Dissipative Processes Research Articles

General framework for quantifying dissipation pathways in open quantum systems. II. Numerical validation and the role of non-Markovianity.

In the previous paper [C. W. Kim and I. Franco, J. Chem. Phys. 160, 214111-1-214111-13 (2024)], we developed a theory called MQME-D, which allows us to decompose the overall energy dissipation process in open quantum system dynamics into contributions by individual components of the bath when the subsystem dynamics is governed by a Markovian quantum master equation (MQME). Here, we contrast the predictions of MQME-D against the numerically exact results obtained by combining hierarchical equations of motion (HEOM) with a recently reported protocol for monitoring the statistics of the bath. Overall, MQME-D accurately captures the contributions of specific bath components to the overall dissipation while greatly reducing the computational cost compared to exact computations using HEOM. The computations show that MQME-D exhibits errors originating from its inherent Markov approximation. We demonstrate that its accuracy can be significantly increased by incorporating non-Markovianity by exploiting time scale separations (TSS) in different components of the bath. Our work demonstrates that MQME-D combined with TSS can be reliably used to understand how energy is dissipated in realistic open quantum system dynamics.

Experimental Observation of a New Attenuation Mechanism in hcp‐Metals That May Operate in the Earth's Inner Core

AbstractSeismic observations show the Earth's inner core has significant and unexplained variation in seismic attenuation with position, depth and direction. Interpreting these observations is difficult without knowledge of the visco‐ or anelastic dissipation processes active in iron under inner core conditions. Here, a previously unconsidered attenuation mechanism is observed in zinc, a low pressure analog of hcp‐iron, during small strain sinusoidal deformation experiments. The experiments were performed in a deformation‐DIA combined with X‐radiography, at seismic frequencies (∼0.003–0.1 Hz), high pressure and temperatures up to ∼80% of melting temperature. Significant dissipation (0.077 ≤ Q−1(ω) ≤ 0.488) is observed along with frequency dependent softening of zinc's Young's modulus and an extremely small activation energy for creep (⩽7 kJ mol−1). In addition, during sinusoidal deformation the original microstructure is replaced by one with a reduced dislocation density and small, uniform, grain size. This combination of behavior collectively reflects a mode of deformation called “internal stress superplasticity”; this deformation mechanism is unique to anisotropic materials and activated by cyclic loading generating large internal stresses. Here we observe a new form of internal stress superplasticity, which we name as “elastic strain mismatch superplasticity.” In it the large stresses are caused by the compressional anisotropy. If this mechanism is also active in hcp‐iron and the Earth's inner‐core it will be a contributor to inner‐core observed seismic attenuation and constrain the maximum inner‐core grain‐size to ≲10 km.

On Parameterization of Dissipative Processes in Turbulent Transport Models for Description of Thermohydrodynamics and Biogeochemistry of Stratified Inland Water Bodies

On Parameterization of Dissipative Processes in Turbulent Transport Models for Description of Thermohydrodynamics and Biogeochemistry of Stratified Inland Water Bodies

Direct Phototransformation of Sulfamethoxazole Characterized by Four-Dimensional Element Compound Specific Isotope Analysis.

The antibiotic sulfamethoxazole (SMX) undergoes direct phototransformation by sunlight, constituting a notable dissipation process in the environment. SMX exists in both neutral and anionic forms, depending on the pH conditions. To discern the direct photodegradation of SMX at various pH levels and differentiate it from other transformation processes, we conducted phototransformation of SMX under simulated sunlight at pH 7 and 3, employing both transformation product (TP) and compound-specific stable isotope analyses. At pH 7, the primary TPs were sulfanilic acid and 3A5MI, followed by sulfanilamide and (5-methylisoxazol-3-yl)-sulfamate, whereas at pH 3, a photoisomer was the dominant product, followed by sulfanilic acid and 3A5MI. Isotope fractionation patterns revealed normal 13C, 34S, and inverse 15N isotope fractionation, which exhibited significant differences between pH 7 and 3. This indicates a pH-dependent transformation process in SMX direct phototransformation. The hydrogen isotopic composition of SMX remained stable during direct phototransformation at both pH levels. Moreover, there was no variation observed in 33S between the two pH levels, indicating that the 33S mass-independent process remains unaffected by changes in pH. The analysis of main TPs and single-element isotopic fractionation suggests varying combinations of bond cleavages at different pH values, resulting in distinct patterns of isotopic fractionation. Conversely, dual-element isotope values at different pH levels did not significantly differ, indicating cleavage of several bonds in parallel. Hence, prudent interpretation of dual-element isotope analysis in these systems is warranted. These findings highlight the potential of multielement compound-specific isotope analysis in characterizing pH-dependent direct phototransformation of SMX, thereby facilitating the evaluation of its natural attenuation through sunlight photolysis in the environment.

Effect of background wind and dissipation processes on the semidiurnal component of atmospheric solar tides

The equations governing the neutral wind velocities and temperature, which depict the oscillations of atmospheric tides, incorporate various elements of the atmosphere. These factors comprise background wind, temperature profiles, and the atmosphere’s composition, including ozone, oxygen, carbon dioxide, water, etc. The current work describes the method of solution of the atmospheric tidal model and investigates the effects of background wind and dissipation processes on the migrating semidiurnal tides during the March equinox. Thermal heating due to water vapour absorption in the troposphere, ozone in the stratosphere and oxygen absorption in the thermosphere is considered in this model. Furthermore, the model takes into account the various dissipation processes (i.e. ion drag and Hall coefficients, Newtonian cooling, divergence of momentum and heat fluxes due to molecular and eddy diffusion) and background wind. The semidiurnal tidal features obtained from the present model match well with the Global Scale Wave Model (GSWM-98). The bar charts are presented to quantify the effects of different parameters on horizontal winds for altitudes below 100 km and above 100 km, separately. It reveals that the background wind plays a vital role in deciding the semidiurnal tidal amplitudes of zonal and meridional winds. Momentum and heat flux coefficients also have a significant influence on semidiurnal tides. At higher altitudes (>100 km), the ion drag reduces the amplitude of semidiurnal tides considerably.

Asymptotic profiles of mean zonal flows generated by thermal convection of Boussinesq fluid in a rapidly rotating thin spherical shell

The asymptotic behavior of the profiles of mean zonal flows generated by thermal convection of a Boussinesq fluid in a rapidly rotating thin spherical shell was studied by extending the integration time of the numerical experiments of a previous study under similar conditions. Calculations were performed for the whole globe, as well as for a one-eighth sector region, with various values of the hyper-diffusion parameters. For sufficiently weak hyper-diffusion, a prograde zonal jet and alternating narrow zonal jets appeared at the equator and in the middle and high latitudes, respectively, as reported in the previous study. However, further integrations showed that the middle and high latitude flows are asymptotically accelerated eastward, jet peaks merge, and the banded zonal flow structure eventually disappears. Thus, the rotating thin spherical shell convection model of a Boussinesq fluid used in the previous studies seems inadequate to explain the surface banded structures of Jovian planetary atmospheres without modification of the standard setups. To maintain the alternating jet structure in the middle and high latitudes over the long-term, some sort of scale-independent dissipation process should be specified that prevents the evolution of zonal flows toward the long-term asymptotic state.

A Direct Entropic Approach to the Thermal Balance of Spontaneous Chemical Reactions.

When working with, and learning about, the thermal balance of a chemical reaction, we need to consider two overlapping but conceptually distinct aspects: one relates to the process of reallocating entropy between reactants and products (because of different specific entropies of the new substances compared to those of the old), and the other to dissipative processes. Together, they determine how much entropy is exchanged between the chemicals and their environment (i.e., in heating and cooling). By making explicit use of (a) the two conjugate pairs chemical amount (i.e., amount of substance) and chemical potential, and entropy and temperature, respectively, (b) the laws of balance of amount of substance on the one hand and entropy on the other, and (c) a generalized approach to the energy principle, it is possible to create both imaginative and formal conceptual tools for modeling thermal balances associated with chemical transformations in general and exothermic and endothermic reactions in particular. In this paper, we outline the concepts and relations needed for a direct approach to chemical and thermal dynamics, create a model of exothermic and endothermic reactions, including numerical examples, and discuss how to relate the direct entropic approach to traditional models of these phenomena.

Pinning Energy and Evidence of Granularity in the AC Susceptibility of an YBa2Cu3O7-x Superconducting Film

The study of granularity in superconducting films by using AC susceptibility has a crucial role in the development of and improvement in the ReBCO-coated conductors, which are a constantly evolving reality in the modern power applications of superconductivity. Specifically, the study of the granularity is essential because the ReBCO superconducting wires and tapes are far from the regularity of a single crystal while they often present an inter- and intragranular contribution to the critical current density. On the other hand, the AC susceptibility is a key part of the characterization of a granular sample because this technique is very sensitive to the presence of granularity in the superconductors and, moreover, the study of its first harmonic allows for determining pivotal properties such as the pinning energy as well as the dissipation processes acting in the sample. The pinning energy values and the granularity of an YBCO thin film have been studied by means of AC susceptibility measurements as a function of the AC amplitude, temperature, and DC field. In particular, the first harmonic imaginary component of the AC susceptibility χ1″ related to the dissipation processes of the sample has been studied. First, starting from the Brandt approach, the critical current density Jc and the pinning energy U of the sample have been extracted at 77 K by using the χ1″ measurements as a function of the AC amplitude at different AC frequencies and DC fields. From these measurements, a first signal of granularity appears. In order to confirm it, the temperature dependence of the χ1″ at different DC fields has been studied and a contribution deriving from the inter- and intragranular part of the sample has emerged. By taking the temperature corresponding to the crossover between the two contributions at the different DC fields, the intergranular and intragranular response has been separated. Successively, the temperature has been fixed to 77 K, together with an AC frequency equal to 1597.9 Hz, and the χ1″ as a function of the DC field at different AC amplitudes has been analyzed showing a clear presence of granularity in all the curves. By drawing the contour plot of the χ1″ with the DC and AC values, it was possible to determine the best parameters to put at 77 K in order to exploit the material for applications.

Spherical Fe3O4 morphology modulation for enhancing infrared emissivity and radiant heat dissipation

Radiant heat dissipation has emerged as a critical technique for effective thermal management, offering energy efficiency, cost-effectiveness, and adaptability to various shapes while promoting environmental sustainability. However, the intricate relationship between material structures and emissivity remains inadequately understood. This study primarily utilizes a straightforward and economical solvothermal method to regulate the synthesis of Fe3O4 ferrite, exploring its structural relationship with infrared radiation performance. By adjusting the solvent ratio of ethylene glycol and diethylene glycol in the solvothermal system, we successfully synthesized uniformly dispersed Fe3O4 nanospheres ranging from 10 to 800 nm. Notably, our research revealed a positive correlation between infrared emissivity and the particle size. Submicron hollow spheres with a size of 800 nm exhibit an exceptionally high emissivity of 0.992 within the 8–13 μm wavelength range. Morphological and size variations induced alterations in lattice strain, oxygen vacancies, and phonon lifetimes, which consequently influencing the lattice vibration absorption and infrared emissivity. Moreover, the device surface with a composite coating formed by combining 15 wt% Fe3O4 particles with organic matter, achieves a temperature difference of 17.6 °C under a 4 W input power condition compared to blank device. This finding contributes to further revealing the process of material radiative heat dissipation, driving the application of high-emissivity materials for heat dissipation in high-power electronic devices.

Tuning the diffusion constant to optimize the readout of positional information of spatial concentration patterns

Abstract Positional information encoded in spatial concentration patterns is crucial for the development of multicellular organisms. However, it is still unclear how such information is affected by the physically dissipative diffusion process. Here we study one-dimensional patterning systems with analytical derivation and numerical simulations. We find that the diffusion constant of the patterning molecules exhibits a nonmonotonic effect on the readout of the positional information from the concentration patterns. Specifically, there exists an optimal diffusion constant that maximizes the positional information. Moreover, we find that the energy dissipation due to the physical diffusion imposes a fundamental upper limit on the positional information.

Thermospheric nitric oxide energy budget during extreme geomagnetic storms: a comparative study

We selected three superstorms (disturbance storm time [Dst] index less than −350 nT) of 2003–04 to study the thermospheric energy budget with a particular emphasis on the thermospheric cooling emission by nitric oxide via a wavelength of 5.3 μm. The nitric oxide radiative emission data are obtained from the Sounding of the Atmosphere by Broadband Emission Radiometry (SABER) instrument onboard the Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) satellite and the thermosphere ionosphere electrodynamic general circulation model (TIEGCM) simulation. Different energy sources for the magnetospheric energy injection and the thermospheric/ionospheric dissipation processes are calculated using empirical formulations, model simulations, and space-borne and ground-based measurements. The Joule heating rates calculated from different sources showed similar variations but significant differences in the magnitude. The nitric oxide cooling power is calculated by zonally and meridionally integrating the cooling flux in the altitude range of 100–250 km. The satellite observed that cooling flux responds faster to the energy input, as compared to the modeled results. The cooling power increases by an order of magnitude during storm time with maximum radiation observed during the recovery phase. Both the satellite-observed and modeled cooling powers show a strong positive correlation with the Joule heating power during the main phase of the storm. It is found that the maximum radiative power does not occur during the strongest storm, and it strongly depends on the duration of the main phase. The model simulation predicts a higher cooling power than that predicted by the observation. During a typical superstorm, on average, a cooling power of 1.87 × 105 GW exiting the thermosphere is estimated by the TIEGCM simulation. On average, it is about 40% higher than the satellite observation.

A collision operator for describing dissipation in noncanonical phase space

The phase space of a noncanonical Hamiltonian system is partially inaccessible due to dynamical constraints (Casimir invariants) arising from the kernel of the Poisson tensor. When an ensemble of noncanonical Hamiltonian systems is allowed to interact, dissipative processes eventually break the phase space constraints, resulting in a thermodynamic equilibrium described by a Maxwell–Boltzmann distribution. However, the time scale required to reach Maxwell–Boltzmann statistics is often much longer than the time scale over which a given system achieves a state of thermal equilibrium. Examples include diffusion in rigid mechanical systems, as well as collisionless relaxation in magnetized plasmas and stellar systems, where the interval between binary Coulomb or gravitational collisions can be longer than the time scale over which stable structures are self-organized. Here, we focus on self-organizing phenomena over spacetime scales such that particle interactions respect the noncanonical Hamiltonian structure, but yet act to create a state of thermodynamic equilibrium. We derive a collision operator for general noncanonical Hamiltonian systems, applicable to fast, localized interactions. This collision operator depends on the interaction exchanged by colliding particles and on the Poisson tensor encoding the noncanonical phase space structure, is consistent with entropy growth and conservation of particle number and energy, preserves the interior Casimir invariants, reduces to the Landau collision operator in the limit of grazing binary Coulomb collisions in canonical phase space, and exhibits a metriplectic structure. We further show how thermodynamic equilibria depart from Maxwell–Boltzmann statistics due to the noncanonical phase space structure, and how self-organization and collisionless relaxation in magnetized plasmas and stellar systems can be described through the derived collision operator.

Study of boundary layer parameterization simulation uncertainties of sand-dust storm windfield using high-resolution three-dimensional Doppler wind lidar data

The processes of dust emission, the development, and dissipation of sand and dust storms all exhibit dynamic and rapidly changing characteristics, posing significant challenges for us to accurately represent their complex behaviors in numerical simulations. In the numerical weather prediction model, the selection and optimization of planetary boundary layer (PBL) parameterization schemes have a direct impact on the simulation results of PBL wind fields, primarily due to the significant uncertainty in the fine-grained simulation of PBL wind field structure. To gain a deeper understanding of the sources of uncertainty in simulating sand-dust storms through vertical wind field structure simulations of the PBL, this study utilizes high-resolution 3D Doppler wind lidar (DWL) data. Focusing on sand-dust storm wind fields with distinctive regional characteristics in Inner Mongolia, China, it examines the turbulent momentum equation and assumptions of the PBL. A comprehensive and systematic evaluation was conducted on the characteristics of horizontal and vertical wind fields, wind shear, as well as the dynamic and thermodynamic differences that arise when simulating typical sand-dust storm wind fields using the Weather Research and Forecasting (WRF) Model with various PBL schemes. This evaluation focused on simulating the typical wind field of sand-dust storms during various stages, including dust generation, occurrence, development, and dissipation. The findings indicate that non-adiabatic heating plays a significant role in the sustainability and evolution of sand-dust storms. The wind shear index holds particular importance as it embodies the dynamic factor in the model. The PBL schemes such as Mellor-Yamada Nakanishi and Niino Level 2.5 (MYNN2), A new version of asymmetric convective model (ACM2), and Grenier-Bretherton-McCaa (GBM), which factor in buoyancy effects, demonstrate more accurate wind shear simulations in sand-dust storms. Furthermore, the ACM2 scheme effectively replicates the fundamental variation characteristics of meteorological elements throughout the sand-dust storm lifecycle.

Effect and mechanism of solution flow rate during interfacial polymerization on morphology and performance of hollow fiber membranes

Interfacial polymerization (IP) using m-phenylenediamine and trimesoyl chloride (TMC) is an essential method for producing thin-film composite (TFC) membranes. Nevertheless, IP is highly sensitive to changes in process parameters, especially the TMC solutions flow rate during inside-out TFC hollow fibers (HF) production. Research on the impact of flow conditions (i.e., laminar vs. turbulent flows) of these organic solutions on the final membrane characteristics and corresponding mechanism is still lacking. This research tested different TMC flow rates (0.1–2 mL/min) to produce inside-out TFC HF. With a TMC flow rate increase, (i) the leaf-like structures of the polyamide (PA) layer decreased in size, (ii) the thickness of the PA layer slightly increased, and (iii) the pure water permeability decreased. However, the elemental composition, chemical bonds, and NaCl rejection remained stable. Different fluid regimes are considered to affect the performance of HF membranes by affecting the heat dissipation process during IP reaction. A TMC flow rate in the laminar flow state (i.e., low flow rates) is recommended for synthesizing HFs. The current research provides new insights into the IP reaction mechanism of HF membrane, thus offering valuable guidance to optimize and produce membrane products with better performance.

A unified two-scale gas–liquid multi-fluid model with capillarity and interface regularization through a mass transfer between scales

In this contribution, we derive a gas–liquid two-scale multi-fluid model with capillarity effects to enable a novel interface regularization approach for multi-fluid models. As this unified modelling is capable of switching from the interface representation of a separated to a disperse regime it lays a new way of modelling regime transitions as it occurs in atomization processes. Above a preset length threshold at large scale, a multi-fluid diffuse interface model resolves the dynamics of the interface while, at small-scale, a set of geometric variables is used to characterize the interface geometry. These variables result from a reduced-order modelling of the small-scale kinetic equation that describes a collection of liquid inclusions. The flow model can be viewed as a two-phase two-scale mixture, and the equations of motion are obtained thanks to the Hamilton’s Stationary Action Principle, which requires to specify the kinetic and potential energies at play. We particularly focus on modelling the effects of capillarity on the mixture’s energy by including dependencies on additional variables accounting for the interface’s geometry at both scales. The regularization of the large-scale interface is then introduced as a local and dissipative process. The local curvature is limited via a relaxation towards a modified Laplace equilibrium such that an inter-scale mass transfer is triggered when the mean curvature is too high. We propose an original numerical method and assess the properties and potential of the modelling strategy on the relevant test-case of a two-dimensional liquid column in a compressible gas flow.

Dissipative processes at the acoustic horizon

A transonic fluid flow generates an acoustic hole that is the hydrodynamic analogue of a gravitational black hole (BH). Acoustic holes emit a detectable thermal radiation of phonons at a characteristic Hawking temperature. The crucial concept is that the spontaneous phonon emission at the horizon produces an irreversible heat increase at the expenses of the bulk fluid kinetic energy. We show that such process can be described in terms of effective shear and bulk viscosities that are defined close to the horizon. We analyze this quantum friction process by resorting to a general kinetic theory approach as well as by the specific description of phonon emission as a tunneling process. The celebrated Kovtun, Son and Starinets (KSS) universal lower bound η/s=1/4π of the shear viscosity coefficient to entropy density ratio, readily follows, and is extended to the longitudinal bulk viscosity at the horizon. We come to the same saturation of the KSS bound after considering the shear viscosity arising from a perturbation of the background metric at the acoustic horizon providing a—in principle testable—realization of the so called BH membrane paradigm.

Conventional sintering of nano-crystalline Yttria-Stabilized Zirconia enables high-strength, highly translucent and opalescent dental ceramics

Development of restorative materials capable of mimicking optical and mechanical performance of natural teeth is a quest in aesthetic density. Yttria-Stabilized Zirconia (YSZ) ceramics represent one of the most popular choices for dental restorations, owing to their biocompatibility, white colour, and the possibility to use CAD-CAM technologies. In particular, YSZ doped with 3 mol. % yttria (3YSZ) is popular because it presents high strength. Nonetheless, the limited light transmission of commercially available high strength 3YSZ does not meet the requirements of highly aesthetic cases. On the other side, YSZ presenting a larger portion of yttria are more translucent but exhibit modest strength. Here, we report on fabrication of dense zirconia nanostructures in bulk form via conventional pressure-less sintering at temperatures down to 1100–1200 °C, achieving highly translucent and strong 3YSZ with significant opalescent behaviour. Both Hall−Petch and inverse Hall-Petch relationship were observed in 3YSZ samples with average grain size in the range of 250 nm and 55 nm, demonstrating the importance of grain size control to enhance both optical and mechanical properties of zirconia ceramics, simultaneously. Maximum biaxial strength of 1980 ± 260 MPa, in-line light transmission of 38% in the visible spectrum and opalescence approaching that of enamel were obtained at optimum grain size of 80 ± 5 nm. The notable optical properties are linked to the miniaturization of the residual pores and refinement of grain size towards the nanoscale while the superior mechanical strength is justified by the activation of different energy dissipation processes at nano and macroscale.

Temperature Intermittent Structures in the Fast Solar Wind

In the solar wind turbulence, proton temperature fluctuations are highly intermittent, especially at small scales in the inertial range. This phenomenon may contain information about solar wind intermittent heating. However, the physical nature of the temperature intermittency is not yet clear. Based on the measurements from Solar Orbiter between 2020 September and 2022 September, we identify 185 temperature intermittent structures at the scale of 24 s with well-defined minimum variance direction in the fast solar wind and study the nature and kinetic effects of them. According to the variations of proton temperature, the intermittent structures are classified into four types, including temperature bump, step, dip, and chain. When considering the magnetic field configuration together with other plasma properties, we find that, among the bump cases, 46 of them are linear magnetic holes (LMHs) that are mainly related to mirror-mode (MM) instability, and 43 of them are current sheets (CSs) with local temperature enhancement. The step cases are found to be associated with shock-like structures (13 cases) and tangential discontinuities (eight cases) that separate two different parcels of plasma. For the dip cases, 21 of them could be associated with the MM instability, and the rest of the 16 ones may prefer soliton scenario. The 38 chain cases are identified as compressive vortex-like structures and a mixture of LMHs and CSs. These results will help to further understand the intermittent dissipation process in the solar wind turbulence.

Experimental and MD simulation study of surface residual stress and surface quality of SiCp/Al in pulsed laser-ultrasonic assisted grinding

The utilisation of aluminium-based silicon carbide composites (SiCp/Al) is on the rise. However, during the grinding process, where abrasive particles compel SiC particles into the aluminium matrix, stress concentration occurs near the interface between the two phases. This phenomenon results in adverse surface deformation accompanied by residual compressive stress (RCS). Consequently, a pulsed laser-ultrasonic-assisted grinding method (PL-UAG) has been proposed. The effectiveness of the method is verified through grinding experiments. Moreover, the influence process of the thermal effect of pulsed laser and the evolution mechanism of microscopic RS near the SiCp/Al two-phase interface are revealed through molecular dynamics simulation. The experimental results show that the surface RCS of SiCp/Al gradually decreases and transforms into residual tensile stress (RTS) with the increase of laser power. The RCS increases with the grinding depth. The RCS is reduced to 17 Mpa and the surface roughness Sa value is reduced to 0.110 μm with the laser power is 40 W and the grinding depth is 1 μm. The simulation results show that the energy dissipation process of pulsed laser helps to reduce thermal deformation and damage. The laser-induced thermal effect reduces the deformation of the aluminium matrix caused by particle extrusion and inhibits the formation and slip of dislocations in the aluminium matrix. In addition, the ultrasonic shock effect effectively weakens the RTS generated by the laser.

Energy Dissipation of Fast Electrons in Polymethylmethacrylate: Toward a Universal Curve for Electron-Beam Attenuation in Solids between ∼0 eV and Relativistic Energies.

Spectroscopy of correlated electron pairs was employed to investigate the energy dissipation process, as well as the transport and the emission of low-energy electrons on a polymethylmethacrylate surface, providing secondary electron spectra causally related to the energy loss of the primary. Two groups are identified in the cascade of slow electrons, corresponding to different stages in the energy dissipation process. The characteristic lengths for attenuation due to collective excitations and momentum relaxation are quantified for both groups and are found to be distinctly different: λ_{1}=(12±2) Å and λ_{2}=(62±11) Å. The results strongly contradict the commonly employed model of exponential attenuation with the electron inelastic mean free path as characteristic length, but they essentially agree with a theory used for decades in astrophysics and neutron transport, albeit with characteristic lengths expressed in units of angstroms rather than light-years.