Search Results (32)

Markov chain perturbation theory is a rapidly developing subfield of the theory of stochastic processes. This review outlines emerging applications of this theory in the analysis of stochastic models of chemical reactions, with a particular focus on biochemistry and molecular biology. We begin by discussing the general problem of approximate modeling in stochastic chemical kinetics. We then briefly review some essential mathematical results pertaining to perturbation bounds for continuous-time Markov chains, emphasizing the relationship between robustness under perturbations and the rate of exponential convergence to the stationary distribution. We illustrate the use of these results to analyze stochastic models of biochemical reactions by providing concrete examples. Particular attention is given to fundamental problems related to approximation accuracy in model reduction. These include the partial thermodynamic limit, the irreversible-reaction limit, parametric uncertainty analysis, and model reduction for linear reaction networks. We conclude by discussing generalizations and future developments of these methodologies, such as the need for time-inhomogeneous Markov models. Full article

Mathematical models and methods serve as fundamental tools for studying ice-related phenomena in the Yellow River. River ice is driven and constrained by hydrometeorological and geographical conditions, creating a complex system. Regarding the Yellow River, there are some uncertainties that manifest in unique features in this context, including ice–water–sediment mixed transport processes and the distribution of sediment both within the ice and on its surface. These distinctive characteristics are considered to different degrees across different scales. Mathematical models for Yellow River ice developed over the past few decades not only encompass models for the large-scale deterministic evolution of river ice formation and melting, but also uncertainty parameter schemes for deterministic mathematical models reflecting the Yellow River’s particular ice-related characteristics. Moreover, there are modern mathematical results quantitatively describing these characteristics with uncertainty, allowing for a better understanding of the unique ice phenomena in the Yellow River. This review summarizes (a) universal equations established according to thermodynamic and hydrodynamic principles in river ice mathematical models, as well as (b) uncertainty sources caused by the river’s characteristics, ice properties, and hydrometeorological conditions, embedded in parametric schemes reflecting the Yellow River’s ice. The intractable uncertainty-related problems in space–sky–ground telemetric image segmentation and the current status of mathematical processing methods are reviewed. In particular, the current status and difficulties faced by various mathematical models in terms of predicting the freeze-up and break-up times, the formation of ice jams and dams, and the early warning of ice disasters are presented. This review discusses the prospects related to the uncertainties in research results regarding the simulation and prediction of Yellow River ice while also exploring potential future trends in research related to mathematical methods for uncertain problems. Full article

The significance of the chiral symmetry restoration is studied by considering the role of the modification of the nucleon mass in nuclear medium at finite density and temperature. Using the Korea-IBS-Daegu-SKKU density functional theory, we can create models that have an identical nuclear matter equation of state but different isoscalar and isovector effective masses at zero temperature. The effect of the effective mass becomes transparent at non-zero temperatures, and it becomes more important as temperature increases. The role of the effective mass is examined thoroughly by calculating the dependence of thermodynamic variables such as free energy, internal energy, entropy, pressure and chemical potential on density, temperature and proton fraction. We find that sensitivity to the isoscalar effective mass is several times larger than that of the isovector effective mass, so the uncertainties arising from the effective mass are dominated by the isoscalar effective mass. In the analysis of the relative uncertainty, we obtain that the maximum uncertainty is less than 2% for free energy, internal energy and chemical potential, but it amounts to 20% for pressure. Entropy shows a behavior completely different from the other four variables that the uncertainty is about 40% at the saturation density and increases monotonically as density increases. The effect of the uncertainty to properties of physical systems is investigated with the proto-neutron star. It is shown that temperature depends strongly on the effective mass at a given density, and substantial swelling of the radius occurs due to the finite temperature. The equation of state is stiffer with smaller isoscalar effective mass, so the effect of the effective mass appears clearly in the mass–radius relation of the proto-neutron star, where a larger radius corresponds to a smaller effective mass. Full article

In previous work, we investigated thermodynamic processes in systems at the mesoscopic level where traditional thermodynamic descriptions (macroscopic or microscopic) may not be fully adequate. The key result is that entropy in such systems does not change continuously, as in macroscopic systems, but rather in discrete steps characterized by the quantization constant $\beta$. This quantization reflects the underlying discrete nature of the collision process in low-dimensional systems and the essential role played by thermodynamic fluctuations at this scale. Thermodynamic variables conjugate to the forces, along with Glansdorff–Prigogine’s dissipative variable can be discretized, enabling a mesoscopic-scale formulation of canonical commutation rules (CCRs). In this framework, measurements correspond to determining the eigenvalues of operators associated with key thermodynamic quantities. This work investigates the quantization parameter $\beta$ in the CCRs using a nano-gas model analyzed through classical statistical physics. Our findings suggest that $\beta$ is not an unknown fundamental constant. Instead, it emerges as the minimum achievable value derived from optimizing the uncertainty relation within the framework of our model. The expression for $\beta$ is determined in terms of the ratio $\chi$, which provides a dimensionless number that reflects the relative scales of volume and mass between entities at the Bohr (atomic level) and the molecular scales. This latter parameter quantifies the relative influence of quantum effects versus classical dynamics in a given scattering process. Full article

As large-span structures, reticulated shells are widely used in large-scale public building and act as emergency shelters in the event of sudden disasters. However, spatial reticulated shells are dynamic-sensitive structures; the effect of the initial structural damage on dynamic stability should be considered. In this study, a new nonlinear dynamic model of cylindrical reticulated shells with initial damage is proposed to investigate the effect of initial damage accurately. Firstly, the damage constitutive relations of the building steels are built based on the irreversible thermodynamic theory; furthermore, its fundamental equations are obtained using simulated shell methods. Then, the nonlinear vibration differential equations with damage are obtained and studied with support. Meanwhile, the nonlinear natural vibration frequency with initial damage is derivatized. After that, a bifurcation problem with initial damage is studied by using Flouquet Index, and the dynamic stability state at the equilibrium point is analyzed in depth. It is found that the local dynamic stability of the system is determined via its initial condition, geometric parameters, and initial damage. Moreover, the initial damage dominates over other influence factors due to its strong randomness and uncertainty for the same structure. The damage accumulation results in the transition of the equilibrium point. In addition, the nonlinear natural vibration frequency decreases to zero with the accumulation of the damage reaching 0.618; the local stability of cylindrical reticulated shells fails and they even lose whole stability. This study provides a theoretical foundation for the future investigation of whole stability with initial damage. Full article

We study the work fluctuations in ergotropic heat engines, namely two-stroke quantum Otto engines where the work stroke is designed to extract the ergotropy (the maximum amount of work by a cyclic unitary evolution) from a couple of quantum systems at canonical equilibrium at two different temperatures, whereas the heat stroke thermalizes back the systems to their respective reservoirs. We provide an exhaustive study for the case of two qutrits whose energy levels are equally spaced at two different frequencies by deriving the complete work statistics. By varying the values of temperatures and frequencies, only three kinds of optimal unitary strokes are found: the swap operator $U_1$, an idle swap $U_2$ (where one of the qutrits is regarded as an effective qubit), and a non-trivial permutation of energy eigenstates $U_3$, which indeed corresponds to the composition of the two previous unitaries, namely $U_3=U_2{U}_1$. While $U_1$ and $U_2$ are Hermitian (and hence involutions), $U_3$ is not. This point has an impact on the thermodynamic uncertainty relations (TURs), which bound the signal-to-noise ratio of the extracted work in terms of the entropy production. In fact, we show that all TURs derived from a strong detailed fluctuation theorem are violated by the transformation $U_3$. Full article

Experimental and theoretical results about entropy limits for macroscopic and single-particle systems are reviewed. All experiments confirm the minimum system entropy $S⩾kln2$. We clarify in which cases it is possible to speak about a minimum system entropy$kln2$ and in which cases about a quantum of entropy. Conceptual tensions with the third law of thermodynamics, with the additivity of entropy, with statistical calculations, and with entropy production are resolved. Black hole entropy is surveyed. Claims for smaller system entropy values are shown to contradict the requirement of observability, which, as possibly argued for the first time here, also implies the minimum system entropy $kln2$. The uncertainty relations involving the Boltzmann constant and the possibility of deriving thermodynamics from the existence of minimum system entropy enable one to speak about a general principle that is valid across nature. Full article

Many dynamical systems consist of multiple, co-evolving subsystems (i.e., they have multiple degrees of freedom). Often, the dynamics of one or more of these subsystems will not directly depend on the state of some other subsystems, resulting in a network of dependencies governing the dynamics. How does this dependency network affect the full system’s thermodynamics? Prior studies on the stochastic thermodynamics of multipartite processes have addressed this question by assuming that, in addition to the constraints of the dependency network, only one subsystem is allowed to change state at a time. However, in many real systems, such as chemical reaction networks or electronic circuits, multiple subsystems can—or must—change state together. Here, we investigate the thermodynamics of such composite processes, in which multiple subsystems are allowed to change state simultaneously. We first present new, strictly positive lower bounds on entropy production in composite processes. We then present thermodynamic uncertainty relations for information flows in composite processes. We end with strengthened speed limits for composite processes. Full article

In a recent paper, we have shown how in Madelung’s hydrodynamic formulation of quantum mechanics, the uncertainties are related to the phase and amplitude of the complex wave function. Now we include a dissipative environment via a nonlinear modified Schrödinger equation. The effect of the environment is described by a complex logarithmic nonlinearity that vanishes on average. Nevertheless, there are various changes in the dynamics of the uncertainties originating from the nonlinear term. Again, this is illustrated explicitly using generalized coherent states as examples. With particular focus on the quantum mechanical contribution to the energy and the uncertainty product, connections can be made with the thermodynamic properties of the environment. Full article

The solutions for a combination of the isotropic harmonic oscillator plus the inversely quadratic potentials and a combination of the pseudo-harmonic with inversely quadratic potentials has not been reported, though the individual potentials have been given attention. This study focuses on the solutions of the combination of the potentials, as stated above using the parametric Nikiforov–Uvarov (PNV) as the traditional technique to obtain the energy equations and their corresponding unnormalized radial wave functions. To deduce the application of these potentials, the expectation values, the uncertainty in the position and momentum, and the thermodynamic properties, such as the mean energy, entropy, heat capacity, and the free mean energy, are also calculated via the partition function. The result shows that the spectra for the PHIQ are higher than the spectra for the IHOIQ. It is also shown that the product of the uncertainties obeyed the Heisenberg uncertainty relation/principle. Finally, the thermal properties of the two potentials exhibit similar behaviours. Full article

A universal upper limit on the entropy contained in a localized quantum system of a given size and total energy is expressed by the so-called Bekenstein bound. In a previous paper [Buoninfante, L. et al. 2022], on the basis of general thermodynamic arguments, and in regimes where the equipartition theorem still holds, the Bekenstein bound has been proved practically equivalent to the Heisenberg uncertainty relation. The smooth transition between the Bekenstein bound and the holographic bound suggests a new pair of canonical non-commutative variables, which could be thought to hold in strong gravity regimes. Full article

Thermodynamic uncertainty relations (TURs) represent one of the few broad-based and fundamental relations in our toolbox for tackling the thermodynamics of nonequilibrium systems. One form of TUR quantifies the minimal energetic cost of achieving a certain precision in determining a nonequilibrium current. In this initial stage of our research program, our goal is to provide the quantum theoretical basis of TURs using microphysics models of linear open quantum systems where it is possible to obtain exact solutions. In paper [Dong et al., Entropy 2022, 24, 870], we show how TURs are rooted in the quantum uncertainty principles and the fluctuation–dissipation inequalities (FDI) under fully nonequilibrium conditions. In this paper, we shift our attention from the quantum basis to the thermal manifests. Using a microscopic model for the bath’s spectral density in quantum Brownian motion studies, we formulate a “thermal” FDI in the quantum nonequilibrium dynamics which is valid at high temperatures. This brings the quantum TURs we derive here to the classical domain and can thus be compared with some popular forms of TURs. In the thermal-energy-dominated regimes, our FDIs provide better estimates on the uncertainty of thermodynamic quantities. Our treatment includes full back-action from the environment onto the system. As a concrete example of the generalized current, we examine the energy flux or power entering the Brownian particle and find an exact expression of the corresponding current–current correlations. In so doing, we show that the statistical properties of the bath and the causality of the system+bath interaction both enter into the TURs obeyed by the thermodynamic quantities. Full article

Thermodynamic uncertainty principles make up one of the few rare anchors in the largely uncharted waters of nonequilibrium systems, the fluctuation theorems being the more familiar. In this work we aim to trace the uncertainties of thermodynamic quantities in nonequilibrium systems to their quantum origins, namely, to the quantum uncertainty principles. Our results enable us to make this categorical statement: For Gaussian systems, thermodynamic functions are functionals of the Robertson-Schrödinger uncertainty function, which is always non-negative for quantum systems, but not necessarily so for classical systems. Here, quantum refers to noncommutativity of the canonical operator pairs. From the nonequilibrium free energy, we succeeded in deriving several inequalities between certain thermodynamic quantities. They assume the same forms as those in conventional thermodynamics, but these are nonequilibrium in nature and they hold for all times and at strong coupling. In addition we show that a fluctuation-dissipation inequality exists at all times in the nonequilibrium dynamics of the system. For nonequilibrium systems which relax to an equilibrium state at late times, this fluctuation-dissipation inequality leads to the Robertson-Schrödinger uncertainty principle with the help of the Cauchy-Schwarz inequality. This work provides the microscopic quantum basis to certain important thermodynamic properties of macroscopic nonequilibrium systems. Full article

In situ aircraft measurements of the sizes and concentrations of liquid cloud droplets and ice crystals with maximum dimensions (D_max) less than ~50 μm have been measured mainly using forward scattering probes over the past half century. The operating principle of forward scattering probes is that the measured intensity of light scattered by a cloud particle at specific forward scattering angles can be related to the size of that particle assuming the shape and thermodynamic phase of the target are known. Current forward-scattering probes assume spherical liquid cloud droplets and use the Lorenz–Mie theory to convert the scattered light to particle size. Uncertainties in sizing ice crystals using forward scattering probes are unavoidable since the single-scattering properties of ice crystals differ from those of spherical liquid cloud droplets and because their shapes can vary. In this study, directional scattering intensities of four different aspect ratios (ARs = 0.25, 0.50, 1.00, and 2.00) of hexagonal ice crystals with random orientations and of spherical liquid cloud droplets were calculated using the discrete dipole approximation (i.e., ADDA) and Lorenz–Mie code, respectively, to quantify the errors in sizing small ice crystals and cloud droplets using current forward scattering probes and to determine the ranges of optimal scattering angles that would be used in future forward scattering probes. The calculations showed that current forward scattering probes have average 5.0% and 17.4% errors in sizing liquid cloud droplets in the forward (4–12°) and backward (168–176°) direction, respectively. For measurements of hexagonal ice crystals, average sizing errors were 42.1% (23.9%) in the forward (backward) direction and depended on the ARs of hexagonal ice crystals, which are larger than those for liquid cloud droplets. A newly developed size conversion table based on the calculated single-scattering properties of hexagonal ice crystals using the ADDA reduced the sizing errors for the hexagonal ice crystals down to 14.2% (21.9%) in the forward (backward) direction. This study is a purely theoretical examination of the operating principle of forward scattering probes and there are several limitations, such as assumed hexagonal ice crystals with smooth surfaces and random orientations. Full article

The traditional field-based measurements of carbon dioxide (pCO₂) for inland waters are a snapshot of the conditions on a particular site, which might not adequately represent the pCO₂ variation of the entire lake. However, these field measurements can be used in the pCO₂ remote sensing modeling and verification. By focusing on inland waters (including lakes, reservoirs, rivers, and streams), this paper reviews the temporal and spatial variability of pCO₂ based on published data. The results indicate the significant daily and seasonal variations in pCO₂ in lakes. Rivers and streams contain higher pCO₂ than lakes and reservoirs in the same climatic zone, and tropical waters typically exhibit higher pCO₂ than temperate, boreal, and arctic waters. Due to the temporal and spatial variations of pCO₂, it can differ in different inland water types in the same space-time. The estimation of CO₂ fluxes in global inland waters showed large uncertainties with a range of 1.40–3.28 Pg C y⁻¹. This paper also reviews existing remote sensing models/algorithms used for estimating pCO₂ in sea and coastal waters and presents some perspectives and challenges of pCO₂ estimation in inland waters using remote sensing for future studies. To overcome the uncertainties of pCO₂ and CO₂ emissions from inland waters at the global scale, more reliable and universal pCO₂ remote sensing models/algorithms will be needed for mapping the long-term and large-scale pCO₂ variations for inland waters. The development of inverse models based on dissolved biogeochemical processes and the machine learning algorithm based on measurement data might be more applicable over longer periods and across larger spatial scales. In addition, it should be noted that the remote sensing-retrieved pCO₂/the CO₂ concentration values are the instantaneous values at the satellite transit time. A major technical challenge is in the methodology to transform the retrieved pCO₂ values on time scales from instant to days/months, which will need further investigations. Understanding the interrelated control and influence processes closely related to pCO₂ in the inland waters (including the biological activities, physical mixing, a thermodynamic process, and the air–water gas exchange) is the key to achieving remote sensing models/algorithms of pCO₂ in inland waters. This review should be useful for a general understanding of the role of inland waters in the global carbon cycle. Full article