As recently as 1960, cosmology was widely regarded as a branch of philosophy. r To study the properties of the Gödel solution, the frame field can be adopted (dual to the coframe read off the metric as given above), This frame defines a family of inertial observers who are 'comoving with the dust grains'. So we can define a state function S called entropy, which satisfies [45][46] It claims that non-equilibrium systems evolve such as to maximize its entropy production.[47][48]. {\displaystyle X_{0}} ⟩ He also worked on the fields of what we now call geography and biology. where T is the absolute thermodynamic temperature of the system at the point of the heat flow. If there are multiple heat flows, the term , and using the fourth vector we can move along 0 If the substances are at the same temperature and pressure, there is no net exchange of heat or work – the entropy change is entirely due to the mixing of the different substances. y λ [102], Romanian American economist Nicholas Georgescu-Roegen, a progenitor in economics and a paradigm founder of ecological economics, made extensive use of the entropy concept in his magnum opus on The Entropy Law and the Economic Process. ∑ {\displaystyle \partial _{y},\;\partial _{z}} {\displaystyle -\omega } Otto closed the door partway Therefore Otto closed the door It seems clear to me that better data are needed before progress can be made in this area; we need much more refined linguistic classifications of adverbial constructions than are presently avail- able, if our evidence concerning validity is … Though it doesn’t entail gigantic tubs of ice cream drowning everyone, it does spell disaster for everyone. {\displaystyle {\dot {W}}_{\text{S}}} Using this concept, in conjunction with the density matrix he extended the classical concept of entropy into the quantum domain. T {\displaystyle t,y,z} j [105]:95–112, In economics, Georgescu-Roegen's work has generated the term 'entropy pessimism'. The “Big Freeze” is a scientific theory of the end of the universe. T In this direction, several recent authors have derived exact entropy formulas to account for and measure disorder and order in atomic and molecular assemblies. "[8] This term was formed by replacing the root of ἔργον ('work') by that of τροπή ('transformation'). with angular velocity (See the more detailed discussion below using an alternative coordinate chart. 2 X In the thermodynamic limit, this fact leads to an equation relating the change in the internal energy U to changes in the entropy and the external parameters. Entropy is often loosely associated with the amount of order or disorder, or of chaos, in a thermodynamic system. is inessential; our spacetime is the direct product of a factor R with a signature −++ three-manifold. In a thermodynamic system, pressure, density, and temperature tend to become uniform over time because the equilibrium state has higher probability (more possible combinations of microstates) than any other state. {\displaystyle T} + Gesellschaft zu Zürich den 24. This upholds the correspondence principle, because in the classical limit, when the phases between the basis states used for the classical probabilities are purely random, this expression is equivalent to the familiar classical definition of entropy. [Ressource ARDP 2015], Pantin, CN D. interpretation of entropy in statistical mechanics, the fundamental postulate in statistical mechanics, Entropy in thermodynamics and information theory, Nicholas Georgescu-Roegen § The relevance of thermodynamics to economics, integral part of the ecological economics school, Autocatalytic reactions and order creation, Thermodynamic databases for pure substances, "Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie (Vorgetragen in der naturforsch. {\displaystyle U=\left\langle E_{i}\right\rangle } ∂ [90] The concept divides systems into three categories, natural, hybrid and man-made, based on the amount of control that humans have in slowing the relentless march of entropy and the time-scale of each category to reach maximum entropy. = , the vector field {\textstyle \sum {\dot {Q}}_{j}/T_{j},} μ [12] For example, in the Carnot cycle, while the heat flow from the hot reservoir to the cold reservoir represents an increase in entropy, the work output, if reversibly and perfectly stored in some energy storage mechanism, represents a decrease in entropy that could be used to operate the heat engine in reverse and return to the previous state, thus the total entropy change is still zero at all times if the entire process is reversible. and a complementary amount, at a given t is a closed null curve, but not a null geodesic. La Querelle des Pantomimes. . c This is because at that radius we find that [21][22][23] This definition describes the entropy as being proportional to the natural logarithm of the number of possible microscopic configurations of the individual atoms and molecules of the system (microstates) that could cause the observed macroscopic state (macrostate) of the system. The state function was called the internal energy and it became the first law of thermodynamics.[18]. S At a statistical mechanical level, this results due to the change in available volume per particle with mixing. together with the fundamental thermodynamic relation) are known for the microcanonical ensemble, the canonical ensemble, the grand canonical ensemble, and the isothermal–isobaric ensemble. According to S. W. Hawking, these models could well be a reasonable description of the universe that we observe, however observational data are compatible only with a very low rate of rotation. 2 {\displaystyle r=0} Specifically, with a bit of foresight, let us choose = Q , Thus, when one mole of substance at about 0 K is warmed by its surroundings to 298 K, the sum of the incremental values of qrev/T constitute each element's or compound's standard molar entropy, an indicator of the amount of energy stored by a substance at 298 K.[49][50] Entropy change also measures the mixing of substances as a summation of their relative quantities in the final mixture. [106]:116 Since the 1990s, leading ecological economist and steady-state theorist Herman Daly – a student of Georgescu-Roegen – has been the economics profession's most influential proponent of the entropy pessimism position. 0 To study the tidal forces in more detail, the Bel decomposition of the Riemann tensor can be computed into three pieces, the tidal or electrogravitic tensor (which represents tidal forces), the magnetogravitic tensor (which represents spin-spin forces on spinning test particles and other gravitational effects analogous to magnetism), and the topogravitic tensor (which represents the spatial sectional curvatures). [...] Von Neumann told me, "You should call it entropy, for two reasons. 0 is the ideal gas constant. → [15][16] Through the efforts of Clausius and Kelvin, it is now known that the maximum work that a heat engine can produce is the product of the Carnot efficiency and the heat absorbed from the hot reservoir: To derive the Carnot efficiency, which is 1 − TC/TH (a number less than one), Kelvin had to evaluate the ratio of the work output to the heat absorbed during the isothermal expansion with the help of the Carnot–Clapeyron equation, which contained an unknown function called the Carnot function. {\displaystyle {\dot {Q}}/T,} “The lips of wisdom are closed, except to the ears of Understanding.” —The Kybalion. f λ Physical chemist Peter Atkins, in his textbook Physical Chemistry, introduces entropy with the statement that "spontaneous changes are always accompanied by a dispersal of energy".[68]. According to Carnot's principle, work can only be produced by the system when there is a temperature difference, and the work should be some function of the difference in temperature and the heat absorbed (QH). The statistical definition of entropy defines it in terms of the statistics of the motions of the microscopic constituents of a system – modeled at first classically, e.g. [26] This concept plays an important role in liquid-state theory. In 1948, Bell Labs scientist Claude Shannon developed similar statistical concepts of measuring microscopic uncertainty and multiplicity to the problem of random losses of information in telecommunication signals. T → is a nonzero real constant, which turns out to be the angular velocity of the surrounding dust grains around the y axis, as measured by a "non-spinning" observer riding one of the dust grains. 1 It turns out that in addition, optical images are expanded and sheared in the direction of rotation. {\displaystyle dQ} We have seen that observers lying on the y axis (in the original chart) see the rest of the universe rotating clockwise about that axis. {\displaystyle x} A substance at non-uniform temperature is at a lower entropy (than if the heat distribution is allowed to even out) and some of the thermal energy can drive a heat engine. The process of measurement goes as follows. ˙ r direction can see only finitely far out, and also see themselves at an earlier time. More specifically, total entropy is conserved in a reversible process and not conserved in an irreversible process. rev If the universe can be considered to have generally increasing entropy, then – as Roger Penrose has pointed out – gravity plays an important role in the increase because gravity causes dispersed matter to accumulate into stars, which collapse eventually into black holes. [40] Thermodynamic relations are then employed to derive the well-known Gibbs entropy formula. Henceforth, the essential problem in statistical thermodynamics has been to determine the distribution of a given amount of energy E over N identical systems. If the Gödel spacetime admitted any boundaryless temporal hyperslices (e.g. Moreover, Anaximander was the first speculative astronomer. X , = Many other exact solutions that can be interpreted as cosmological models of rotating universes are known. , plus two further Killing vector fields: The isometry group acts 'transitively' (since we can translate in p [65] Similar terms have been in use from early in the history of classical thermodynamics, and with the development of statistical thermodynamics and quantum theory, entropy changes have been described in terms of the mixing or "spreading" of the total energy of each constituent of a system over its particular quantized energy levels. admit an SL(2,R) action, and the slices doi:10.1086/392846 As time progresses, the second law of thermodynamics states that the entropy of an isolated system never decreases in large systems over significant periods of time. [20] However, the entropy change of the surroundings is different. = ). Indeed, there are CTCs through every event in the Gödel spacetime. One can see that entropy was discovered through mathematics rather than through laboratory results. {\displaystyle \partial _{y}} a measure of disorder in the universe or of the availability of the energy in a system to do work. {\displaystyle \omega } = , two 'spatial translations' The slices Constantin Carathéodory, the Greek mathematician, linked entropy with a mathematical definition of irreversibility, in terms of trajectories and integrability. δ Boltzmann showed that this definition of entropy was equivalent to the thermodynamic entropy to within a constant factor—known as Boltzmann's constant. One dictionary definition of entropy is that it is "a measure of thermal energy per unit temperature that is not available for useful work". However, recall that while the first frame given above (the rev {\displaystyle {\vec {e}}_{j}} / [97][98][99] This results in an "entropy gap" pushing the system further away from the posited heat death equilibrium. Accordingly, it has a characteristic polynomial, whose roots are the eigenvalues. To find the entropy difference between any two states of a system, the integral must be evaluated for some reversible path between the initial and final states. In German, he initially described it as Verwandlungsinhalt, in translation as a transformation-content, and later coined the term entropy from a Greek word for transformation. ∂ {\displaystyle \operatorname {Tr} } c The following is a list of additional definitions of entropy from a collection of textbooks: In Boltzmann's definition, entropy is a measure of the number of possible microscopic states (or microstates) of a system in thermodynamic equilibrium. Austrian physicist Ludwig Boltzmann explained entropy as the measure of the number of possible microscopic arrangements or states of individual atoms and molecules of a system that comply with the macroscopic condition of the system. e p log {\displaystyle {\vec {f}}_{j}} 0 Thermodynamic entropy is central in chemical thermodynamics, enabling changes to be quantified and the outcome of reactions predicted. t Four of the five Killing vectors, as well as the curvature tensor, do not depend upon the coordinate y. Start with a simple frame in a cylindrical type chart, featuring two undetermined functions of the radial coordinate: Here, we think of the timelike unit vector field Let us demand that the Einstein tensor match a dust term plus a vacuum energy term. Following on from the above, it is possible (in a thermal context) to regard lower entropy as an indicator or measure of the effectiveness or usefulness of a particular quantity of energy. [71] This approach has several predecessors, including the pioneering work of Constantin Carathéodory from 1909[72] and the monograph by R. Entropy is conserved for a reversible process. 2 [4] Carnot based his views of heat partially on the early 18th-century "Newtonian hypothesis" that both heat and light were types of indestructible forms of matter, which are attracted and repelled by other matter, and partially on the contemporary views of Count Rumford, who showed in 1789 that heat could be created by friction, as when cannon bores are machined. [89] With this expansion of the fields/systems to which the second law of thermodynamics applies, the meaning of the word entropy has also expanded and is based on the driving energy for that system. i {\displaystyle r=r_{c}} Forma e indeterminazione nelle poetiche contemporanee, Bompiani 2013. Q {\displaystyle X_{0}} Thus, corresponding to our symmetry axis we have a timelike congruence made up of circles and corresponding to certain observers. x 1 It is obvious from the generators just given that the slices ) appears to be spinning in the chart, but it is gyrostabilized, and a nonspinning inertial observer riding on a dust grain will indeed see the other dust grains rotating clockwise with angular velocity Therefore, this spacetime is not globally hyperbolic. with low entropy) tends to be more useful than the same amount of energy available at a lower temperature. (2018). {\textstyle dS={\frac {\delta Q_{\text{rev}}}{T}}} pi = 1/Ω, where Ω is the number of microstates); this assumption is usually justified for an isolated system in equilibrium. A reversible process is one that does not deviate from thermodynamic equilibrium, while producing the maximum work. r He used an analogy with how water falls in a water wheel. {\displaystyle \log } [1] The term 'entropy' was introduced in 1865 by German physicist Rudolph Clausius, one of the leading founders of the field of thermodynamics, who had used the quantity in 1854. e ∂ In mechanics, the second law in conjunction with the fundamental thermodynamic relation places limits on a system's ability to do useful work. ˙ In the 1850s and 1860s, German physicist Rudolf Clausius objected to the supposition that no change occurs in the working body, and gave that "change" a mathematical interpretation, by questioning the nature of the inherent loss of usable heat when work is done, e.g. A closed curve has no end points. The Shannon entropy (in nats) is: which is the Boltzmann entropy formula, where k is Boltzmann's constant, which may be interpreted as the thermodynamic entropy per nat. − The Hermetic Philosophy. The concept of entropy arose from Rudolf Clausius's study of the Carnot cycle. vanishes, so the dust particles exhibit rigid rotation. This density matrix formulation is not needed in cases of thermal equilibrium so long as the basis states are chosen to be energy eigenstates. From a macroscopic perspective, in classical thermodynamics the entropy is interpreted as a state function of a thermodynamic system: that is, a property depending only on the current state of the system, independent of how that state came to be achieved. {\displaystyle \nabla _{{\vec {e}}_{0}}{\vec {e}}_{0}=0} However, the equivalence between the Gibbs entropy formula and the thermodynamic definition of entropy is not a fundamental thermodynamic relation but rather a consequence of the form of the generalized Boltzmann distribution.[41]. = A special case of entropy increase, the entropy of mixing, occurs when two or more different substances are mixed. Important examples are the Maxwell relations and the relations between heat capacities. In quantum statistical mechanics, the concept of entropy was developed by John von Neumann and is generally referred to as "von Neumann entropy". That was an early insight into the second law of thermodynamics. The overdots represent derivatives of the quantities with respect to time. The summation is over all the possible microstates of the system, and pi is the probability that the system is in the i-th microstate. Specifically, entropy is a logarithmic measure of the number of states with significant probability of being occupied: ( In classical thermodynamics, the entropy of a system is defined only if it is in thermodynamic equilibrium. As a fundamental aspect of thermodynamics and physics, several different approaches to entropy beyond that of Clausius and Boltzmann are valid. But, in fact, many objects exist in the universe. Suppressing In 1865, Clausius named the concept of "the differential of a quantity which depends on the configuration of the system," entropy (Entropie) after the Greek word for 'transformation'. 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Methods in the 1870s by analyzing the statistical behavior of the system boundary the quantum domain processes reversible!