THE MINISTRY OF EDUCATION AND SCIENCE OF THE KYRGYZ REPUBLIC KYRGYZ NATIONAL UNIVERSITY named after J.BALASAGYN FACULTY: MATHEMATICS COMPUTER SCIENCE AND CYBERNETICS DEPARTMENTS : MATHEMATICS COMPUTER SCIENCE AND COMPUTER TECHNOLOGY INDEPENDENT WORK Theme: Entropy problems Checked by: Kalenderova T Done by: Kriuchkova K. Group: MCHM-1-22 Bishkek 2022 Оглавление Abstract ..................................................................................................................... 3 Introduction ............................................................................................................... 3 Entropy ...................................................................................................................... 5 History ....................................................................................................................... 7 References ............................................................................................................... 10 2 Abstract Еarlier in several papers the question on minimizing of the entropy increment in the achievement of some goals was arisen. As an application of the partial differential equations theory, here the problem of minimum of entropy increment is considered when straightening of plastic rod depends on time. Based on the analysis of the processes discussed in this paper and in the hypotheses on entropy have been put forward. An accurate estimate of the entropy increment is obtained with unlimited braking of the point, and on based on - an accurate estimate of the entropy increment when the plastic rod is straightened with unlimited braking. The problem of the minimum of entropy increment in straightening a plastic rod has been solved depending on time. Introduction The concept of entropy is interesting and meaningful. Many have focused on the grim picture of growing disorder and thermal death that characterizes equilibrium thermodynamics and isolated systems. In this regard, entropy is used to indicate and measure the decrease in the availability of high-quality resources (i.e. resources with low entropy and high free energy), the increase in pollution due to the release of waste, chemicals and heat into the environment. , the growth of social disorder due to the deterioration of living conditions in megacities around the world, the "collapse" of the economy, etc. Others identify entropy with the style of nature: since the biosphere is an open system supported by a constant influx of solar energy, its structures and life phenomena undergo a continuous process of self-organization. Geological processes, atmospheric systems, ecosystems, and societies are interconnected through a series of infinitely different and changing relationships, each receiving energy and materials from the other, returning them, and acting through feedback mechanisms to self-organize the whole in grand interaction. space, time, energy and information. During this process of self-organization, entropy is generated and then released into outer space. Living structures do not violate the second law, but 3 are fed by input sources that continuously supply low-entropy material and energy (free energy) for the development of the system. While the first point of view requires increased attention to preventing the misuse of resources and preventing the degradation of both the natural and human environment (i.e. disordered materials, due to the influx of resources from outside. This point of view requires adaptation to the style of nature, recognizing the existence of fluctuations (growth and fall) and resource constraints, within which, however, many variations and new models are possible. Both points of view are interesting and stimulate and explain the richness of the concept of entropy. The latter, first introduced in the field of energy conversion, very soon acquired citizenship in a number of other areas. Based on many experiments, it has been proven that heat passes from the warmer components of the system to colder ones, but not vice versa. The paper considers the problem of minimizing the increase in entropy when certain goals are achieved and the problem of developing a general methodology for research and proof of existence, along with establishing the properties of solutions to nonlinear differential and integro-differential equations in partial derivatives, put forward hypotheses about entropy. Such a technique is necessary to obtain quantitative lower bounds for the increase in entropy in almost closed systems described by differential equations with control. A lower estimate is obtained for the increment of entropy under controlled transformation of an extended object as a function of time. The second law of thermodynamics and hypotheses about the lower estimate for the increase in entropy are presented. The practical significance of this law lies, in particular, in the fact that the increase in entropy in a certain sense corresponds to the concept of environmental pollution. 4 Entropy Entropy is the state function of a thermodynamic system. There is no concept of the absolute value of entropy. In any processes, only the magnitude of its change can be determined. In thermodynamics, the degree of scattering is measured by the entropy value, i.e. transition to thermal energy, any other form of energy contained in the system. Any thermodynamic system isolated from the external world tends to equalize the temperatures of all its parts, i.e. to the maximum increase in entropy in it. A system that had a non-equilibrium thermal state passes to an equilibrium one when the heat transfer processes stop. In statistical physics, entropy is interpreted as a measure of the probability of a system being in a given state. The more disorder, the more entropy. Any system gradually moves to its more probable state. In the process of this, disorder increases in it, chaos grows, and hence entropy increases. The concept of entropy is used in chemistry, biology, computer science, etc. Examples of an increase in entropy are well known. These include: the processes of cooling and heating of objects until the heat flows die out; any kind of destruction; overgrowing gardens with weeds; cases of loss of information on the hard drive of a computer under the influence of viruses and even an increase in domestic chaos, requiring us to periodically clean the apartment. It is generally accepted in science that, in addition to the initial stages of the formation of the Universe , up to the formation of galaxies , entropy increases in all natural processes in inanimate nature . Living objects (see Life ), producing similar ones to themselves, streamline the surrounding inanimate matter , building living organisms out of it. Reasonable human activity most often leads to the creation of unlikely states of matter. These include almost all the products, works, structures, etc. created by him. 5 Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the microscopic description of nature in statistical physics, and to the principles of information theory. It has found far-ranging applications in chemistry and physics, in biological systems and their relation to life, in cosmology, economics, sociology, weather science, climate change, and information systems including the transmission of information in telecommunication. The thermodynamic concept was referred to by Scottish scientist and engineer William Rankine in 1850 with the names thermodynamic function and heat-potential. In 1865, German physicist Rudolf Clausius, one of the leading founders of the field of thermodynamics, defined it as the quotient of an infinitesimal amount of heat to the instantaneous temperature. He initially described it as transformation-content, in German Verwandlungsinhalt, and later coined the term entropy from a Greek word for transformation. Referring to microscopic constitution and structure, in 1862, Clausius interpreted the concept as meaning disgregation. A consequence of entropy is that certain processes are irreversible or impossible, aside from the requirement of not violating the conservation of energy, the latter being expressed in the first law of thermodynamics. Entropy is central to the second law of thermodynamics, which states that the entropy of isolated systems left to spontaneous evolution cannot decrease with time, as they always arrive at a state of thermodynamic equilibrium, where the entropy is highest. 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. He thereby introduced the concept of statistical disorder and probability distributions into a new field of thermodynamics, called statistical mechanics, and 6 found the link between the microscopic interactions, which fluctuate about an average configuration, to the macroscopically observable behavior, in form of a simple logarithmic law, with a proportionality constant, the Boltzmann constant, that has become one of the defining universal constants for the modern International System of Units (SI). 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. Upon John von Neumann's suggestion, Shannon named this entity of missing information in analogous manner to its use in statistical mechanics as entropy, and gave birth to the field of information theory. This description has been identified as a universal definition of the concept of entropy. History In his 1803 paper, Fundamental Principles of Equilibrium and Movement, the French mathematician Lazare Carnot proposed that in any machine, the accelerations and shocks of the moving parts represent losses of moment of activity; in any natural process there exists an inherent tendency towards the dissipation of useful energy. In 1824, building on that work, Lazare's son, Sadi Carnot, published Reflections on the Motive Power of Fire, which posited that in all heat-engines, whenever "caloric" (what is now known as heat) falls through a temperature difference, work or motive power can be produced from the actions of its fall from a hot to cold body. He used an analogy with how water falls in a water wheel. That was an early insight into the second law of thermodynamics.[5] 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.[6] Carnot reasoned that if the body of the working substance, such as a body of steam, is returned to its original state at the 7 end of a complete engine cycle, "no change occurs in the condition of the working body". The first law of thermodynamics, deduced from the heat-friction experiments of James Joule in 1843, expresses the concept of energy, and its conservation in all processes; the first law, however, is unsuitable to separately quantify the effects of friction and dissipation. 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., heat produced by friction.[7] He described his observations as a dissipative use of energy, resulting in a transformationcontent (Verwandlungsinhalt in German), of a thermodynamic system or working body of chemical species during a change of state.[7] That was in contrast to earlier views, based on the theories of Isaac Newton, that heat was an indestructible particle that had mass. Clausius discovered that the non-usable energy increases as steam proceeds from inlet to exhaust in a steam engine. From the prefix en-, as in 'energy', and from the Greek word τροπή [tropē], which is translated in an established lexicon as turning or change[8] and that he rendered in German as Verwandlung, a word often translated into English as transformation, in 1865 Clausius coined the name of that property as entropy.[9] The word was adopted into the English language in 1868. Later, scientists such as Ludwig Boltzmann, Josiah Willard Gibbs, and James Clerk Maxwell gave entropy a statistical basis. In 1877, Boltzmann visualized a probabilistic way to measure the entropy of an ensemble of ideal gas particles, in which he defined entropy as proportional to the natural logarithm of the number of microstates such a gas could occupy. The proportionality constant in this definition, called the Boltzmann constant, has become one of the defining universal constants for the modern International System of Units (SI). Henceforth, the essential problem in statistical 8 thermodynamics has been to determine the distribution of a given amount of energy E over N identical systems. Constantin Carathéodory, a Greek mathematician, linked entropy with a mathematical definition of irreversibility, in terms of trajectories and integrability. 9 1. 2. 3. 4. 5. 6. 7. 8. References Wehrl, Alfred (1 April 1978). "General properties of entropy". Reviews of Modern Physics. 260. Bibcode:1978RvMP...50..221W. doi:10.1103/RevModPhys.50.221. Truesdell, C. (1980). The Tragicomical History of Thermodynamics, 1822– 1854. New York: Springer-Verlag. p. 215. ISBN 0387904034 – via Internet Archive. Brush, S.G. (1976). The Kind of Motion We Call Heat: a History of the Kinetic Theory of Gases in the 19th Century, Book 2, Statistical Physics and Irreversible Processes, Elsevier, Amsterdam, ISBN 0-444-87009-1, pp. 576– 577. Ben-Naim, Arieh (2008). A Farewell to Entropy: Statistical Thermodynamics Based on Information. Singapore: World-Scientific Publishing. ISBN 9789812707062. "Carnot, Sadi (1796–1832)". Wolfram Research. 2007. Retrieved 24 February 2010. McCulloch, Richard, S. (1876). Treatise on the Mechanical Theory of Heat and its Applications to the Steam-Engine, etc. D. Van Nostrand. Jump up to:a b Clausius, Rudolf (1850). "Über die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen". Annalen der Physik. 155 (3): 368– 397. Bibcode:1850AnP...155..368C. doi:10.1002/andp.18501550306. 10
