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The filtration of stationary Gaussian statistical experiments is determined by a solution of the equation of optimum filtration, which is characterized by the two-dimensional matrix of covariances. The parameters of a filtered signal are set by empiric covariances.
Since data for statistical analysis are always given in a discretized form, observations contain not only measurement errors but also rounding errors which are determined by the discretization step. In this paper we consider situations where the rounding errors are considerable: they are comparable to or even greater (in average) than the measurement errors. It is shown that it can be reasonable to increase the measurement errors in order to reduce the error of the final result.
We study theoretically the full counting statistics of electron transport through side-coupled double quantum dot (QD) based on an efficient particle-number-resolved master equation. It is demonstrated that the high-order cumulants of transport current are more sensitive to the quantum coherence than the average current, which can be used to probe the quantum coherence of the considered double QD system. Especially, quantum coherence plays a crucial role in determining whether the super-Poissonian noise occurs in the weak inter-dot hopping coupling regime depending on the corresponding QD-lead coupling, and the corresponding values of super-Poissonian noise can be relatively enhanced when considering the spins of conduction electrons. Moreover, this super-Poissonian noise bias range depends on the singly-occupied eigenstates of the system, which thus suggests a tunable super-Poissonian noise device. The occurrence-mechanism of super-Poissonian noise can be understood in terms of the interplay of quantum coherence and effective competition between fast-and-slow transport channels. -- Highlights: •The FCS can be used to probe the quantum coherence of side-coupled double QD system. •Probing quantum coherence using FCS may permit experimental tests in the near future. •The current noise characteristics depend on the quantum coherence of this QD system. •The super-Poissonian noise can be enhanced when considering conduction electron spin. •The side-coupled double QD system suggests a tunable super-Poissonian noise device.
We study the full counting statistics of transport electrons through a semiconductor two-level quantum dot with Rashba spin–orbit (SO) coupling, which acts as a nonabelian gauge field and thus induces the electron transition between two levels along with the spin flip. By means of the quantum master equation approach, shot noise and skewness are obtained at finite temperature with two-body Coulomb interaction. We particularly demonstrate the crucial effect of SO coupling on the super-Poissonian fluctuation of transport electrons, in terms of which the SO coupling can be probed by the zero-frequency cumulants. While the charge currents are not sensitive to the SO coupling.
We study theoretically the full counting statistics of electron transport through a quantum dot weakly coupled to two ferromagnetic leads, in which an effective nuclear-spin magnetic field originating from the configuration of nuclear spins is considered. We demonstrate that the quantum coherence between the two singly-occupied eigenstates and the spin polarization of two ferromagnetic leads play an important role in the formation of super-Poissonian noise. In particular, the orientation and magnitude of the effective field have a significant influence on the variations of the values of high-order cumulants, and the variations of the skewness and kurtosis values are more sensitive to the orientation and magnitude of the effective field than the shot noise. Thus, the high-order cumulants of transport current can be used to qualitatively extract information on the orientation and magnitude of the effective nuclear-spin magnetic field in a single quantum dot. - Highlights: • The effective nuclear-spin magnetic field gives rise to the off-diagonal elements of the reduced density matrix of single QD. • The off-diagonal elements of reduced density matrix of the QD have a significant impact on the high-order current cumulants. • The high-order current cumulants are sensitive to the orientation and magnitude of the effective nuclear-spin magnetic field. • The FCS can be used to detect the orientation and magnitude of the effective nuclear-spin magnetic field in a single QD.
The role of quantum statistics in the decay dynamics of a multi-particle state, which is suddenly released from a confining potential, is investigated. For an initially confined double particle state, the exact dynamics is presented for both bosons and fermions. The time-evolution of the probability to measure two-particle is evaluated and some counterintuitive features are discussed. For instance, it is shown that although there is a higher chance of finding the two bosons (as oppose to fermions, and even distinguishable particles) at the initial trap region, there is a higher chance (higher than fermions) of finding them on two opposite sides of the trap as if the repulsion between bosons is higher than the repulsion between fermions. The results are demonstrated by numerical simulations and are calculated analytically in the short-time approximation. Furthermore, experimental validation is suggested.
We analyze the photon statistics of a weakly driven cavity quantum electrodynamics system and discuss the effects of photon blockade and photon-induced tunneling by effectively utilizing instead of avoiding the center-of-mass motion of a two-level atom trapped in the cavity. With the resonant interaction between atom, photon and phonon, it is shown that the bunching and anti-bunching of photons can occur with properly driving frequency. Our study shows the influence of the imperfect cooling of atom on the blockade and provides an attempt to take advantage of the center-of-mass motion.
Highlights: •The quantum coherence dependence of FCS depends on the SOC-tuned hopping strength. •The spin polarization rates determine the formation of super-Poissonian shot noise. •The spin polarization rates and magnitude of magnetic field can induce a strong NDC. •The obvious energy-level detuning dependence of FCS can reflect the magnitude of SOC. •The dip position of the skewness can be used to qualitatively extract SOC strength. -- Abstract: We study the full counting statistics of electron transport through a serially coupled double quantum dot (QD) system with spin–orbit coupling (SOC) weakly coupled to two electrodes. We demonstrate that the spin polarizations of the source and drain electrodes determine whether the shot noise maintains super-Poissonian distribution, and whether the sign transitions of the skewness from positive to negative values and of the kurtosis from negative to positive values take place. In particular, the interplay between the spin polarizations of the source and drain electrodes and the magnitude of the external magnetic field, can give rise to a gate-voltage-tunable strong negative differential conductance (NDC) and the shot noise in this NDC region is significantly enhanced. Importantly, for a given SOC parameter, the obvious variation of the high-order current cumulants as a function of the energy-level detuning in a certain range, especially the dip position of the Fano factor of the skewness can be used to qualitatively extract the information about the magnitude of the SOC.
Highlights: • Standard deviation of stresses in C-S-H has the order of magnitude of the stresses applied at concrete scale. • Fluctuations increase with the stiffness and contrast of phases. • Fluctuations are dependent of mechanical load. • Transformation fields also cause mechanical fluctuations. - Abstract: Analytical homogenization provides information about the average mechanical fields but can also be used to obtain information about the statistical variability or fluctuations of these fields. Here, we present analytical estimations of the fluctuations of elastic stresses within cement-based materials (CBM). Mori-Tanaka, self-consistent and generalized self-consistent schemes are combined to represent the multiscale character of CBM. The second moment of mechanical fields is derived from the internal energy. The presence of transformation fields and resulting fluctuations are considered. We account for the evolution of the volume fractions of phases due to cement hydration. Scenarios considering or not the ITZ are studied. Our estimations are dependent on load and representation CBM microstructure. The obtained standard deviation of stresses in C-S-H are on the order of the stresses applied at concrete scale. We provide estimations of the compressive strength of a cement paste at early-age using our results of fluctuation of stresses.
First-principles statistical mechanics enables the prediction of thermodynamic and kinetic properties of materials, but is computationally expensive. Many approaches require surrogate models to calculate energies within Monte Carlo or molecular dynamics simulations. Inexpensive surrogates such as cluster expansions enable otherwise intractable calculations by interpolating data from higher accuracy methods, such as Density Functional Theory (DFT). Surrogate models introduce uncertainty into downstream calculations, in addition to any uncertainty inherent to DFT calculations. Bayesian frameworks address this by quantifying uncertainty and incorporating expert knowledge through priors. However, constructing effective priors remains challenging. This work introduces and describes practical strategies for building Bayesian cluster expansions, focusing on basis truncation, hyperparameter selection, and ground state replication. We analyze multiple basis truncation schemes, compare cross-validation to the evidence-approximation for hyperparameter optimization, and provide methods to find and enforce ground-state-preserving models through priors. Additionally, we compare the uncertainties between different approximations to DFT (LDA, PBE, SCAN) against the uncertainty introduced with the use of cluster expansion surrogate models. These approaches are demonstrated on the BCC Li x Mg 1-x and Li x Al 1-x alloys, which are both of interest for solid-state Li batteries. Our results provide guidelines for constructing and utilizing Bayesian cluster expansions, thereby improving the transparency of materials modeling. Furthermore, the approaches and insights developed in this work can be transferred to a wide range of cluster expansion surrogate models, including the atomic cluster expansion and related machine-learned interatomic potential architectures.
Purpose: To derive and validate a statistical model of motion and deformation for the clinical target volume (CTV) of early-stage rectal cancer patients. Methods and Materials: For 16 patients, 4 to 5 magnetic resonance images (MRI) were acquired before each fraction was administered. The CTV was delineated on each MRI. Using a leave-one-out methodology, we constructed a population-based principal component analysis (PCA) model of the CTV motion and deformation of 15 patients, and we tested the model on the left-out patient. The modeling error was calculated as the amount of the CTV motion-deformation of the left-out-patient that could not be explained by the PCA model. Next, the PCA model was used to construct a PCA target volume (PCA-TV) by accumulating motion-deformations simulated by the model. A PCA planning target volume (PTV) was generated by expanding the PCA-TV by uniform margins. The PCA-PTV was compared with uniform and nonuniform CTV-to-PTV margins. To allow comparison, geometric margins were determined to ensure adequate coverage, and the volume difference between the PTV and the daily CTV (CTV-to-PTV volume) was calculated. Results: The modeling error ranged from 0.9 ± 0.5 to 2.9 ± 2.1 mm, corresponding to a reduction of the CTV motion-deformation between 6% and 60% (average, 23% ± 11%). The reduction correlated with the magnitude of the CTV motion-deformation (P<.001, R=0.66). The PCA-TV and the CTV required 2-mm and 7-mm uniform margins, respectively. The nonuniform CTV-to-PTV margins were 4 mm in the left, right, inferior, superior, and posterior directions and 8 mm in the anterior direction. Compared to uniform and nonuniform CTV-to-PTV margins, the PCA-based PTV significantly decreased (P<.001) the average CTV-to-PTV volume by 128 ± 20 mL (49% ± 4%) and by 35 ± 6 mL (20% ± 3.5%), respectively. Conclusions: The CTV motion-deformation of a new patient can be explained by a population-based PCA model. A PCA model-generated PTV significantly improved sparing of organs at risk compared to uniform and nonuniform CTV-to-PTV margins.
Adaptive multilevel splitting algorithms have been introduced rather recently for estimating tail distributions in a fast and efficient way. In particular, they can be used for computing the so-called reactive trajectories corresponding to direct transitions from one metastable state to another. The algorithm is based on successive selection–mutation steps performed on the system in a controlled way. It has two intrinsic parameters, the number of particles/trajectories and the reaction coordinate used for discriminating good or bad trajectories. We investigate first the convergence in law of the algorithm as a function of the timestep for several simple stochastic models. Second, we consider the average duration of reactive trajectories for which no theoretical predictions exist. The most important aspect of this work concerns some systems with two degrees of freedom. They are studied in detail as a function of the reaction coordinate in the asymptotic regime where the number of trajectories goes to infinity. We show that during phase transitions, the statistics of the algorithm deviate significatively from known theoretical results when using non-optimal reaction coordinates. In this case, the variance of the algorithm is peaking at the transition and the convergence of the algorithm can be much slower than the usual expected central limit behaviour. The duration of trajectories is affected as well. Moreover, reactive trajectories do not correspond to the most probable ones. Such behaviour disappears when using the optimal reaction coordinate called committor as predicted by the theory. We finally investigate a three-state Markov chain which reproduces this phenomenon and show logarithmic convergence of the trajectory durations.
We present an efficient numerical algorithm to approximate the statistical moments of stochastic problems, in the presence of models with different fidelities. The method extends the multi-fidelity approximation method developed in . By combining the efficiency of low-fidelity models and the accuracy of high-fidelity models, our method exhibits fast convergence with a limited number of high-fidelity simulations. We establish an error bound of the method and present several numerical examples to demonstrate the efficiency and applicability of the multi-fidelity algorithm.
To design increasingly tough, resilient, and fatigue-resistant elastomers and hydrogels, the relationship between controllable network parameters at the molecular level (bond type, non-uniform chain length, entanglement density, etc.) to macroscopic quantities that govern damage and failure must be established. Many of the most successful constitutive models for elastomers have been rooted in statistical mechanical treatments of polymer chains. Typically, such constitutive models have used variants of the freely jointed chain model with rigid links. However, since the free energy state of a polymer chain is dominated by enthalpic bond distortion effects as the chain approaches its rupture point, bond extensibility ought to be accounted for if the model is intended to capture chain rupture. To that end, a new bond potential is supplemented to the freely jointed chain model (as derived in the u FJC framework of Buche and Silberstein (2021) and Buche et al. (2022)), which we have extended to yield a tractable, closed-form model of single chain behavior that should be amenable to continuum-level constitutive model development. Inspired by the asymptotically matched u FJC model response in both the low/intermediate chain force and high chain force regimes, a simple, quasi-polynomial bond potential energy function is derived. This bond potential exhibits harmonic behavior near the equilibrium state and anharmonic behavior for large bond stretches tending to a characteristic energy plateau (akin to the Lennard-Jones and Morse bond potentials). Using this bond potential, approximate yet highly-accurate analytical functions for bond stretch and chain force dependent upon chain stretch are established. Then, using this polymer chain model, a stochastic thermal fluctuation-driven chain rupture framework is developed. This framework is based upon a force-modified tilted bond potential that accounts for distortional bond potential energy, allowing for the derivation and subsequent calculation of the dissipated chain scission energy. Here, the cases of rate-dependent and rate-independent scission are accounted for throughout the rupture framework. The impact of Kuhn segment number on chain rupture behavior is also investigated. The model is fit to single chain mechanical response data collected from atomic force microscopy tensile tests for validation and to glean deeper insight into the molecular physics taking place. Due to their analytical nature, this polymer chain model and the associated rupture framework can in the future be implemented in finite element models accounting for fracture and fatigue in polydisperse elastomer networks.
Dislocations are line defects in crystals that multiply and self-organize into a complex network during strain hardening. The length of dislocation links, connecting neighboring nodes within this network, contains crucial information about the evolving dislocation microstructure. By analyzing data from Discrete Dislocation Dynamics (DDD) simulations in face-centered cubic (fcc) Cu, we characterize the statistical distribution of link lengths of dislocation networks during strain hardening on individual slip systems. Here, our analysis reveals that link lengths on active slip systems follow a double-exponential distribution, while those on inactive slip systems conform to a single-exponential distribution. The distinctive long tail observed in the double-exponential distribution is attributed to the stress-induced bowing out of long links on active slip systems, a feature that disappears upon removal of the applied stress. We further demonstrate that both observed link length distributions can be explained by extending a one-dimensional Poisson process to include different growth functions. Specifically, the double-exponential distribution emerges when the growth rate for links exceeding a critical length becomes super-linear, which aligns with the physical phenomenon of long links bowing out under stress. This work advances our understanding of dislocation microstructure evolution during strain hardening and elucidates the underlying physical mechanisms governing its formation.
Metal oxide dopants, such as titanium and chromium oxides, have garnered considerable attention for their potential to increase grain size (≥ 30 µm) in UO 2 fuel, purportedly enhancing fission gas retention during reactor operation. Fuel performance is significantly impacted by fuel fracture behavior, so it is important to understand the effects of enhanced grain size and dopant content on UO 2 fuel fracture. UO 2 pellets were doped with 0.1 wt% TiO 2 and 0.3 wt% Cr 2 O 3 to alter density and grain size. Inductively coupled plasma mass spectroscopy measured dopant levels pre- and post-sintering. X-ray diffraction revealed lattice changes and microstrain via Rietveld refinement. Field emission scanning electron microscopy determined grain sizes of approximately 30 µm for TiO 2 doping and 7 µm for Cr 2 O 3 doping. Transverse rupture strength tests were performed on over 30 samples per dataset to obtain characteristic strength and Weibull modulus. Results indicate no statistical difference in fracture strength between 0.1 wt% TiO 2 doped UO 2 and undoped UO 2 , while 0.3 wt% Cr 2 O 3 doped UO 2 exhibited a 20% decrease in fracture strength. Doped UO 2 samples also showed reduced Weibull modulus compared to undoped UO 2 , suggesting increased scatter in fracture strength. This study's findings suggest that titanium and chromium oxide doping in UO 2 , regardless of grain size, induce residual stresses, decreasing fracture strength and increasing variability in fracture behavior.
Li 1.03 (Ni 0.5 Mn 0.3 Co 0.2 ) 0.97 O 2 (NMC)-based coin cells containing the electrolyte additives vinylene carbonate (VC) and tris(trimethylsilyl)phosphite (TMSPi) in the range of 0-2 wt% were cycled between 3.0 and 4.4 V. The changes in capacity at rates of C/10 and C/1 and resistance at 60% state of charge were found to follow linear-with-time kinetic rate laws. Further, the C/10 capacity and resistance data were amenable to modeling by a statistics of mixtures approach. Applying physical meaning to the terms in the empirical models indicated that the interactions between the electrolyte and additives were not simple. For example, there were strong, synergistic interactions between VC and TMSPi affecting C/10 capacity loss, as expected, but there were other, more subtle interactions between the electrolyte components. In conclusion, the interactions between these components controlled the C/10 capacity decline and resistance increase.
Effective thermal management systems, designed to handle the impacts of thermal runaway (TR) and to prevent cell-to-cell propagation, are key to safe operation of lithium-ion (Li-ion) battery assemblies. Critical factors for optimizing these systems include the total energy released during a single cell TR event and the fraction of the total energy that is released through the cell casing versus through the ejecta material. A unique fractional thermal runaway calorimeter (FTRC), designed to characterize said critical factors, was utilized to examine the TR behavior of 18650-format Li-ion cells representing a variety of manufacturers, chemistries, capacities, and safety features. Primarily, the impacts of bottom vent (BV) safety features, varied cell casing thickness, and Dreamweaver cellulose based separators were assessed for select cell types. A subset of cells also had an imbedded internal short circuiting (ISC) device to allow examination of TR behavior initiated at lower temperatures (i.e. closer to field failure conditions). The impact of bottom rupture on TR behavior was also examined for experiments that resulted in this non-standard failure mechanism. Statistical analysis of the results for each cell configuration reveals that a lognormal distribution effectively characterizes the variation of total TR energy release. Typically 20%-30% of the total energy yield is released through the cell casing with the remainder through the ejecta material. Higher energy cells tend to exhibit more violent ejections and less predictable TR events. Inclusion of a BV feature reduces the overall severity of the TR event and can increase the predictability. Results also suggest that the magnitude of the TR event may not be directly proportional to the stored electrical energy, but rather has additional dependence on other design and failure mode factors.