non-Boltzmann Distribution Research Articles

I2BODIPY as a new photoswitchable spin label for light-induced pulsed EPR dipolar spectroscopy exploiting magnetophotoselection.

Electron paramagnetic resonance (EPR) pulsed dipolar spectroscopy (PDS) using triplet states of organic molecules is a growing area of research due to the favourable properties that these transient states may afford over stable spin centers, such as switchability, increased signal intensity when the triplet is formed in a non-Boltzmann distribution and the triplet signal is used for detection, and high orientation selection, when the triplet signal is probed by microwave pulses. This arises due to the large spectral width at low fields, a result of the large zero field splitting, and limited bandwidth of microwave pulses used. Here we propose the triplet state of a substituted BODIPY moiety as a spin label in light induced PDS, coupled to a nitroxide, in a model peptide with a rigid structure. Orientation selection allows information on the relative position of the centres of the two labels to be obtained with respect to the nitroxide reference frame. Additionally, magnetophotoselection effects are employed to introduce optical selection and additional constraints for the determination of the relative orientation of the spin labels considering the reference frame of the triplet state.

Encoding prior knowledge in ensemble refinement.

The proper balancing of information from experiment and theory is a long-standing problem in the analysis of noisy and incomplete data. Viewed as a Pareto optimization problem, improved agreement with the experimental data comes at the expense of growing inconsistencies with the theoretical reference model. Here, we propose how to set the exchange rate a priori to properly balance this trade-off. We focus on gentle ensemble refinement, where the difference between the potential energy surfaces of the reference and refined models is small on a thermal scale. By relating the variance of this energy difference to the Kullback-Leibler divergence between the respective Boltzmann distributions, one can encode prior knowledge about energy uncertainties, i.e., force-field errors, in the exchange rate. The energy uncertainty is defined in the space of observables and depends on their type and number and on the thermodynamic state. We highlight the relation of gentle refinement to free energy perturbation theory. A balanced encoding of prior knowledge increases the quality and transparency of ensemble refinement. Our findings extend to non-Boltzmann distributions, where the uncertainty in energy becomes an uncertainty in information.

Theory and simulation of shock waves freely propagating through monoatomic non-Boltzmann gas

The effect of non-Boltzmann energy distributions on the free propagation of shock waves through a monoatomic gas is investigated via theory and simulation. First, the non-Boltzmann heat capacity ratio γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}, as a key property for describing shock waves, is derived from first principles via microcanonical integration. Second, atomistic molecular dynamics simulations resembling a shock tube setup are used to test the theory. The presented theory provides heat capacity ratios ranging from the well-known γ=5/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma = 5/3$$\\end{document} for Boltzmann energy-distributed gas to γ→1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma \\rightarrow 1$$\\end{document} for delta energy-distributed gas. The molecular dynamics simulations of Boltzmann and non-Boltzmann driven gases suggest that the shock wave propagates about 9% slower through the non-Boltzmann driven gas, while the contact wave appears to be about 4% faster if it trails non-Boltzmann driven gas. The observed slowdown of the shock wave through applying a non-Boltzmann energy distribution was found to be consistent with the classical shock wave equations when applying the non-Boltzmann heat capacity ratio. These fundamental findings provide insights into the behavior of non-Boltzmann gases and might help to improve the understanding of gas dynamical phenomena.Graphical abstract

Spectral analysis and kinetic modeling of radioluminescence in air and nitrogen.

In this article we present a quantitative analysis of the second positive system of molecular nitrogen and the first negative system of the molecular nitrogen cation excited in the presence of ionizing radiation. Optical emission spectra of atmospheric air and nitrogen surrounding 210Po sources were measured from 250 to 400 nm. Multi-Boltzmann and non-Boltzmann vibrational distribution spectral models were used to determine the vibrational temperature and vibrational distribution function of the emitting N2(C3Πu) and N2+(B2Σ+u) states. A zero-dimensional kinetic model, based on the electron energy distribution function (EEDF) and steady-state excitation and de-excitation of N2(X1Σ+g), N2+(B2Σ+u), N2+(X2Σ+g), N4+, O2+, and N2(C3Πu, v), was developed for the prediction of the relative spectral intensity of both the N2+(B2Σ+u → X2Σ+g) emission band and the vibrational bands of N2(C3Πu → B3Πg) for comparison with the experimental data.

High-fidelity state-to-state modeling of hypersonic flow over a double cone

The state-of-the-art high-fidelity state-to-state (StS) model is performed to investigate the hypersonic shock wave/laminar boundary layer interaction over a 25°–55° double cone. This work aims to clarify whether the shortcomings of thermochemical models are the underlying source for the discrepancies between the simulations and experiments. A high-enthalpy nitrogen flow with a Mach number of 11.54 and a unit Reynolds number of 4.394×105/m is considered. We first find that the StS and widely used two-temperature models yield two different shock reflection patterns (i.e., the regular reflection and Mach reflection, respectively). However, the surface pressure and heat flux distributions predicted by the two models are generally consistent, which are not influenced by the differences in the shock patterns, dissociation rates, and non-Boltzmann vibrational distributions in the flowfields. Moreover, the StS model fails to match the experiments in spite of fairly limited improvement. Our findings indicate that the shortcomings of thermochemical models are not the main reason for the discrepancies in the simulations and experiments for the high-enthalpy nitrogen double-cone flow.

Characterization of non-Boltzmann CN X2Σ+ behind shock waves in CH4-N2 via broadband ultraviolet femtosecond absorption spectroscopy.

This article describes the temporal evolution of rotationally and vibrationally non-Boltzmann CN X2Σ+ formed behind reflected shock waves in N2-CH4 mixtures at conditions relevant to atmospheric entry into Titan. A novel ultrafast (i.e., femtosecond) laser absorption spectroscopy diagnostic was developed to provide broadband (≈400 cm-1) spectrally resolved (0.02nm resolution) measurements of CN absorbance spectra belonging to its B2Σ+ ← X2Σ+ electronic system and its first four Δv = 0 vibrational bands (v″ = 0, 1, 2, 3). Measurements were acquired behind reflected shock waves in a mixture with 5.65% CH4 and 94.35% N2 at initial chemically and vibrationally frozen temperatures and pressures of 4400-5900K and 0.55-0.75bar, respectively. A six-temperature line-by-line absorption spectroscopy model for CN was developed to determine the rotational temperature of CN in v″ = 0, 1, 2, and 3, as well as two vibrational temperatures via least-squares fitting. The measured CN spectra revealed rotationally and vibrationally non-Boltzmann population distributions that strengthened with increasing shock speed and persisted for over 100 µs. The measured vibrational temperatures of CN initially increase in time with the increasing CN mole fraction and eventually exceed the expected post-shock rotational temperature of N2. The results suggest that strong chemical pumping is ultimately responsible for these trends and that, at the conditions studied, CN is primarily formed in high vibrational states within the A2Π or B2Σ+ state at characteristic rates, which are comparable to or exceed those of key vibrational equilibration processes.

DeepStSNet: Reconstructing the quantum state-resolved thermochemical nonequilibrium flowfield using deep neural operator learning with scarce data

The hypersonic flow is in a thermochemical nonequilibrium state due to the high-temperature caused by the strong shock compression. In a thermochemical nonequilibrium flow, the distribution of molecular internal energy levels strongly deviates from the equilibrium distribution (i.e., the Boltzmann distribution). It is intractable to directly obtain the microscopic nonequilibrium distribution from existed experimental measurements usually described by macroscopic field variables such as temperature or velocity. Motivated by the idea of deep multi-scale multi-physics neural network (DeepMMNet) proposed in [1], we develop in this paper a data assimilation framework called DeepStSNet to accurately reconstruct the quantum state-resolved thermochemical nonequilibrium flowfield by using sparse experimental measurements of vibrational temperature and pre-trained deep neural operator networks (DeepONets). In particular, we first construct several DeepONets to express the coupled dynamics between field variables in the thermochemical nonequilibrium flow and to approximate the state-to-state (StS) approach, which traces the variation of each vibrational level of molecule accurately. These proposed DeepONets are then trained by using the numerical simulation data, and would later be served as building blocks for the DeepStSNet. We demonstrate the effectiveness and accuracy of DeepONets with different test cases showing that the density and energy of vibrational groups as well as the temperature and velocity fields are predicted with high accuracy. We then extend the architectures of DeepMMNet by considering a simplified thermochemical nonequilibrium model, i.e., the 2T model, showing that the entire thermochemical nonequilibrium flowfield is well predicted by using scattered measurements of full or even partial field variables. We next consider a more accurate and complex thermochemical nonequilibrium model, i.e., the StS-CGM model, and develop a DeepStSNet for this model. In this case, we employ the coarse-grained method, which divides the vibrational levels into groups (vibrational bins), to alleviate the computational cost for the StS approach in order to achieve a fast but reliable prediction with DeepStSNet. We test the present DeepStSNet framework with sparse numerical simulation data showing that the predictions are in excellent agreement with the reference data for test cases. We further employ the DeepStSNet to assimilate a few experimental measurements of vibrational temperature obtained from the shock tube experiment, and the detailed non-Boltzmann vibrational distribution of molecule oxygen is reconstructed by using the sparse experimental data for the first time. Moreover, by considering the inevitable uncertainty in the experimental data, an average strategy in the predicting procedure is proposed to obtain the most probable predicted fields. The present DeepStSNet is general and robust, and can be applied to build a bridge from sparse measurements of macroscopic field variables to a microscopic quantum state-resolved flowfield. This kind of reconstruction is beneficial for exploiting the experimental measurements and uncovering the hidden physicochemical processes in hypersonic flows.

Two-Temperature Modeling of Nonequilibrium Relaxation and Dissociation in Shock-Heated Oxygen

Two-temperature models for coupled vibrational relaxation and dissociation in shock-heated oxygen are assessed using low-uncertainty measured data from reflected shock tube experiments. A computationally efficient multistep technique is developed to model the unsteady dynamics of shock reflection in a relaxing and dissociating gas. The developed technique is then benchmarked through comparison with unsteady computational fluid dynamic simulations. Results from the benchmarking effort demonstrate that the adopted multistep modeling procedure accurately captures the dominant gas dynamic effects influencing the state of the test gas at the measurement location. A parametric study is then performed to assess several combinations of possible two-temperature modeling approaches for nonequilibrium oxygen dissociation. The current assessment demonstrates that the widely adopted Park model is inconsistent with the measured data, while the recently developed modified Marrone and Treanor (MMT) model demonstrates promising agreement with the data. The results of the present study clearly indicate that the MMT model is more appropriate for two-temperature modeling of nonequilibrium oxygen dissociation than the legacy Park model. Patterns in the parametric comparison also suggest that the approximate treatment of non-Boltzmann vibrational state distributions within the MMT model may require improvement.

Stochastic localization and non-Boltzmann distribution

We investigate the microscopical root and macroscopical representation of the stochastic localization, which is a nonergodic motion in oppositive to the other limit of ballistic diffusion. In order to produce such anomalous kinetics, we consider that a tagged particle is linearly coupled to a series connection bath or the terminal of a coupled-oscillator-chain. By means of generalized Langevin equation formalism, we obtain the coordinate autocorrelation function. The localization emerged of a particle at finite temperature, is due to the spectrum of driving noise diverging at zero frequency. Consequently, the steady distribution of system depends on its initial coordinate preparation.

Non-equilibrium dissociation rate coefficient of O2+O2 collision at high temperatures

We focus on the collision dissociation process of O2+O2→O+O+O2 at high temperatures (8000 K–30000 K). The research method is the new general non-equilibrium dynamics model proposed by Singh et al. The model parameters are determined by calculating the dissociation probability under the specific vibrational energy and the energy distribution functions in the quasi-steady state (QSS). It is found that the dissociation rate is more easily at high vibrational energy levels. Moreover, the energy distribution function deviates from the Boltzmann distribution in the QSS state. We obtain the dissociation rate coefficients at high temperatures by integrating the dissociation probability and the internal energy distribution functions (the Boltzmann distribution, the QSS state distribution, and the non-Boltzmann distribution). Our results demonstrate that the rate coefficients depend on the vibrational temperature and more strongly at a lower translational temperature.

Nonequilibrium steady state for harmonically confined active particles.

We study the full nonequilibrium steady-state distribution P_{st}(X) of the position X of a damped particle confined in a harmonic trapping potential and experiencing active noise whose correlation time τ_{c} is assumed to be very short. Typical fluctuations of X are governed by a Boltzmann distribution with an effective temperature that is found by approximating the noise as white Gaussian thermal noise. However, large deviations of X are described by a non-Boltzmann steady-state distribution. We find that, in the limit τ_{c}→0, they display the scaling behavior P_{st}(X)∼e^{-s(X)/τ_{c}}, where s(X) is the large-deviation function. We obtain an expression for s(X) for a general active noise and calculate it exactly for the particular case of telegraphic (dichotomous) noise.

Numerical study of pure rotational fs/ps CARS coherence beating at high pressure and for multi-species rotation-vibration non-equilibrium thermometry

Coherent anti-Stokes Raman scattering (CARS) has long been the gold standard for non-intrusively measuring gas temperature in reacting flows such as flames and plasmas. Recently, the development of ultrafast hybrid fs/ps CARS has enabled the exploitation of coherence beating between neighboring spectral lines to simultaneously measure rotational and vibrational temperatures from a single pure rotational spectrum. However, the influence of non-Boltzmann vibrational state distributions and limitations due to collisional dephasing at pressures greater than 1 atm remains unclear. In this work, we use spectral simulations to investigate the effects of non-Boltzmann vibrational state distributions and the applicability of coherence beating at pressures up to 10 atm. We show that short probe pulses can be leveraged to quantify non-Boltzmann vibrational state distributions of N2. Furthermore, we demonstrate that fs/ps CARS coherence beating can simultaneously provide sensitive measurements of rotational and vibrational temperatures of both O2 and N2 in air. A sensitivity analysis was conducted to qualitatively explain the accuracy and precision comparisons between probe delays.

First-passage time of run-and-tumble particles with noninstantaneous resetting

We study the statistics of the first-passage time of a single run-and-tumble particle (RTP) in one spatial dimension, with or without resetting, to a fixed target located at L>0. First, we compute the first-passage time distribution of a free RTP, without resetting or in a confining potential, but averaged over the initial position drawn from an arbitrary distribution p(x). Recent experiments used a noninstantaneous resetting protocol that motivated us to study in particular the case where p(x) corresponds to the stationary non-Boltzmann distribution of an RTP in the presence of a harmonic trap. This distribution p(x) is characterized by a parameter ν>0, which depends on the microscopic parameters of the RTP dynamics. We show that the first-passage time distribution of the free RTP, drawn from this initial distribution, develops interesting singular behaviors, depending on the value of ν. We then switch on resetting, mimicked by relaxation of the RTP in the presence of a harmonic trap. Resetting leads to a finite mean first-passage time and we study this as a function of the resetting rate for different values of the parameters ν and b=L/c, where c is the position of the right edge of the initial distribution p(x). In the diffusive limit of the RTP dynamics, we find a rich phase diagram in the (b,ν) plane, with an interesting reentrance phase transition. Away from the diffusive limit, qualitatively similar rich behaviors emerge for the full RTP dynamics.

Absence of stationary states and non-Boltzmann distributions of fractional Brownian motion in shallow external potentials

We study the diffusive motion of a particle in a subharmonic potential of the form U(x) = |x| c (0 < c < 2) driven by long-range correlated, stationary fractional Gaussian noise ξ α (t) with 0 < α ⩽ 2. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent α. While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation c > 2(1 − 1/α) holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function. Moreover we discuss analogies of non-stationarity of Lévy flights in shallow external potentials.

State-Specific Study of Air in the Expansion Tunnel Nozzle and Test Section

Thermochemical nonequilibrium in expansion tunnel nozzles is investigated numerically using a state-to-state description in one dimension for representative air conditions. Limiting the multiquantum jumps of vibration–vibration–translation (VVT) transitions to three in both and can accurately simulate the nonequilibrium nozzle flow. The reduction of VVT transitions to vibration–translation transitions works well. State-to-state modeling of an actual expansion tunnel nozzle condition yielded agreement with the measured static pressure. A study on the influence of different thermochemical excitations in the freestream at the test section shows that the postshock radiation emissions can differ by more than 50%. However, the non-Boltzmann distributions in the freestream have no influence. An evaluation of the discrepancy between the two-temperature and state-to-state models shows that the former generally predicts a faster thermochemical relaxation. Furthermore, the state-to-state results indicate that, in general, the molecular species all have a different vibrational temperature.

Simultaneous single-shot rotation-vibration non-equilibrium thermometry using pure rotational fs/ps CARS coherence beating.

We report the development of a simple and sensitive two-beam hybrid femtosecond/picosecond pure rotational coherent anti-Stokes Raman scattering (fs/ps CARS) method to simultaneously measure the rotational and vibrational temperatures of diatomic molecules. Rotation-vibration non-equilibrium plays a key role in the chemistry and thermalization in low-temperature plasmas as well as thermal loading of hypersonic vehicles. This approach uses time-domain interferences between ground state and vibrationally excited N2 molecules to intentionally induce coherence beating that leads to apparent non-Boltzmann distributions in the pure rotational spectra. These distortions enable simultaneous inference of both the rotational and vibrational temperatures. Coherence beating effects were observed in single-shot fs/ps CARS measurements of a 75 Torr N2 DC glow discharge and were successfully modeled for rotational and vibrational temperature extraction. We show that this method can be more sensitive than a pure rotational fs/ps CARS approach using a spectrally narrow probe pulse. Lastly, we experimentally measured the beat frequencies via Fourier transform of the time-domain response and obtained excellent agreement with the model.

Deformed Butler–Volmer Models for Convex Semilogarithmic Current-Overpotential Profiles of Li-ion Batteries

The Butler-Volmer (BV) equation links the current flux crossing an electrochemical interface to the electric potential drop across it with the assumption of Arrhenius kinetics and the Boltzmann factor. Applying the semilogarithmic Tafel analysis in which the logarithm of current is plotted vs. the overpotential one expects straight lines from which the fundamental reaction rate of the kinetic process can be computed. However, some Li-ion battery data, which is the focus here, show nonlinear convex profiles that cannot be adequately fitted with the standard BV model. We propose instead two deformed BV models for the analysis of such types of behaviors constructed from the superposition of cells exhibiting only local equilibrium and thus giving rise to the power-law $q$-exponential and $\kappa$-exponential functions. Non-Boltzmann distributions have been successfully employed for the modeling of a wide spectrum of physical systems in nonequilibrium situations, but not yet for batteries. We verify the validity of the deformed BV models on experimental data obtained from \ce{LiFePO4} and \ce{Li}-\ce{O2} batteries.

Rate constants and molecular recombination pathways of oxygen from quasi-classical trajectory simulations of theO3system

• Investigation of recombination mechanisms using quasi-classical trajectory method. • Direct collisions are more effective for recombination. • Indirect recombinations proceed through orbiting pairs. • Recombination products result in strong non-Boltzmann internal energy distributions. • Recombination products exhibit large vibrational and low rotational quantum numbers. A lack of understanding of the dynamics of molecular recombination has led to the development of a large number of recombination models and methods to quantify recombination rate constants, each with their own set of assumptions and ranges of applicability. This work aims to unravel the complex dynamics behind the recombination of atomic oxygen to molecular oxygen using the quasi-classical trajectory method for the O 3 system. Recombination pathways considered include a single-step “direct” reaction, and a two-step mechanism that involves the formation of a long-lived intermediate which is stabilized through collision with a third body. The latter is further classified as Lindemann or Chaperon recombination pathways depending on the identity of the intermediate and the final product. The classical versions of all three mechanisms are explored in detail for the O 3 system. A time-lag parameter is introduced to quantify the time interval between the intermediate formation and third-body stabilization steps, and the direct collisions are treated as a limiting case when the time lag is zero. It is found that the recombination probabilities reduce as the time-lag is increased. Direct recombination events are dominated by head-on collisions, while the two step recombination events are characterized by a combination of glancing and head-on collisions. The resulting recombination products follow a strongly non-Boltzmann distribution. Recombination rate coefficients are computed through a novel rate coefficient formula that takes into consideration the Poisson distribution in time of third body stabilizing collisions with the orbiting pair characterized by finite lifetimes (on the order of picoseconds) that lead to recombination. Good agreement is obtained with recombination rates computed from detailed balance based on known dissociation rate coefficients. This approach thus enables the direct calculation of recombination rate coefficients at low temperatures, without the need for dissociation rate coefficients and detailed balance.

Fractional Brownian motion in superharmonic potentials and non-Boltzmann stationary distributions

We study the stochastic motion of particles driven by long-range correlated fractional Gaussian noise (FGN) in a superharmonic external potential of the form U(x) ∝ x 2n (). When the noise is considered to be external, the resulting overdamped motion is described by the non-Markovian Langevin equation for fractional Brownian motion. For this case we show the existence of long time, stationary probability density functions (PDFs) the shape of which strongly deviates from the naively expected Boltzmann PDF in the confining potential U(x). We analyse in detail the temporal approach to stationarity as well as the shape of the non-Boltzmann stationary PDF. A typical characteristic is that subdiffusive, antipersistent (with negative autocorrelation) motion tends to effect an accumulation of probability close to the origin as compared to the corresponding Boltzmann distribution while the opposite trend occurs for superdiffusive (persistent) motion. For this latter case this leads to distinct bimodal shapes of the PDF. This property is compared to a similar phenomenon observed for Markovian Lévy flights in superharmonic potentials. We also demonstrate that the motion encoded in the fractional Langevin equation driven by FGN always relaxes to the Boltzmann distribution, as in this case the fluctuation-dissipation theorem is fulfilled.

Analytical Non-Boltzmann vibrational distributions and their influence on vibrational partition function

The analytical forms of non-Boltzmann vibrational distributions are studied by means of the vibrational partition function which, being the cumulative quantity, allows to detect general differences in behavior of vibrational distributions. Various forms of the Treanor distributions are studied in both discrete (sum of quantum vibrational levels) and continuous manners (classical values from integration over the phase space). The advantages of the continuous forms of distributions are pointed out (in both classical and quantum vibrational temperature regimes). The typical distribution is not properly described by the Treanor forms so the values based on continuous form of the piecewise Treanor-Hyperbolic distribution are given. The influence of vibrational excitation on rotational partition function is quantified.