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The dynamics of dissipative fluids in Eulerian variables may be derived from an algebra of Leibniz brackets of observables, the metriplectic algebra, that extends the Poisson algebra of the frictionless limit of the sytem via a symmetric semidefinite component, that generates dissipative forces. The metriplectic algebra includes the conserved total Hamiltonian $H$, generating the non-dissipative part of dynamics, and the entropy S of those microscopic degrees of freedom draining energy irreversibly, that generates dissipation. This S is a Casimir invariant of the Poisson algebra to which the metriplectic algebra reduces in the frictionless limit. The role of S is as paramount as that of H, but this fact may be underestimated in the Eulerian formulation because S is not the only Casimir of the symplectic non-canonical part of the algebra. Instead, when the dynamics of the non-ideal fluid is written through the parcel variables of the Lagrangian formulation, the fact that entropy is symplectically invariant appears to be clearly related to its the microscopic degrees of freedom of the fluid, that do not participate at all to the symplectic canonical part of the algebra (which, indeed, involves and evolves only the macroscopic degrees of freedom of the fluid parcel).

In the intention of its creator, Maxwell's demon was thought to be an intelligent being able to perform work at the expense of the entropy reduction of a closed operating system. The perplexing notion of the demon's intelligence was formalized in terms of the information processing by Szilard, who pointed out that, in order for the system to be consistent with the second law of thermodynamics, the entropy reduction should be compensated for by, at least, the same entropy increase related to the demon's information on the operating system state. A mechanical (non-informational) formulation of the problem was proposed by Smoluchowski and popularized by Feynman as the ratchet and pawl machine, which can operate only in agreement with the second law. A. F. Huxley and followers adopted this way of thinking to propose numerous thermal ratchet mechanisms of the biological molecular machines action. Here we show that, because of the necessary energy dissipation, both for the thermal ratchet and the concurrent power stroke models, no entropy reduction takes place. It is possible only for protein machine models with a number of conformational states organized in a network of transitions that allow the performance of work in a variety of ways. For such models, no information processing is necessary for the generalized fluctuation theorem to be satisfied. A computer realization is investigated of the conformational network, displaying, like networks of the systems biology, a transition from the fractal organization on a small length-scale to the small-world organization on the large length-scale. This model is able to explain a surprising observation to Yanagida and co-workers that the myosin II head can take several steps along the actin filament per ATP molecule hydrolysed. From a broader perspective, of especial importance could be the supposition that the mechanism of the action of small G-proteins, having a common ancestor with the myosin II, is, after a malignant transformation, similar. Presumably, also transcription factors look actively and not passively for their target on the genome.

Using mathematical modeling to address large scale problems in the world of biological regulatory networks has become increasingly necessary given the sheer quantity of data made available by improved technology. In the most general sense, modeling approaches can be thought of as being either quantitative or qualitative. Quantitative methods such as ordinary differential equations or the chemical master equation are widespread in the literature; when the model is well developed, the detail therein can be incredibly informative. However, they require an in depth knowledge of the reaction kinetics and generally fail as the problem size grows. The alternative approach, qualitative models, does not possess the same amount of detail but captures the essential dynamics of the system. Gene regulation, as a sub-genre of biological regulatory networks, is characterized by large numbers of interconnected species whose influences depend on passing some threshold, thus, largely sigmoidal behaviors. The application of qualitative methods to these systems can be highly advantageous to the modeler. As just mentioned realistic models in gene regulation are immense and highly interconnected, such that the simply enumeration of the possible states of the resulting system creates a combinatorial explosion. There are some questions for which one must access the underlying probability distribution associated with the Markov transitions of the qualitative model, as for example a qualitative and intuitive analysis of the system as a whole. The most pervasive methods have historically been simulation-based. Here, we propose a method to solve the system by treating the Markov equations of a Process Hitting model with numerical techniques. Proper Generalized Decomposition (PGD) can be used to overcome the curse of dimensionality, providing fast and accurate solutions to an otherwise intractable problem. Moreover PGD allows considering unknown parameters as a model extra-coordinate to obtain a parametric solution.

By suitable reformulations, we cast the mathematical frameworks of several well-known different approaches to the description of non-equilibrium dynamics into a unified formulation valid in all these contexts, which extends to such frameworks the concept of Steepest Entropy Ascent (SEA) dynamics introduced by the present author in previous works on quantum thermodynamics. Actually, the present formulation constitutes a generalization also for the quantum thermodynamics framework. The analysis emphasizes that in the SEA modeling principle a key role is played by the geometrical metric with respect to which to measure the length of a trajectory in state space. In the near thermodynamic equilibrium limit, the metric tensor turns is directly related to the Onsager's generalized resistivity tensor. Therefore, through the identification of a suitable metric field which generalizes the Onsager generalized resistance to the arbitrarily far non-equilibrium domain, most of the existing theories of non-equilibrium thermodynamics can be cast in such a way that the state exhibits the spontaneous tendency to evolve in state space along the path of SEA compatible with the conservation constraints and the boundary conditions. The resulting unified family of SEA dynamical models are all intrinsically and strongly consistent with the second law of thermodynamics. The nonnegativity of the entropy production is a general and readily proved feature of SEA dynamics. In several of the different approaches to non-equilibrium description we consider here, the SEA concept has not been investigated before. We believe it defines the precise meaning and the domain of general validity of the so-called Maximum Entropy Production principle. Therefore, it is hoped that the present unifying approach may prove useful in providing a fresh basis for effective, thermodynamically consistent, numerical models and theoretical treatments of irreversible conservative relaxation towards equilibrium from far non-equilibrium states. The mathematical frameworks are: A) Statistical or Information Theoretic Models of Relaxation; B) Small-Scale and Rarefied Gases Dynamics (i.e., kinetic models for the Boltzmann equation); C) Rational Extended Thermodynamics, Macroscopic Non-Equilibrium Thermodynamics, and Chemical Kinetics; D) Mesoscopic Non-Equilibrium Thermodynamics, Continuum Mechanics with Fluctuations; E) Quantum Statistical Mechanics, Quantum Thermodynamics, Mesoscopic Non-Equilibrium Quantum Thermodynamics, and Intrinsic Quantum Thermodynamics.

Let L1, L2, L3 be three well established (i.e. well tested with experimental observations) levels of description, ordered from the most micro- scopic to the least microscopic, on which mesoscopic dynamics of macro- scopic systems is formulated. Let Eqs1; Eqs2; Eqs3 be the time evolution equations on the three levels. By comparing solutions to these three systems of equations we find reductions L1 → L2 → L3 and L1 → L3 consisting of: (i) relations Eqs1 → Eqs2 → Eqs3 and Eqs1 → Eqs3, (ii) relations P1 → P2 → P3 and P1 → P3, where P stands for material parameters, i.e. the parameters with which the individual nature of the system under consideration is expressed in the time evolution equations, and (iii) six entropies, namely s^(1→2), s^(1→3), s^(2→3) and S^(3←1), S^(3←2) ,S^(2←1). The entropies s^(i→j); i < j are potentials generating the approach of the level Li to the level Lj and S^(i←j); i > j are the entropies s^(j→i) evaluated at the states on the level Li that are reached in the approach Lj → Li. These six entropies represent the multiscale thermodynamics corresponding to the sequence of levels L1, L2, L3. In the particular case when L3 is the level used in the classical equilibrium thermodynamics then S^(3←2) and S^(3←1) are the classical equilibrium entropies. I will illustrate such multiscale thermodynamics (and provide some of its applications) on the example of L1  level of description used in the Catteneo heat conduction theory, L2  level of description used in the Fourier heat conduction theory, and L3  level of description used in the classical equilibrium thermodynamics.

In this paper, cubic boron nitride (cBN) single crystals were synthesized with lithium nitride as catalyst under high pressure and high temperature. A lot of nanometer-sized cubic boron nitride nucleuses were found in the near surface layer of cBN crystals by high resolution transmission electron microscopy examinations. Based on the experiment results, the transformation kinetics is described by a nucleation and growth process in the thermodynamic stability region of cBN. The theoretical description is based on the heterogeneous nucleation and layer growth mechanism, and the relevant parameters are estimated and discussed. The results show that critical crystal radius, r*, is increasing with the elevated temperature under the same pressure, and temperature is the main influence on it under lower pressure. At the same time, the results show that crystal growth velocity has different changing tendency with different pressure under proper synthesized scope. The effect of the catalyst is suggested to lower the activation enthalpy for nucleation. These results are well consistent with experimental data.

The reliability of machineries and the mortality of individuals are topics of great interest for scientists and common people as well. The reliability theory of aging and longevity is a scientific approach aimed to gain theoretical insights into engineering and biology. However the vast majority of researchers make conclusions about population based on information extracted from random samples; in short theorists follow inductive logic. A mature discipline instead complies with the deductive logic, that is to say theorists derive the results from principles and axioms using theorems. After decades of enquiries, it would be desirable that the reliability theory becomes a mature scientific sector in accordance to the style inaugurated by Gnedenko’s seminal book.   The second law of thermodynamics claims that the entropy of an isolated system will increase as the system goes forward in time. This entails – in a way – that physical objects have an inherent tendency towards disorder, and a general predisposition towards decay. Such a wide-spreading process of annihilation hints an intriguing parallel with the decadence of biological and artificial systems to us. The issues in reliability theory are not so far from some issues inquired by thermodynamics and this closeness suggested us to introduce the entropy function for the study of reliable/reparable systems. We consider that the states of the stochastic system S can be more or less reversible and mean to calculate the state A_i of the system S using the Boltzmann-like entropy H_i where P_i is the probability of A_i. H_i = ln (P_i).   We confine our attention to the reliability entropy H_f of the functioning state A_f and the recovery entropy H_r of the recovery state A_r  whose meanings can be described as follows. When the functioning state is irreversible, the system S works steadily. In particular, the more A_f is irreversible, the more H_f is high and S is reliable. On the other hand, when H_f is low, S often abandons A_f and switches to A_r since it fails and we say that S is unreliable. The recovery entropy calculates the irreversibility of the recovery state, this implies that the more H_r is high, the more A_r is stable and in practice S is hard to be repaired and/or cured in the world. In sum H_r expresses the aptitude of S to work or to live without failures; the entropy H_r illustrates the disposition of S toward reparation or restoration to health.   Universal experience brings evidence how the components of the functioning state A_f degenerate by time passing and at last impede the correct functioning to S. Thus we assume that the entropy H_fg of the generic component g of the functioning state A_f decreases linearly as time goes by; and from this assumption a theorem demonstrates that the hazard rate (or mortality rate) of S is constant with time. When the system is rather old, an endangered part of S can harm to close components and starts a cascade effect while the machine proceeds to run. The cascade effect accelerates the evolution of S toward definitive stop. Now we face two alternative models of system. If the system is linear, one can prove that the hazard rate is power of time. If the system is a mesh, the hazard rate is exponential of time. One can map the reliability entropy H_f with the recovery entropy H_r using the reparability function     This function demonstrates four basic properties of repairable systems. In conclusion, fundamental laws tested in the reliability domain can be deduced from precise assumptions using the Boltzmann-like entropy. The theorems provide deep insights on how systems degenerate. The assumptions make clear the causes of the system failures which instead cannot be justified using usual statistical inference. The present frame seems to be a promising approach for developing a deductive theory of aging integrating mathematical methods with engineering notions and specific biological knowledge.

This paper discusses the major assumptions of influential ecological approaches on the human movement variability in sports and how it can be analyzed by benefiting from well-known measures of entropy. These measures are exploited so as to further understand the performance of athletes from a dynamical and chaotic perspective. Based on the presented evidences, entropy-based techniques will be considered to measure, analyze and evaluate the human performance variability under three different case studies: i) golf; ii) tennis; and iii) soccer. At a first stage, the athletes' performance will be analyzed at the individual level by considering the golf putting (pendulum movement) and the tennis serve (ballistic movement). Under these gestures, the approximate entropy is considered to extract the variability inherent to the process variables. Afterwards, the athletes' performance will be analyzed at the collective level by considering the soccer case (team sport). To that end, both approximate entropy and Shannon's entropy are mutually considered to assess the variability of football players' trajectory. To outline the applicability of entropy-based measures to analyze sports, this article ends with an overall reflection about the potential of such measures towards an increased understanding on the overall human performance. This methodology proves to be useful to provide decisive information and feedback for coaches, sports analysts and even for the athletes.

The current work is applied optimization process with multi objective on the solar-powered Stirling engine with high temperature differential. On the basis of finite –time thermodynamic, new mathematical approach was evolved. Furthermore, thermal efficiency of the solar Stirling system with rate of finite heat transfer, regenerative heat loss, the output power, finite regeneration process time and conductive thermal bridging loss are specified. The power output and thermal efficiency and entransy loss rate are specified at Maximum condition for a dish-Stirling system and entropy generation's rate in the engine Minimized by carrying out thermodynamic analysis and NSGAǁ approach. Three well known decision making methods are carried out to indicate optimum values of outputs obtained with optimization process. Finally, with the aim of error analysis the error of the aforementioned results are determined.