydrous acetonitrile solvent.

A decomposition of irreversible diffusion processes without detailed balance

As a generalization of deterministic, nonlinear conservative dynamical systems, a notion of canonical conservative dynamics with respect to a positive, differentiable stationary density ρ(x) is introduced: \documentclass[12pt]{minimal}\begin{document}$\dot{x}=j(x)$\end{document}ẋ=j(x) in which ∇·(ρ(x)j(x)) = 0. Such systems have a conserved “generalized free energy function” F[u] = ∫u(x, t)ln (u(x, t)/ρ(x))dx in phase space with a density flow u(x, t) satisfying ∂ut = −∇·(ju). Any general stochastic diffusion process without detailed balance, in terms of its Fokker-Planck equation, can be decomposed into a reversible diffusion process with detailed balance and a canonical conservative dynamics. This decomposition can be rigorously established in a function space with inner product defined as ⟨ϕ, ψ⟩ = ∫ρ−1(x)ϕ(x)ψ(x)dx. Furthermore, a law for balancing F[u] can be obtained: The non-positive dF[u(x, t)]/dt = Ein(t) − ep(t) where the “source” Ein(t) ⩾ 0 and the “sink” ep(t) ⩾ 0 are known as house-keeping heat and entropy production, respectively. A reversible diffusion has Ein(t) = 0. For a linear (Ornstein-Uhlenbeck) diffusion process, our decomposition is equivalent to the previous approaches developed by Graham and Ao, as well as the theory of large deviations. In terms of two different formulations of time reversal for a same stochastic process, the meanings of dissipative and conservative stationary dynamics are discussed.

First-principles investigation of diffusion behaviours of H isotopes: From W(110) surface into bulk and in bulk W

The diffusion behaviours of hydrogen (H), deuterium (D), and tritium (T) from W(110) surface into bulk and in bulk W are investigated using first-principles calculations combined with simplified models. The diffusion energy barrier is shown to be 1.87 eV from W(110) surface to the subsurface, along with a much reduced barrier of 0.06 eV for the reverse diffusion process. After H enters into the bulk, its diffusion energy barrier with quantum correction is 0.19 eV. In terms of the diffusion theory presented by Wert and Zener, the diffusion pre-exponential factor of H is calculated to be 1.57×10−7 m2·s−1, and it is quantitatively in agreement with the experimental value of 4.1×10−7 m2·s−1. Subsequently, according to mass dependence () of H isotope effect, the diffusion pre-exponential factors of D and T are estimated to be 1.11×10−7 m2·s−1 and 0.91×10−7 m2·s−1, respectively.

Modeling epistemic subsurface reservoir uncertainty using a reverse Wiener jump–diffusion process

This paper develops a mathematical model of how subsurface reservoir uncertainty, in particular estimated ultimate recovery (EUR), can evolve over time. The model assumes that epistemic uncertainty reduces over time as information from seismic surveys, appraisal wells and production logs is used to improve EUR estimates. A reverse Wiener diffusion process with superimposed jumps is developed to capture the exponential decrease in estimate volatility due to learning but also the existence of sudden jumps in estimates due to unexpected discoveries such as reservoir fault lines or aquifer support. The model can be applied to quantify the evolution of reservoir uncertainty over time during appraisal and planning of new oil and gas development projects. Appreciation Factor data from 34 North Sea fields is used to calibrate and validate the model showing that the evolution of EUR estimates is predicted with 82.4% of validation data points within the simulated P10 and P90 uncertainty envelope, which should theoretically cover 80% of data points if there is no model error. The key parameters in the model are the initial EUR distribution, as well as the exponential decay rates for EUR volatility and the likelihood of occurrence of discrete jumps.