The evidence here provided shows that the thermodynamics of the second law, as currently understood, originated in a correction of the flaws affecting Clausius original work on this matter. The body of knowledge emerg...The evidence here provided shows that the thermodynamics of the second law, as currently understood, originated in a correction of the flaws affecting Clausius original work on this matter. The body of knowledge emerging from this correction has been here called post-Clausius’ thermodynamics. The said corrections, carried on with the intended goal of preserving the validity of Clausius’ main result, namely the law of increasing entropy, made use of a number of counterintuitive or logically at fault notions. A joint revision of Clausius’ and post-Clausius’ work on the second law, carried on retaining some of Clausius original notions, and disregarding others introduced by post-Clausius thermodynamics, led this author to results in direct contradiction to the law of increasing entropy. Among the key results coming out of this work we find the one stating that the total-entropy change for spontaneous thermodynamic processes is the result of the summation of the opposite-sign contributions coming from the entropic (energy degrading) and negentropic (energy upgrading) changes subsumed by any such process. These results also show, via the total-entropy change for a non-reversible heat engine, that negentropic thermodynamics subsumes post-Clausius thermodynamics as a special case.展开更多
Discrete dynamical systems are given by the pair (X,f) where X is a compact metric space and f: X→X is a continuous map. During years, a long list of results have appeared to precise and understand what is the comple...Discrete dynamical systems are given by the pair (X,f) where X is a compact metric space and f: X→X is a continuous map. During years, a long list of results have appeared to precise and understand what is the complexity of the systems. Among them, one of the most popular is that of topological entropy. In modern applications, other conditions on X and f have been considered. For example, X can be non-compact or f can be discontinuous (only in a finite number of points and with bounded jumps on the values of f or even non-bounded jumps). Such systems are interesting from theoretical point of view in Topological Dynamics and appear frequently in applied sciences such as Electronics and Control Theory. In this paper, we are reviewing the origins of the notion of entropy and studying some developing of it leading to modern notions of entropies. At the same time, we will incorporate some mathematical foundations of such old and new ideas until the appearance of Shannon entropy. To this end, we start with the introduction for the first time of the notion of entropy in thermodynamics by R. Clausius and its evolution by L. Boltzmann until the appearing in the twenty century of Shannon and Kolmogorov-Sinai entropies and the subsequent topological entropy. In turn, such notions have evolved to other recent situations where it is necessary to give some extended versions of them adapted to new problems. Of special interest is to appreciate the connexions of the notions of entropy from Boltzmann and Shannon. Since this history is long, we will not deal with the Kolmogorov-Sinai entropy or with topological entropy and modern approaches.展开更多
It is impossible that proving the internal energy change has the relations with volume and pressure. About the second law of thermodynamics, many mistakes of formulations need to be put right and modified, and many ne...It is impossible that proving the internal energy change has the relations with volume and pressure. About the second law of thermodynamics, many mistakes of formulations need to be put right and modified, and many new concepts are surveyed too. The equality and inequality on the ratios of internal energy change to temperature and work to temperature are discussed. The relation between the reversible paths and their realistic paths is also researched. In an isothermal process, the internal energy change for the gases is equal to zero, but the internal energy change is not equal to zero for the phase transition or chemical reaction. The Clausius inequality can be derived from the equation calculating the internal energy change in mathematics;it is the new method proving the Clausius inequality. These change laws of thermodynamics could be applied to the gravitational field and mechanical motion and so on.展开更多
Applying Clausius relation with energy-supply defined by the unified first law of thermodynamics formalism to the apparent horizon of a massive gravity model in cosmology proposed lately, the corrected entropic formul...Applying Clausius relation with energy-supply defined by the unified first law of thermodynamics formalism to the apparent horizon of a massive gravity model in cosmology proposed lately, the corrected entropic formula of the apparent horizon is obtaJned with the help of the modified Friedmann equations. This entropy-area relation, together with the identified Misner-Sharp internal energy, verifies the first law of thermodynamics for the apparent horizon with a volume change term for consistency. On the other hand, by means of the corrected entropy-area formula and the Clausius relation δQ = T dS, where the heat flow δQ is the energy-supply of pure matter projecting on the vector ξ tangent to the apparent horizon and should be looked on as the amount of energy crossing the apparent horizon during the time interval dt and the temperature of the apparent horizon for energy crossing during the same interval is 1/（2πτA）, the modified Friedmann equations governing the dynamical evolution of the universe are reproduced with the known energy density and pressure of massive graviton. The integration constant is found to correspond to a cosmological term which could be absorbed into the energy density of matter. Having established the correspondence of massive cosmology with the unified first law of thermodynamics on the apparent horizon, the validity of the generalized second law of thermodynamics is also discussed by assuming the thermal equilibrium between the apparent horizon and the matter field bounded by the apparent horizon. It is found that, in the limit Hc → 0, which recovers the Minkowski reference metric solution in the fiat case, the generalized second law of thermodynamics holds if α3 ＋ 4α4 〈 0. Without this condition, even for the simplest model of dRGT massive cosmology with α3= α4 = 0, the generalized second law of thermodynamics could be violated.展开更多
We investigate the unified first law and the generalized second law in a modified holographic dark energy model. The thermodynamical analysis on the apparent horizon can work and the corresponding entropy formula is e...We investigate the unified first law and the generalized second law in a modified holographic dark energy model. The thermodynamical analysis on the apparent horizon can work and the corresponding entropy formula is extracted from the systematic algorithm. The entropy correction term depends on the extra-dimension number of the brane as expected, but the interplay between the correction term and the extra dimensions is more complicated. With the unified first law of thermodynamics well-founded, the generalized second law of thermodynamics is discussed and it is found that the second law can be violated in certain circumstances. Particularly, if the number of the extra dimensions is larger than one, the generalized law of thermodynamics is always satisfied; otherwise, the validity of the second law can only be guaranteed with the Hubble radius greatly smaller than the crossover scale rcof the 5-dimensional DGP model.展开更多
The change of extreme precipitation with temperature has regional characteristics in the context of global warming.In this study, radiosonde data, co-located rain gauge(RG) observations, and Tropical Rainfall Measurin...The change of extreme precipitation with temperature has regional characteristics in the context of global warming.In this study, radiosonde data, co-located rain gauge(RG) observations, and Tropical Rainfall Measuring Mission(TRMM) precipitation radar(PR) products are used to explore the relationship between extreme precipitation intensity and near-surface temperature in Middle–East China(MEC) and the eastern Tibetan Plateau(TP) during1998–2012. The results show that extreme precipitation intensity increases with increasing temperature at an approximate Clausius–Clapeyron(C–C) rate(i.e., water vapor increases by 7% as temperature increases by 1°C based on the C–C equation) in MEC and TP, but the rate of increase is larger in TP than in MEC. This is probably because TP(MEC) is featured with deep convective(stratiform) precipitation, which releases more(less) latent heat and strengthens the convection intensity on a shorter(longer) timescale. It is also found that when temperature is higher than 25°C(15°C) in MEC(TP), the extreme precipitation intensity decreases with rise of temperature, suggesting that the precipitation intensity does not always increase with warming. In this case, the limited atmospheric humidity and precipitable water could be the primary factors for the decrease in extreme precipitation intensity at higher temperatures.展开更多
The classical thermodynamics reflects the significant relationship between the heat and the temperature. On the basis of the relationships, according to the mathematical derivation, this paper structures the conceptio...The classical thermodynamics reflects the significant relationship between the heat and the temperature. On the basis of the relationships, according to the mathematical derivation, this paper structures the conceptions of generalized heat, generalized thermodynamic temperature, generalized entropy and so on. The series of conceptions in the classical thermodynamics is merely a special case of the generalized thermodynamics. Based on these conceptions of generalized thermodynamics, this paper presents the new expressions of the first law and the second law of thermodynamics. In other words, these expressions are endued with new explanations. The Eq. LZ = kTS given by this paper provides theoretical basis for these new expressions.展开更多
文摘The evidence here provided shows that the thermodynamics of the second law, as currently understood, originated in a correction of the flaws affecting Clausius original work on this matter. The body of knowledge emerging from this correction has been here called post-Clausius’ thermodynamics. The said corrections, carried on with the intended goal of preserving the validity of Clausius’ main result, namely the law of increasing entropy, made use of a number of counterintuitive or logically at fault notions. A joint revision of Clausius’ and post-Clausius’ work on the second law, carried on retaining some of Clausius original notions, and disregarding others introduced by post-Clausius thermodynamics, led this author to results in direct contradiction to the law of increasing entropy. Among the key results coming out of this work we find the one stating that the total-entropy change for spontaneous thermodynamic processes is the result of the summation of the opposite-sign contributions coming from the entropic (energy degrading) and negentropic (energy upgrading) changes subsumed by any such process. These results also show, via the total-entropy change for a non-reversible heat engine, that negentropic thermodynamics subsumes post-Clausius thermodynamics as a special case.
文摘Discrete dynamical systems are given by the pair (X,f) where X is a compact metric space and f: X→X is a continuous map. During years, a long list of results have appeared to precise and understand what is the complexity of the systems. Among them, one of the most popular is that of topological entropy. In modern applications, other conditions on X and f have been considered. For example, X can be non-compact or f can be discontinuous (only in a finite number of points and with bounded jumps on the values of f or even non-bounded jumps). Such systems are interesting from theoretical point of view in Topological Dynamics and appear frequently in applied sciences such as Electronics and Control Theory. In this paper, we are reviewing the origins of the notion of entropy and studying some developing of it leading to modern notions of entropies. At the same time, we will incorporate some mathematical foundations of such old and new ideas until the appearance of Shannon entropy. To this end, we start with the introduction for the first time of the notion of entropy in thermodynamics by R. Clausius and its evolution by L. Boltzmann until the appearing in the twenty century of Shannon and Kolmogorov-Sinai entropies and the subsequent topological entropy. In turn, such notions have evolved to other recent situations where it is necessary to give some extended versions of them adapted to new problems. Of special interest is to appreciate the connexions of the notions of entropy from Boltzmann and Shannon. Since this history is long, we will not deal with the Kolmogorov-Sinai entropy or with topological entropy and modern approaches.
文摘It is impossible that proving the internal energy change has the relations with volume and pressure. About the second law of thermodynamics, many mistakes of formulations need to be put right and modified, and many new concepts are surveyed too. The equality and inequality on the ratios of internal energy change to temperature and work to temperature are discussed. The relation between the reversible paths and their realistic paths is also researched. In an isothermal process, the internal energy change for the gases is equal to zero, but the internal energy change is not equal to zero for the phase transition or chemical reaction. The Clausius inequality can be derived from the equation calculating the internal energy change in mathematics;it is the new method proving the Clausius inequality. These change laws of thermodynamics could be applied to the gravitational field and mechanical motion and so on.
基金Supported by the National Natural Science Foundation of China under Grant Nos. 10747155, 11205131, 11175270, 11005164, and 10935013, ChongqingChongqing Science and Technology Commission under Grant No. 2010BB0408Local Support from Argonne National Laboratory
文摘Applying Clausius relation with energy-supply defined by the unified first law of thermodynamics formalism to the apparent horizon of a massive gravity model in cosmology proposed lately, the corrected entropic formula of the apparent horizon is obtaJned with the help of the modified Friedmann equations. This entropy-area relation, together with the identified Misner-Sharp internal energy, verifies the first law of thermodynamics for the apparent horizon with a volume change term for consistency. On the other hand, by means of the corrected entropy-area formula and the Clausius relation δQ = T dS, where the heat flow δQ is the energy-supply of pure matter projecting on the vector ξ tangent to the apparent horizon and should be looked on as the amount of energy crossing the apparent horizon during the time interval dt and the temperature of the apparent horizon for energy crossing during the same interval is 1/（2πτA）, the modified Friedmann equations governing the dynamical evolution of the universe are reproduced with the known energy density and pressure of massive graviton. The integration constant is found to correspond to a cosmological term which could be absorbed into the energy density of matter. Having established the correspondence of massive cosmology with the unified first law of thermodynamics on the apparent horizon, the validity of the generalized second law of thermodynamics is also discussed by assuming the thermal equilibrium between the apparent horizon and the matter field bounded by the apparent horizon. It is found that, in the limit Hc → 0, which recovers the Minkowski reference metric solution in the fiat case, the generalized second law of thermodynamics holds if α3 ＋ 4α4 〈 0. Without this condition, even for the simplest model of dRGT massive cosmology with α3= α4 = 0, the generalized second law of thermodynamics could be violated.
基金Supported by National Foundation of China under Grant Nos.10747155 and 11205131the Ministry of Science and Technology of the National Natural Science Foundation of China key project under Grant Nos.11175270,11005164,11073005,and 10935013Chongqing Science and Technology Commission under Grant No.2010BB0408
文摘We investigate the unified first law and the generalized second law in a modified holographic dark energy model. The thermodynamical analysis on the apparent horizon can work and the corresponding entropy formula is extracted from the systematic algorithm. The entropy correction term depends on the extra-dimension number of the brane as expected, but the interplay between the correction term and the extra dimensions is more complicated. With the unified first law of thermodynamics well-founded, the generalized second law of thermodynamics is discussed and it is found that the second law can be violated in certain circumstances. Particularly, if the number of the extra dimensions is larger than one, the generalized law of thermodynamics is always satisfied; otherwise, the validity of the second law can only be guaranteed with the Hubble radius greatly smaller than the crossover scale rcof the 5-dimensional DGP model.
基金Supported by the National Natural Science Foundation of China(91837310)National Key Research and Development Program of China(2017YFC1501402 and 2018YFC1507200)
文摘The change of extreme precipitation with temperature has regional characteristics in the context of global warming.In this study, radiosonde data, co-located rain gauge(RG) observations, and Tropical Rainfall Measuring Mission(TRMM) precipitation radar(PR) products are used to explore the relationship between extreme precipitation intensity and near-surface temperature in Middle–East China(MEC) and the eastern Tibetan Plateau(TP) during1998–2012. The results show that extreme precipitation intensity increases with increasing temperature at an approximate Clausius–Clapeyron(C–C) rate(i.e., water vapor increases by 7% as temperature increases by 1°C based on the C–C equation) in MEC and TP, but the rate of increase is larger in TP than in MEC. This is probably because TP(MEC) is featured with deep convective(stratiform) precipitation, which releases more(less) latent heat and strengthens the convection intensity on a shorter(longer) timescale. It is also found that when temperature is higher than 25°C(15°C) in MEC(TP), the extreme precipitation intensity decreases with rise of temperature, suggesting that the precipitation intensity does not always increase with warming. In this case, the limited atmospheric humidity and precipitable water could be the primary factors for the decrease in extreme precipitation intensity at higher temperatures.
文摘The classical thermodynamics reflects the significant relationship between the heat and the temperature. On the basis of the relationships, according to the mathematical derivation, this paper structures the conceptions of generalized heat, generalized thermodynamic temperature, generalized entropy and so on. The series of conceptions in the classical thermodynamics is merely a special case of the generalized thermodynamics. Based on these conceptions of generalized thermodynamics, this paper presents the new expressions of the first law and the second law of thermodynamics. In other words, these expressions are endued with new explanations. The Eq. LZ = kTS given by this paper provides theoretical basis for these new expressions.
