EntropyFrom CryptoDox, The Online Encyclopedia on Cryptography and Information SecurityTemplate:Dablink Template:Seeintro In physics, entropy, symbolized by S, from the Greek μετατροπή (metatropi) meaning "transformation",<ref name="Clausius">Clausius, Rudolf. (1879). Mechanical Theory of Heat (pg. 107), 2nd Edition. London: Macmillan & Co.</ref><ref>The etymology of entropy, in modern terms, according to Perrot’s A to Z of Thermodynamics, can be interpreted to mean, “from the Greek root εντροπη, the act of turning around (τροπη, change of direction), implying the idea of reversibility”.</ref> is a measure of the unavailability of a system’s energy to do work.<ref name="Daintith" >{{{author}}}, 2005, Oxford Dictionary of Physics, Oxford University Press, {{{location}}}</ref> Entropy is central to the second law of thermodynamics and the combined law of thermodynamics, which deal with physical processes and whether they occur spontaneously. Spontaneous changes, in isolated systems, occur with an increase in entropy. Spontaneous changes tend to smooth out differences in temperature, pressure, density, and chemical potential that may exist in a system, and entropy is thus a measure of how far this smoothingout process has progressed. The concept of entropy was developed in the 1850s by German physicist Rudolf Clausius who described it as the transformationcontent, i.e. dissipative energy use, of a thermodynamic system or working body of chemical species during a change of state.<ref name="Clausius" /> In contrast, the first law of thermodynamics, formalized through the heatfriction experiments of James Joule in 1843, deals with the concept of energy, which is conserved in all processes; the first law, however, lacks in its ability to quantify the effects of friction and dissipation. Entropy change has often been defined as a change to a more disordered state at a molecular level. In recent years, entropy has been interpreted in terms of the "dispersal" of energy. Entropy is an extensive state function that accounts for the effects of irreversibility in thermodynamic systems. Quantitatively, entropy is defined by the differential quantity <math>dS = \delta Q/T</math>, where <math>\delta Q</math> is the amount of heat absorbed in an isothermal and reversible process in which the system goes from one state to another, and T is the absolute temperature at which the process is occurring.<ref name="Perrot" >{{{author}}}, 1998, A to Z of Thermodynamics, Oxford University Press, {{{location}}}</ref> Entropy is one of the factors that determines the free energy of the system. This thermodynamic definition of entropy is only valid for a system in equilibrium (because temperature is defined only for a system in equilibrium), while the statistical definition of entropy (see below) applies to any system. Thus the statistical definition is usually considered the fundamental definition of entropy. When a system's energy is defined as the sum of its "useful" energy, (e.g. that used to push a piston), and its "useless energy", i.e. that energy which cannot be used for external work, then entropy may be (most concretely) visualized as the "scrap" or "useless" energy whose energetic prevalence over the total energy of a system is directly proportional to the absolute temperature of the considered system. (Note the product "TS" in the Gibbs free energy or Helmholtz free energy relations). In terms of statistical mechanics, the entropy describes the number of the possible microscopic configurations of the system. The statistical definition of entropy is the more fundamental definition, from which all other definitions and all properties of entropy follow. Although the concept of entropy was originally a thermodynamic construct, it has been adapted in other fields of study, including information theory, psychodynamics, thermoeconomics, and evolution.<ref>{{{author}}}, 2003, Information Theory and Evolution, World Scientific, {{{location}}}</ref><ref>{{{author}}}, 2005, Information Theory, Evolution, and the Origin of Life., Cambridge University Press, {{{location}}}</ref><ref name="Brooks" >{{{author}}}, 1988, Entropy as Evolution – Towards a Unified Theory of Biology, University of Chicago Press, {{{location}}}</ref> HistoryMain article: History of entropy
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 recent 1789 views of Count Rumford who showed that heat could be created by friction as when cannon bores are machined.<ref>{{{author}}}, 1876, Treatise on the Mechanical Theory of Heat and its Applications to the SteamEngine, etc., D. Van Nostrand, {{{location}}}</ref> Accordingly, Carnot reasoned that if the body of the working substance, such as a body of steam, is brought back to its original state (temperature and pressure) at the end of a complete engine cycle, that "no change occurs in the condition of the working body." This latter comment was amended in his foot notes, and it was this comment that led to the development of entropy. In the 1850s and 60s, German physicist Rudolf Clausius gravely objected to this latter supposition, i.e. that no change occurs in the working body, and gave this "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.<ref name="Clausius" >{{{author}}}, 1850, On the Motive Power of Heat, and on the Laws which can be deduced from it for the Theory of Heat, Poggendorff's Annalen der Physick, LXXIX (Dover Reprint), {{{location}}}</ref> This was in contrast to earlier views, based on the theories of Isaac Newton, that heat was an indestructible particle that had mass. Later, scientists such as Ludwig Boltzmann, Willard Gibbs, and James Clerk Maxwell gave entropy a statistical basis. Carathéodory linked entropy with a mathematical definition of irreversibility, in terms of trajectories and integrability. Definitions and descriptionsIn science, the term "entropy" is generally interpreted in three distinct, but semirelated, ways, i.e. from macroscopic viewpoint (classical thermodynamics), a microscopic viewpoint (statistical thermodynamics), and an information viewpoint (information theory). Entropy in information theory is a fundamentally different concept from thermodynamic entropy. However, at a philosophical level, some argue that thermodynamic entropy can be interpreted as an application of the information entropy concept to a highly specific set of physical questions. The statistical definition of entropy (see below) is the fundamental definition because the other two can be mathematically derived from it, but not vice versa. All properties of entropy (including second law of thermodynamics) follow from this definition. Macroscopic viewpoint (classical thermodynamics)Template:Conjugate variables (thermodynamics)
Main article: Entropy (classical thermodynamics)
In a thermodynamic system, a "universe" consisting of "surroundings" and "systems" and made up of quantities of matter, its pressure differences, density differences, and temperature differences all tend to equalize over time  simply because equilibrium state has higher probability (more possible combinations of microstates) than any other  see statistical mechanics. In the ice melting example, the difference in temperature between a warm room (the surroundings) and cold glass of ice and water (the system and not part of the room), begins to be equalized as portions of the heat energy from the warm surroundings become spread out to the cooler system of ice and water. Over time the temperature of the glass and its contents and the temperature of the room become equal. The entropy of the room has decreased as some of its energy has been dispersed to the ice and water. However, as calculated in the example, the entropy of the system of ice and water has increased more than the entropy of the surrounding room has decreased. In an isolated system such as the room and ice water taken together, the dispersal of energy from warmer to cooler always results in a net increase in entropy. Thus, when the 'universe' of the room and ice water system has reached a temperature equilibrium, the entropy change from the initial state is at a maximum. The entropy of the thermodynamic system is a measure of how far the equalization has progressed. A special case of entropy increase, the entropy of mixing, occurs when two or more different substances are mixed. If the substances are at the same temperature and pressure, there will be no net exchange of heat or work  the entropy increase will be entirely due to the mixing of the different substances.<ref>See, e.g., Notes for a “Conversation About Entropy” for a brief discussion of both thermodynamic and "configurational" ("positional") entropy in chemistry.</ref> From a macroscopic perspective, in classical thermodynamics the entropy is interpreted simply 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. The state function has the important property that, when multiplied by a reference temperature, it can be understood as a measure of the amount of energy in a physical system that cannot be used to do thermodynamic work; i.e., work mediated by thermal energy. More precisely, in any process where the system gives up energy ΔE, and its entropy falls by ΔS, a quantity at least T_{R} ΔS of that energy must be given up to the system's surroundings as unusable heat (T_{R} is the temperature of the system's external surroundings). Otherwise the process will not go forward. In 1862, Clausius stated what he calls the “theorem respecting the equivalencevalues of the transformations” or what is now known as the second law of thermodynamics, as such:
Quantitatively, Clausius states the mathematical expression for this theorem is as follows. Let δQ be an element of the heat given up by the body to any reservoir of heat during its own changes, heat which it may absorb from a reservoir being here reckoned as negative, and T the absolute temperature of the body at the moment of giving up this heat, then the equation:
must be true for every reversible cyclical process, and the relation:
must hold good for every cyclical process which is in any way possible. This is the essential formulation of the second law and one of the original forms of the concept of entropy. It can be seen that the dimensions of entropy are energy divided by temperature, which is the same as the dimensions of Boltzmann's constant (k_{B}) and heat capacity. The SI unit of entropy is "joule per kelvin" (J•K^{−1}). In this manner, the quantity "ΔS" is utilized as a type of internal energy, which accounts for the effects of irreversibility, in the energy balance equation for any given system. In the Gibbs free energy equation, i.e. ΔG = ΔH  TΔS, for example, which is a formula commonly utilized to determine if chemical reactions will occur, the energy related to entropy changes TΔS is subtracted from the "total" system energy ΔH to give the "free" energy ΔG of the system, as during a chemical process or as when a system changes state. Microscopic definition of entropy (statistical mechanics)Main article: Entropy (statistical thermodynamics)
In statistical thermodynamics the entropy is defined as the number of microscopic configurations that result in the observed macroscopic description of the thermodynamic system:
where
This definition is considered to be the fundamental definition of entropy (as all other definitions can be mathematically derived from it, but not vice versa). In Boltzmann's 1896 Lectures on Gas Theory, he showed that this expression gives a measure of entropy for systems of atoms and molecules in the gas phase, thus providing a measure for the entropy of classical thermodynamics. In 1877, thermodynamicist Ludwig Boltzmann visualized a probabilistic way to measure the entropy of an ensemble of ideal gas particles, in which he defined entropy to be proportional to the logarithm of the number of microstates such a gas could occupy. Henceforth, the essential problem in statistical thermodynamics, i.e. according to Erwin Schrödinger, has been to determine the distribution of a given amount of energy E over N identical systems. Statistical mechanics explains entropy as the amount of uncertainty (or "mixedupness" in the phrase of Gibbs) which remains about a system, after its observable macroscopic properties have been taken into account. For a given set of macroscopic quantities, like temperature and volume, the entropy measures the degree to which the probability of the system is spread out over different possible quantum states. The more states available to the system with higher probability, and thus the greater the entropy. In essence, the most general interpretation of entropy is as a measure of our ignorance about a system. The equilibrium state of a system maximizes the entropy because we have lost all information about the initial conditions except for the conserved quantities; maximizing the entropy maximizes our ignorance about the details of the system.<ref>EntropyOrderParametersComplexity.pdf</ref> On the molecular scale, the two definitions match up because adding heat to a system, which increases its classical thermodynamic entropy, also increases the system's thermal fluctuations, so giving an increased lack of information about the exact microscopic state of the system, i.e. an increased statistical mechanical entropy. Entropy in chemical thermodynamicsMain article: Chemical thermodynamics
Entropy is equally essential in predicting the extent of complex chemical reactions, i.e. whether a process will go as written or proceed in the opposite direction. For such applications, ΔS must be incorporated in an expression that includes both the system and its surroundings, Δ S_{universe} = ΔS_{surroundings} + Δ S _{system}. This expression becomes, via some steps, the Gibbs free energy equation for reactants and products in the system: Δ G [the Gibbs free energy change of the system] = Δ H [the enthalpy change] – T Δ S [the entropy change].<ref name=ctms/> The second lawMain article: Second law of thermodynamics
In general, according to the second law, the entropy of a system that is not isolated may decrease. An air conditioner, for example, cools the air in a room, thus reducing the entropy of the air. The heat, however, involved in operating the air conditioner always makes a bigger contribution to the entropy of the environment than the decrease of the entropy of the air. Thus the total entropy of the room and the environment increases, in agreement with the second law. Entropy balance equation for open systemsIn chemical engineering, the principles of thermodynamics are commonly applied to "open systems", i.e. those in which heat, work, and mass flow across the system boundary. In a system in which there are flows of both heat (<math>\dot{Q}</math>) and work, i.e. <math>\dot{W}_S</math> (shaft work) and P(dV/dt) (pressurevolume work), across the system boundaries, the heat flow, but not the work flow, causes a change in the entropy of the system. This rate of entropy change is <math>\dot{Q}/T</math>, where T is the absolute thermodynamic temperature of the system at the point of the heat flow. If, in addition, there are mass flows across the system boundaries, the total entropy of the system will also change due to this convected flow. To derive a generalized entropy balanced equation, we start with the general balance equation for the change in any extensive quantity Θ in a thermodynamic system, a quantity that may be either conserved, such as energy, or nonconserved, such as entropy. The basic generic balance expression states that dΘ/dt, i.e. the rate of change of Θ in the system, equals the rate at which Θ enters the system at the boundaries, minus the rate at which Θ leaves the system across the system boundaries, plus the rate at which Θ is generated within the system. Using this generic balance equation, with respect to the rate of change with time of the extensive quantity entropy S, the entropy balance equation for an open thermodynamic system is:<ref>{{{author}}}, 1989, Chemical and Engineering Thermodynamics, John Wiley & Sons, {{{location}}}</ref>
where
Note, also, that if there are multiple heat flows, the term <math>\dot{Q}/T</math> is to be replaced by <math>\sum \dot{Q}_j/T_j</math>, where <math>\dot{Q}_j</math> is the heat flow and <math>T_j</math> is the temperature at the jth heat flow port into the system. Entropy in quantum mechanics (von Neumann entropy)In quantum statistical mechanics, the concept of entropy was developed by John von Neumann and is generally referred to as "von Neumann entropy". Von Neumann established the correct mathematical framework for quantum mechanics with his work Mathematische Grundlagen der Quantenmechanik. He provided in this work a theory of measurement, where the usual notion of wave collapse is described as an irreversible process (the so called von Neumann or projective measurement). Using this concept, in conjunction with the density matrix he extended the classical concept of entropy into the quantum domain. It is well known that a Shannon based definition of information entropy leads in the classical case to the Boltzmann entropy. It is tempting to regard the Von Neumann entropy as the corresponding quantum mechanical definition. But the latter is problematic from quantum information point of view. Consequently Stotland, Pomeransky, Bachmat and Cohen have introduced a new definition of entropy that reflects the inherent uncertainty of quantum mechanical states. This definition allows to distinguish between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.<ref> The information entropy of quantum mechanical states, Europhysics Letters 67, 700 (2004) </ref> Standard textbook definitionsNote that textbook definitions are not always the most helpful definitions, but they are an important aspect of the culture surrounding the concept of entropy.
Approaches to understanding entropyOrder and disorderMain article: Entropy (order and disorder)
Similarly, the total amount of "order" in the system is given by:
In which C_{D} is the "disorder" capacity of the system, which is the entropy of the parts contained in the permitted ensemble, C_{I} is the "information" capacity of the system, an expression similar to Shannon's channel capacity, and C_{O} is the "order" capacity of the system.<ref name="Brooks" /> Energy dispersalMain article: Entropy (energy dispersal)
The concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature.<ref>Frank L. Lambert, A Student’s Approach to the Second Law and Entropy</ref> 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. Ambiguities in the terms disorder and chaos, which usually have meanings directly opposed to equilibrium, contribute to widespread confusion and hamper comprehension of entropy for most students.<ref>Carson, E. M. and J. R. Watson (Department of Educational and Professional Studies, Kings College, London), Undergraduate students' understandings of entropy and Gibbs Free energy, University Chemistry Education  2002 Papers, Royal Society of Chemistry.</ref> As the second law of thermodynamics shows, in an isolated system internal portions at different temperatures will tend to adjust to a single uniform temperature and thus produce equilibrium. A recently developed educational approach avoids ambiguous terms and describes such spreading out of energy as dispersal, which leads to loss of the differentials required for work even though the total energy remains constant in accordance with the first law of thermodynamics.<ref>Frank L. Lambert, JCE 2002 (79) 187 [Feb DisorderA Cracked Crutch for Supporting Entropy Discussions]</ref> Physical chemist Peter Atkins, for example, who previously wrote of dispersal leading to a disordered state, now writes that "spontaneous changes are always accompanied by a dispersal of energy", and has discarded 'disorder' as a description.<ref>{{{author}}}, 1984, The Second Law, Scientific American Library, {{{location}}}</ref><ref name=atkins/> Entropy and Information theoryMain article: Information entropy
In information theory, entropy is the measure of the amount of information that is missing before reception and is sometimes referred to as Shannon entropy.<ref>Balian, Roger (2003). Entropy – Protean Concept (PDF). Poincare Seminar 2: 11945.</ref> Shannon entropy is a broad and general concept which finds applications in information theory as well as thermodynamics. It was originally devised by Claude Shannon in 1948 to study the amount of information in a transmitted message. The definition of the information entropy is, however, quite general, and is expressed in terms of a discrete set of probabilities <math>p_i</math>. In the case of transmitted messages, these probabilities were the probabilities that a particular message was actually transmitted, and the entropy of the message system was a measure of how much information was in the message. For the case of equal probabilities (i.e. each message is equally probable), the Shannon entropy (in bits) is just the number of yes/no questions needed to determine the content of the message. The question of the link between information entropy and thermodynamic entropy is a hotly debated topic. Many authors argue that there is a link between the two,<ref>{{{author}}}, 1956, Science and Information Theory, name, {{{location}}}</ref><ref>{{{author}}}, 1971, The Entropy Law and the Economic Process, Harvard University Press, {{{location}}}</ref><ref>{{{author}}}, 2005, The Physical Foundation of Economics  an Analytical Thermodynamic Theory, World Scientific, {{{location}}}</ref> while others will argue that they have absolutely nothing to do with each other.<ref>Lin, ShuKun. (1999). “Diversity and Entropy.” Entropy (Journal), 1[1], 13.</ref> The expressions for the two entropies are very similar. The information entropy H for equal probabilities <math>p_i=p</math> is:
where K is a constant which determines the units of entropy. For example, if the units are bits, then K=1/\ln(2). The thermodynamic entropy S , from a statistical mechanical point of view was first expressed by Boltzmann:
where p is the probability of a system being in a particular microstate, given that it is in a particular macrostate, and k is Boltzmann's constant. It can be seen that one may think of the thermodynamic entropy as Boltzmann's constant, divided by ln(2), times the number of yes/no questions that must be asked in order to determine the microstate of the system, given that we know the macrostate. The link between thermodynamic and information entropy was developed in a series of papers by Edwin Jaynes beginning in 1957.<ref>http://bayes.wustl.edu/etj/node1.html</ref> The problem with linking thermodynamic entropy to information entropy is that in information entropy the entire body of thermodynamics which deals with the physical nature of entropy is missing. The second law of thermodynamics which governs the behavior of thermodynamic systems in equilibrium, and the first law which expresses heat energy as the product of temperature and entropy are physical concepts rather than informational concepts. If thermodynamic entropy is seen as including all of the physical dynamics of entropy as well as the equilibrium statistical aspects, then information entropy gives only part of the description of thermodynamic entropy. Some authors, like Tom Schneider, argue for dropping the word entropy for the H function of information theory and using Shannon's other term "uncertainty" instead.<ref>Schneider, Tom, Delila Syste (Deoxyrobonucleic acid Library Language), (Information Theory Analysis of binding sites), Laboratory of Mathematical Biology, National Cancer Institute, FCRDC Bldg. 469. Rm 144, P.O. Box. B Frederick, MD 217021201, USA.</ref> Ice melting exampleMain article: disgregation
It is important to realize that the decrease in the entropy of the surrounding room is less than the increase in the entropy of the ice and water: the room temperature of 298 K is larger than 273 K and therefore the ratio, (entropy change), of δQ/298 K for the surroundings is smaller than the ratio (entropy change), of δQ/273 K for the ice+water system. To find the entropy change of our 'universe', we add up the entropy changes for its constituents: the surrounding room, and the ice+water. The total entropy change is positive; this is always true in spontaneous events in a thermodynamic system and it shows the predictive importance of entropy: the final net entropy after such an event is always greater than was the initial entropy. As the temperature of the cool water rises to that of the room and the room further cools imperceptibly, the sum of the δQ/T over the continuous range, at many increments, in the initially cool to finally warm water can be found by calculus. The entire miniature "universe", i.e. this thermodynamic system, has increased in entropy. Energy has spontaneously become more dispersed and spread out in that "universe" than when the glass of ice water was introduced and became a "system" within it. Topics in entropyEntropy and lifeMain article: Entropy and life
In the popular 1982 textbook Principles of Biochemistry by noted American biochemist Albert Lehninger, for example, it is argued that the order produced within cells as they grow and divide is more than compensated for by the disorder they create in their surroundings in the course of growth and division. In short, according to Lehninger, "living organisms preserve their internal order by taking from their surroundings free energy, in the form of nutrients or sunlight, and returning to their surroundings an equal amount of energy as heat and entropy."<ref name="Lehninger" >{{{author}}}, 1993, Principles of Biochemistry, 2nd Ed., Worth Publishers, {{{location}}}</ref> Evolution related definitions:
The arrow of timeMain article: Entropy (arrow of time)
Entropy and cosmologyMain article: Black hole thermodynamics
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. Jacob Bekenstein and Stephen Hawking have shown that black holes have the maximum possible entropy of any object of equal size. This makes them likely end points of all entropyincreasing processes, if they are totally effective matter and energy traps. Hawking has, however, recently changed his stance on this aspect. The role of entropy in cosmology remains a controversial subject. Recent work has cast extensive doubt on the heat death hypothesis and the applicability of any simple thermodynamic model to the universe in general. Although entropy does increase in the model of an expanding universe, the maximum possible entropy rises much more rapidly  thus entropy density is decreasing with time. This results in an "entropy gap" pushing the system further away from equilibrium. Other complicating factors, such as the energy density of the vacuum and macroscopic quantum effects, are difficult to reconcile with thermodynamical models, making any predictions of largescale thermodynamics extremely difficult. Template:Fact Miscellaneous definitions
Other relationsOther mathematical definitions
Sociological definitionsThe concept of entropy has also entered the domain of sociology, generally as a metaphor for chaos, disorder or dissipation of energy, rather than as a direct measure of thermodynamic or information entropy:
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