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Search Results (1,358)

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Keywords = non-equilibrium systems

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18 pages, 17519 KB  
Article
Sodium-Oxide Fluxed Slag Design, Phase Chemistry and Thermochemistry Calculations for Aluminium Recycling from Aluminothermic Reduction of Manganese Ore
by Theresa Coetsee and Frederik De Bruin
Crystals 2026, 16(6), 401; https://doi.org/10.3390/cryst16060401 (registering DOI) - 20 Jun 2026
Abstract
A novel sodium-oxide-fluxed slag is applied in the aluminothermic reduction of manganese ore. The slag’s high Al2O3 solubility facilitates the recycling of Al2O3 through hydrometallurgical processes, where NaAlO2 serves as a water-leachable compound. Aluminothermic reduction is [...] Read more.
A novel sodium-oxide-fluxed slag is applied in the aluminothermic reduction of manganese ore. The slag’s high Al2O3 solubility facilitates the recycling of Al2O3 through hydrometallurgical processes, where NaAlO2 serves as a water-leachable compound. Aluminothermic reduction is gaining renewed interest as an alternative processing route for the circular economy. In addition, CO2 emissions in aluminium production via the electrochemical Hall–Héroult process can be reduced if the process electricity is sourced from non-fossil fuels. The unique Na2O-fluxed MnO2 ore formulation includes a small quantity of carbon reductant to ensure rapid pre-reduction to MnO. This approach negates the need for a pre-roasting step. Feed mixture variations with different collector metal additions (Si, Cr, Cu) were made to improve alloy–slag separation efficiency. The collector metals may influence the chemistry of the slag. This work compares the phase chemistry of slags formed during aluminothermic reduction to equilibrium phase chemistries calculated for the Na2O-SiO2-Al2O3-MnO-CaO system. The slag phase morphology consists of distinct alumina-rich strands (1.5% to 2.1%) embedded within a Na2O-SiO2-Al2O3-MnO-CaO glass matrix. The alumina-rich strands appear molten, indicating that the processing temperatures were higher than their liquidus temperatures (1537 °C to 1655 °C), as high as 1921 °C and 2053 °C. These findings contribute to sustainable practices in the circular economy through the production of low-carbon ferro-manganese complex alloys. Full article
(This article belongs to the Section Inorganic Crystalline Materials)
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20 pages, 631 KB  
Article
Developing ‘Integral GenAI Innovation Ecosystems’ in the Chinese Higher Education Context
by Ken Spours and Liying Rong
Systems 2026, 14(6), 703; https://doi.org/10.3390/systems14060703 (registering DOI) - 19 Jun 2026
Abstract
This article provides the theoretical foundation for upcoming primary research on the formation of ‘integral generative AI (GenAI) innovation ecosystems’ in the Chinese higher education context. Based on an adaptation of Gramsci’s idea of the ‘integral state’, which informs the move beyond Western [...] Read more.
This article provides the theoretical foundation for upcoming primary research on the formation of ‘integral generative AI (GenAI) innovation ecosystems’ in the Chinese higher education context. Based on an adaptation of Gramsci’s idea of the ‘integral state’, which informs the move beyond Western civil society/market-led and Chinese political state-led innovation ecosystem models, key features of an integral innovation GenAI ecosystem are elaborated upon. An expanded framework builds on previously published work on socialised GenAI systems comprising a multi-level approach, with particular emphasis on ‘thickened’ meso-institutional layers (e.g., supportive local investment, institutional governance frameworks and critical practices) mediating between an enhanced macro-strategic direction and upscaled micro-level practices. Theorising the institutional meso-system helps analyse challenges facing non-elite Chinese universities in moving from a ‘low-technological-baseline equilibrium’ (LTBE) constraining GenAI development to demonstrating features of GenAI innovation ecosystem ‘readiness’. The framework also draws on Lury’s ‘problem space’ research methodology, with a particular focus on its ‘within/without’ contextual factors, while also contributing a chrono-dimension to reinforce its conceptual role over time. The article concludes with an outline of a primary research strategy to investigate the challenges of building integral GenAI innovation ecosystems in Chinese higher education institutions more broadly. Full article
(This article belongs to the Section Complex Systems and Cybernetics)
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28 pages, 5059 KB  
Article
Study on the Non-Equilibrium Dynamic Phase Transition Model for Oil–Gas Systems
by Hanmin Tu, Yi Peng, Ping Guo, Zhouhua Wang, Shuoshi Wang, Yu Li, Wei Chen, Lidong Wang and Xiang Deng
Energies 2026, 19(12), 2902; https://doi.org/10.3390/en19122902 (registering DOI) - 18 Jun 2026
Abstract
In gas-condensate reservoirs, the phase behavior of reservoir fluids is inherently dynamic during pressure depletion. When the rate of external pressure decline exceeds the intrinsic relaxation rate governing phase equilibrium, the system deviates from thermodynamic equilibrium and exhibits pronounced non-equilibrium effects. These transient [...] Read more.
In gas-condensate reservoirs, the phase behavior of reservoir fluids is inherently dynamic during pressure depletion. When the rate of external pressure decline exceeds the intrinsic relaxation rate governing phase equilibrium, the system deviates from thermodynamic equilibrium and exhibits pronounced non-equilibrium effects. These transient behaviors significantly influence fluid properties; meanwhile, conventional equilibrium models neglect phase transition lag, resulting in inaccurate phase behavior and biased production predictions. In this study, a non-equilibrium dynamic phase transition model is developed to quantitatively couple the pressure depletion rate with the relaxation kinetics of the system. This model, established based on controlled non-equilibrium phase transition experiments performed on the condensate-gas fluid investigated in this work, provides an analytical framework for describing the temporal evolution of phase behavior under dynamic conditions. Model validation through integrated experimental measurements and numerical simulations shows good agreement between calculated and measured results for the studied condensate-gas system, with average relative errors below 5%. Results reveal that accelerated pressure depletion strengthens non-equilibrium effects. At a rate of 15 MPa/h, the relative volume and retrograde condensate saturation decrease by 9.09% and 5.38%, respectively, while condensate recovery improves by 13.85%. Moreover, the characteristic relaxation time toward equilibrium exhibits a strong dependence on the depletion rate, increasing as the depletion rate rises. This work provides an experimentally constrained analytical framework for describing rate-dependent non-equilibrium phase behavior during pressure depletion and for interpreting its impact on condensate recovery in the specific condensate-gas system studied. Although the governing framework may be transferable to other rate-sensitive hydrocarbon systems after fluid-specific recalibration, the parameterized analytical model and validation presented in this study are limited to the investigated condensate-gas fluid, and its applicability to other hydrocarbon fluid types remains to be evaluated in future studies. Full article
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19 pages, 3554 KB  
Article
The Bifurcation Characteristics and Dynamical Evolution Rule of Non-Isothermal Seepage Mechanical Model in Fractured Rock Mass
by Zhengzheng Cao
Processes 2026, 14(12), 1985; https://doi.org/10.3390/pr14121985 - 18 Jun 2026
Abstract
Aiming at the non-isothermal seepage phenomena in fractured rock mass, this paper conducts nonlinear dynamic research on the coupled seepage problem. Based on solid–fluid heat conduction energy equations and the mutual coupling of temperature and seepage fields, the non-isothermal seepage constitutive relation of [...] Read more.
Aiming at the non-isothermal seepage phenomena in fractured rock mass, this paper conducts nonlinear dynamic research on the coupled seepage problem. Based on solid–fluid heat conduction energy equations and the mutual coupling of temperature and seepage fields, the non-isothermal seepage constitutive relation of fractured rock is derived, and a one-dimensional nonlinear dynamic governing model is established. Theoretical analysis indicates the equilibrium solution of non-isothermal seepage is more complex than that under the isothermal condition. Numerical calculations reveal that temperature variation shifts equilibrium positions and alters the occurrence conditions of hysteresis bifurcation, verifying temperature as a core regulatory factor for seepage dynamic responses. Successive sub-relaxation iteration stability analysis demonstrates obvious differentiated convergence speeds: the seepage field converges markedly faster than the temperature field when the coupled system reaches steady state. Compared with the isothermal seepage, the temperature effect changes the location of abrupt transition points and critical threshold of control parameters, rendering fractured rock seepage systems easier to trigger abrupt structural mutation even at low rock fragmentation degrees. This study clarifies the internal nonlinear dynamic mechanism of thermal–fluid coupled seepage, identifies potential mutation risks in petroleum exploitation and geothermal development, and supplies essential theoretical support for related engineering applications. Full article
(This article belongs to the Section Petroleum and Low-Carbon Energy Process Engineering)
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32 pages, 428 KB  
Article
Green Transition in Europe: The Effectiveness of Environmental Taxes and Green Innovation in Reducing CO2 Emissions
by Jafar Babakhonov, Hilola Qosimova, Samariddin Makhmudov, Yuldoshboy Sobirov, Feruza Murodkhujayeva, Daniyor Kurbanov and Bakhodir Ruzmetov
Economies 2026, 14(6), 231; https://doi.org/10.3390/economies14060231 - 15 Jun 2026
Viewed by 190
Abstract
This study examines the determinants of carbon dioxide (CO2) emissions across 25 European Union countries over the period 2000–2021, with particular emphasis on the roles of environmental taxation and green innovation in shaping environmental sustainability. The analysis is grounded in ecological [...] Read more.
This study examines the determinants of carbon dioxide (CO2) emissions across 25 European Union countries over the period 2000–2021, with particular emphasis on the roles of environmental taxation and green innovation in shaping environmental sustainability. The analysis is grounded in ecological modernization theory, endogenous growth theory, and the Environmental Kuznets Curve hypothesis, which collectively explain the long-run and dynamic interactions between environmental policy, economic activity, structural transformation, and environmental outcomes. To ensure robust empirical inference, this study applies a comprehensive econometric framework that accounts for cross-sectional dependence, heterogeneity, non-stationarity, cointegration, and endogeneity. The empirical strategy begins with Pesaran cross-sectional dependence tests and slope heterogeneity diagnostics, followed by second-generation panel unit root tests (Pesaran CADF/CIPS) and Westerlund cointegration tests to establish the existence of long-run equilibrium relationships among the variables. Long-run coefficients are estimated using Fully Modified Ordinary Least Squares (FMOLS), Dynamic Ordinary Least Squares (DOLS), Canonical Cointegrating Regression (CCR), and Common Correlated Effects Mean Group (CCEMG) estimators. In addition, the Panel Autoregressive Distributed Lag (ARDL) model is employed to capture both short-run dynamics and long-run adjustment processes, while the System Generalized Method of Moments (System GMM) estimator addresses potential endogeneity, reverse causality, omitted variable bias, and dynamic persistence in CO2 emissions. The empirical results indicate that environmental taxation has a positive and statistically significant association with CO2 emissions, suggesting that current fiscal environmental policies in EU-25 countries may not yet be sufficiently effective in discouraging pollution-intensive activities. In contrast, green innovation is found to significantly reduce CO2 emissions, underscoring the critical role of innovation-driven environmental investment and technological progress in improving environmental quality. Economic growth, exports, and urbanization are associated with higher emissions, while imports contribute to emission reductions, reflecting differences between domestic production-based effects and trade-related structural adjustments. The System GMM results further confirm the persistence of CO2 emissions over time and validate the robustness of the long-run relationships identified by alternative estimators. Likewise, the CCEMG and Panel ARDL results support the stability and consistency of the findings under conditions of cross-sectional dependence and heterogeneous country dynamics. Taken together, the results highlight the importance of integrating environmental taxation with green innovation policies, innovation-driven investment, and sustainable trade policies to achieve long-term emission reductions in the European Union. This study contributes to the environmental economics literature by providing robust empirical evidence using second-generation panel econometric techniques that explicitly address cross-sectional dependence, heterogeneity, and endogeneity in the analysis of environmental sustainability. Full article
23 pages, 1401 KB  
Article
User-Centric Analysis of Time-Consistent Strategies in Car-Sharing and Rental Platforms
by Hui Jiang, Ye Gao, Ping Sun, Yang Yu and Hongwei Gao
Mathematics 2026, 14(12), 2140; https://doi.org/10.3390/math14122140 - 15 Jun 2026
Viewed by 71
Abstract
The rapid growth of the sharing economy has improved resource utilization in car-sharing, yet it has also sharpened market competition and diversified user demand. A persistent obstacle is the low coordination efficiency between asset-heavy operating companies and traffic-driven platforms, whose misaligned objectives waste [...] Read more.
The rapid growth of the sharing economy has improved resource utilization in car-sharing, yet it has also sharpened market competition and diversified user demand. A persistent obstacle is the low coordination efficiency between asset-heavy operating companies and traffic-driven platforms, whose misaligned objectives waste social resources. This paper uses differential game theory to analyze their dynamic coordination strategies and benefit allocation mechanisms. The Nerlove–Arrow model captures the evolution of brand goodwill, while the company’s decisions on station layout, vehicle dispatch, and pricing, together with the platform’s advertising investment, form the core decision variables in a two-party game framework linking the asset side and the traffic side. Compared with the non-cooperative Nash equilibrium, the cooperative mode removes the double marginalization effect, strengthens the investment incentives of both parties, and raises the system’s steady-state goodwill and total profit, achieving a Pareto improvement. To ground the cooperative framework in rigorous theory, we supply a verification theorem confirming that the linear candidate value functions satisfy the Hamilton–Jacobi–Bellman equations over the entire admissible state space. A formal proof of instantaneous rationality ensures that neither party falls into a cooperation trap on the horizon [0,T], and the asymptotic stability of the steady-state goodwill trajectory is established. We further endogenize the revenue-sharing coefficient through a generalized Nash bargaining model that admits asymmetric bargaining structures, and introduce a Stackelberg leadership benchmark as a third comparative regime. Sensitivity analyses with respect to the discount rate and user heterogeneity confirm the robustness of the findings. A dedicated discussion section bridges the gap between idealized parameterization and data-driven calibration, describing practical pathways via A/B testing, user churn metrics, and econometric estimation of demand parameters. The results offer a scientific decision-making reference for strategic cooperation in the car-sharing industry. Full article
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53 pages, 5818 KB  
Review
Multiscale Thermodynamic and Exergetic Assessment of Tri-Reforming of Methane for CO2 Valorization and Process Intensification
by Parisa Ebrahimi, Methene Briones Cutad, Anand Kumar and Mohammed J. Al-Marri
Energies 2026, 19(12), 2832; https://doi.org/10.3390/en19122832 - 14 Jun 2026
Viewed by 136
Abstract
Tri-reforming of methane (TRM) has emerged as a promising pathway for low-carbon syngas production by integrating steam reforming, dry reforming, and partial oxidation within a single process. This coupling enables simultaneous CH4 utilization and CO2 valorization while enabling internal heat generation [...] Read more.
Tri-reforming of methane (TRM) has emerged as a promising pathway for low-carbon syngas production by integrating steam reforming, dry reforming, and partial oxidation within a single process. This coupling enables simultaneous CH4 utilization and CO2 valorization while enabling internal heat generation and flexible adjustment of the H2/CO ratio for downstream synthesis. However, TRM performance cannot be adequately evaluated using conversion or energy efficiency alone, because the process involves complex interactions among competing reaction pathways, transport phenomena, catalyst stability, and thermodynamic irreversibility. This review provides a multiscale critical assessment of TRM from both first-law energy and second-law exergy perspectives, linking reaction-network fundamentals to reactor-level behavior and system-level performance. The literature evidence shows that although high temperatures and near-autothermal operation can enhance CH4 conversion and reduce external heat demand, these conditions may simultaneously intensify deep oxidation, hotspot formation, carbon-forming tendencies, and exergy destruction. While equilibrium analyses help define feasible operating windows, they are insufficient without kinetic modeling and reactor-scale studies that capture spatial non-uniformities and pathway competition. Across reported TRM systems, exergy destruction is consistently concentrated within the reformer, identifying the reacting core as the dominant thermodynamic bottleneck. Accordingly, the key challenge in TRM is not simply to maximize conversion but to preserve chemical work potential while maintaining syngas quality and operational stability. Viewed from this perspective, TRM is better understood as an irreversibility-aware multiscale design problem in which optimal performance depends on the integrated optimization of catalyst functionality, reactor architecture, heat management, and system-level operation. Full article
(This article belongs to the Special Issue Reforming of Methane for Hydrogen Energy and Synthesis Gas)
24 pages, 4516 KB  
Article
Analytical and Asymptotic Modeling of Coupled Transient Gas Redistribution Induced by Simultaneous Injection and Withdrawal in Transmission Pipelines
by Ahad Mammadov, Firangiz Mammadrzayeva and Ilgar G. Aliyev
Math. Comput. Appl. 2026, 31(3), 103; https://doi.org/10.3390/mca31030103 - 11 Jun 2026
Viewed by 126
Abstract
This study develops an analytical and computational framework for coupled transient gas redistribution induced by simultaneous localized injection and withdrawal in transmission pipelines. The aim is to describe source–sink interactions within a single transmission system, unlike conventional approaches that treat inflow and outflow [...] Read more.
This study develops an analytical and computational framework for coupled transient gas redistribution induced by simultaneous localized injection and withdrawal in transmission pipelines. The aim is to describe source–sink interactions within a single transmission system, unlike conventional approaches that treat inflow and outflow processes independently. The governing equations of one-dimensional non-stationary isothermal compressible gas flow are transformed into a diffusion-type formulation using Charny regularization. The pipeline is divided into three interacting regions connected through pressure-continuity and mass-flux coupling conditions. Closed-form Laplace-domain solutions are derived for the dimensionless pressure field, and a practical Laplace-domain approximation is used for computational evaluation of transient pressure profiles. The results reveal a characteristic balancing point separating injection-dominated and withdrawal-dominated regions and show rapid convergence toward a quasi-steady redistribution regime. A pressure-deviation-based objective function is introduced to evaluate hydraulic disturbance, and the optimization analysis shows that the minimum disturbance occurs under a near-balanced source–sink operating condition. The obtained pressure profiles, asymptotic behavior, and regional redistribution patterns confirm the physical consistency of the proposed model. The framework provides a mathematically interpretable basis for analyzing coupled redistribution dynamics, hydraulic stabilization, and asymptotic equilibrium in gas transmission systems. Full article
(This article belongs to the Section Engineering)
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19 pages, 26846 KB  
Article
Numerical Investigation of Stall Flutter of a Pitching Airfoil at Low Reynolds Number
by Maria Adele Cecchini, Giulio Soldati, Peter Jordan and Sergio Pirozzoli
Fluids 2026, 11(6), 149; https://doi.org/10.3390/fluids11060149 - 11 Jun 2026
Viewed by 120
Abstract
The present work investigates fluid–structure instabilities and flow-induced oscillations of a pitching NACA0012 airfoil through numerical simulations. The flow is modeled using the compressible Navier–Stokes equations in a non-inertial rotating reference frame, while the structural dynamics are represented by a torsional spring–mass–damper system. [...] Read more.
The present work investigates fluid–structure instabilities and flow-induced oscillations of a pitching NACA0012 airfoil through numerical simulations. The flow is modeled using the compressible Navier–Stokes equations in a non-inertial rotating reference frame, while the structural dynamics are represented by a torsional spring–mass–damper system. The analysis focuses on the effects of reduced velocity, equilibrium angle of attack, and elastic axis position on the aeroelastic behavior at low Reynolds number (Re=1000). Particular attention is devoted to characterizing the transition from vortex-shedding-dominated oscillations to fully developed limit-cycle oscillations and to assessing its sensitivity to aerodynamic and structural parameters. The results show a transition from steady flow to vortex shedding and, at higher reduced velocities, to limit-cycle oscillations. Increasing the equilibrium angle of attack promotes an earlier onset of instability and stronger aerodynamic forcing, while moving the elastic axis downstream has a similar destabilizing effect due to the larger aerodynamic moment arm (up to approximately 20% reduction of the critical reduced velocity). The nature of the transition is found to depend strongly on the equilibrium angle of attack, with distinct behaviors observed at low and high incidence. Frequency analysis highlights the progressive coupling between fluid and structural dynamics: vortex shedding dominates in the weakly coupled regime, whereas the structural frequency governs the response in the limit-cycle regime. The study provides a consistent description of the mechanisms driving flow-induced oscillations and of the parameters controlling aeroelastic stability. Full article
(This article belongs to the Special Issue Industrial CFD and Fluid Modelling in Engineering, 4th Edition)
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23 pages, 4985 KB  
Article
Engineering Performance of Expansive Soil Stabilized with Cement and Montmorillonite Adsorption Modifier
by Aiping Chen, Yong Cao, Wei Qi, Lihong Shu, Feiyang Liu, Ge Yang, Jianbiao Du and Tengfei Wang
Materials 2026, 19(12), 2522; https://doi.org/10.3390/ma19122522 - 11 Jun 2026
Viewed by 179
Abstract
To enhance the strength and water stability of stabilized expansive soil, this study investigates the use of cement, montmorillonite adsorption modifier (MAM), and their composite system. Laboratory tests evaluated compaction characteristics, swell–shrink behavior, and mechanical performance. The results show that MAM more effectively [...] Read more.
To enhance the strength and water stability of stabilized expansive soil, this study investigates the use of cement, montmorillonite adsorption modifier (MAM), and their composite system. Laboratory tests evaluated compaction characteristics, swell–shrink behavior, and mechanical performance. The results show that MAM more effectively regulates compaction by reducing optimum water content and increasing maximum dry density; 6% MAM increases maximum dry density by ≈0.04 g/cm3 and reduces optimum water content by ≈2%. In terms of swell–shrink behavior, MAM reduces both swelling and linear shrinkage more effectively than cement. The incorporation of 5% MAM reduces the free swelling ratio by 40% and the equilibrium moisture absorption by 2.7%, lowering the swelling classification to non-expansive. Furthermore, 5% MAM decreases the unloaded and loaded swelling ratio by 14.7% and 5%, respectively, while increasing MAM from 2% to 6% further reduces linear shrinkage by 1.12%. Cement significantly enhances compressive strength, with 7–28 d values reaching 2.2–2.7 times those of untreated soil at 9% content; however, its water stability under wet–dry cycles is limited. In contrast, the cement–MAM composite system achieves balanced improvement by simultaneously suppressing swelling and enhancing both strength and water stability. These findings provide a reference for the treatment and engineering application of expansive soils. Full article
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25 pages, 877 KB  
Perspective
Rethinking Immovable Cultural Heritage Within One Health: An Ecophysical Perspective
by Marco Casazza
Environments 2026, 13(6), 329; https://doi.org/10.3390/environments13060329 - 9 Jun 2026
Viewed by 394
Abstract
Immovable cultural heritage, including archaeological sites, historic buildings, and long-standing landscape structures, is typically interpreted through historical, aesthetic, and identity-based perspectives. This paper proposes an alternative reading, situating heritage within the broader context of coupled environmental, biological, and human systems. Grounded in non-equilibrium [...] Read more.
Immovable cultural heritage, including archaeological sites, historic buildings, and long-standing landscape structures, is typically interpreted through historical, aesthetic, and identity-based perspectives. This paper proposes an alternative reading, situating heritage within the broader context of coupled environmental, biological, and human systems. Grounded in non-equilibrium thermodynamics and system ecology, the study advances an ecophysical perspective in which heritage is understood as a persistent structural and informational component of the human niche. Drawing on evidence from building physics, landscape ecology, environmental psychology, and health-related research, this paper discusses the scientific plausibility of heritage-mediated effects, including environmental buffering, habitat stabilization, and cognitive and physiological regulation. These heterogeneous processes are reinterpreted within a unified conceptual framework, HEROS (HERitage One Health System), which links observable indicators to underlying mechanisms of organization and dissipation. A simplified stock–flow formulation, consistent with ecophysics and system ecology literature, is introduced to illustrate how heritage may influence dissipation across environmental, animal, and human subsystems. Rather than presenting a fully operational model, this perspective aims to reposition heritage within One Health and sustainability frameworks, highlighting its potential role in supporting system stability, resilience, and long-term continuity. Full article
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14 pages, 270 KB  
Article
Expectation Identities for Dynamical Systems: A Classical Analog of the Ehrenfest Theorem
by Abiam Tamburrini, Sergio Davis, Diego González and Pablo S. Moya
Entropy 2026, 28(6), 654; https://doi.org/10.3390/e28060654 - 9 Jun 2026
Viewed by 190
Abstract
In this work, we formulate a systematic expectation-value framework for dynamical systems whose probability densities evolve according to linear partial differential equations, such as the Fokker-Planck and Liouville equations. The approach is based on expectation-calculus identities associated with the Fluctuation-Dissipation Theorem and the [...] Read more.
In this work, we formulate a systematic expectation-value framework for dynamical systems whose probability densities evolve according to linear partial differential equations, such as the Fokker-Planck and Liouville equations. The approach is based on expectation-calculus identities associated with the Fluctuation-Dissipation Theorem and the Conjugate Variables Theorem, allowing the derivation of evolution equations directly for arbitrary observables and fluctuations without explicitly solving the full probability-density equation. The resulting relations provide a classical Ehrenfest-type formulation for observable dynamics and fluctuations under linear probability-density evolution. While the resulting equations are not closed in general, since they typically involve higher-order moments, correlations, or derivatives, the formalism offers a unified operational framework for studying observable dynamics under suitable approximations or closure assumptions. We illustrate the procedure with examples involving Fokker–Planck and Liouville dynamics and discuss the scope, limitations, and possible applications of the framework in nonequilibrium statistical mechanics. In particular, we emphasize that the method is intended as a systematic observable-based formulation for systems governed by linear evolution equations, rather than as a universal closure scheme for arbitrary nonequilibrium dynamics. Full article
(This article belongs to the Section Statistical Physics)
28 pages, 2857 KB  
Article
Entropy Production from Spin–Vibrational Coupling in Endohedral-Fullerene Qubits Encapsulated in Suspended Carbon Nanotubes
by Cristian Staii
Entropy 2026, 28(6), 646; https://doi.org/10.3390/e28060646 - 8 Jun 2026
Viewed by 116
Abstract
Hybrid carbon nanotube–fullerene architectures provide a controllable setting in which to study irreversibility and information flow in strongly structured quantum environments. We analyze entropy generation in a platform where paramagnetic endohedral fullerenes (PEFs), such as N@C60 and P@C60, are encapsulated [...] Read more.
Hybrid carbon nanotube–fullerene architectures provide a controllable setting in which to study irreversibility and information flow in strongly structured quantum environments. We analyze entropy generation in a platform where paramagnetic endohedral fullerenes (PEFs), such as N@C60 and P@C60, are encapsulated inside a suspended carbon nanotube (CNT) resonator, such that selected multi-level PEF spin states define an effective qubit coupled to quantized CNT flexural modes. Motivated by prior work on fullerene-filled CNTs, on spin–phonon manipulation in suspended nanotubes, and on exact phase-space propagators for damped driven oscillators, we formulate a hybrid open-system description that combines a driven quantum Brownian description of the CNT resonator with an effective Jaynes–Cummings type spin–vibrational interaction. The resonator dynamics are represented in phase space through the Wigner function, whose time evolution can be written analytically in terms of the initial Wigner distribution and a Gaussian propagator. This representation makes it possible to separate drive-induced phase space displacement, diffusion, and damping, and to connect these features directly to entropy flow. The coupled spin–mechanical dynamics are then embedded in a Lindblad quantum master equation that includes mechanical damping, spin relaxation, pure dephasing, and thermally activated excitation channels. Within this framework we derive the entropy balance equation—identifying entropy flux and non-negative entropy production—and examine how hybridization between the molecular spin and the nanotube vibration redistributes irreversibility between coherent exchange and dissipative channels. We show that spin–phonon coupling enhanced by a magnetic field gradient, resonant driving, and moderate thermal occupation can produce identifiable crossovers between entropy–production regimes dominated by the oscillator and those dominated by the spin. The resulting framework provides a quantitative basis for using CNT–PEF hybrids as nanoscale platforms for studying nonequilibrium quantum thermodynamics, decoherence, and information loss in structured vibrational environments. Full article
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15 pages, 1087 KB  
Article
Periodic Temperature Fluctuations as an Energy Source for RNA Evolution
by Christian Mayer
Life 2026, 16(6), 958; https://doi.org/10.3390/life16060958 - 5 Jun 2026
Viewed by 250
Abstract
The effect of periodic temperature variation on short and partially matching interacting RNA strands is demonstrated using three pairs of competing RNA duplexes as simplified model systems. They represent random base-pairing interactions between short RNA chains at very early stages of RNA evolution. [...] Read more.
The effect of periodic temperature variation on short and partially matching interacting RNA strands is demonstrated using three pairs of competing RNA duplexes as simplified model systems. They represent random base-pairing interactions between short RNA chains at very early stages of RNA evolution. The molecular interaction kinetics are simulated based on the experimental thermodynamic data obtained by M. E. Christiansen et al. and on the Eyring theory. The simulated time developments demonstrate the impact of shifting reaction kinetics and a state of continuous non-equilibrium. The product mix that develops slowly at low temperatures releases chemical (free) energy quickly at high temperatures, and the product mix that develops quickly at high temperatures releases chemical (free) energy slowly at low temperatures. Regarding heat flow and energy storage, the chosen model system represents a generalized Carnot engine, similar to most comparable reactions during RNA evolution. Its action provides a perpetual driving force for selection processes, creating ideal conditions for an ongoing molecular evolution. Full article
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21 pages, 13943 KB  
Article
Tunable Dynamics of Memristive Chaotic Systems and Its Application in Water Facility Image Encryption
by Xuehui Lu, Tingting Wang, Hongzhi Wang, Shaohua Zhang and Cong Wang
Mathematics 2026, 14(11), 1945; https://doi.org/10.3390/math14111945 - 2 Jun 2026
Viewed by 147
Abstract
Nonlinear memristors frequently contribute to enhancing the dynamical richness of chaotic systems, yet their complexity and flexibility have often been overlooked. In this work, a piecewise non-smooth threshold memristor model is proposed, which is coupled as a nonlinear term into the Sprott C [...] Read more.
Nonlinear memristors frequently contribute to enhancing the dynamical richness of chaotic systems, yet their complexity and flexibility have often been overlooked. In this work, a piecewise non-smooth threshold memristor model is proposed, which is coupled as a nonlinear term into the Sprott C system, yielding a novel four-dimensional memristive chaotic dynamical system. From a theoretical perspective, stability analysis reveals that unstable index-2 saddle-focus equilibrium points are governed by the memristive piecewise parameter, and the topological invariance of the system is verified. In numerical simulations, bifurcation diagrams, Lyapunov exponents, and phase portraits are employed to reveal the mechanism of novel tunable chaotic dynamics. The results demonstrate that memristive coupling strength can induce the system to generate double-scroll, double-wing, and double-butterfly chaotic attractors; the piecewise parameter of the memristor can control the system to produce multi-structure attractors with expanded quantity, and the initial condition of the memristor can regulate the system to generate offset-boosted chaotic attractors. Finally, the novel tunable dynamics is applied to water facility image encryption. Experimental results demonstrate that the proposed algorithm possesses a key space of 2100, a correlation coefficient of only 0.0002, and information entropy close to the ideal value of eight. The NPCR and UACI reach 99.6161% and 33.4669%, respectively, the key sensitivity is up to 1016, and all p-values from the NIST tests are greater than 0.01, confirming that the algorithm achieves excellent security performance. Full article
(This article belongs to the Special Issue Chaotic Systems and Their Applications, 3rd Edition)
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