The concept of entropy developed in response to the observation that a certain amount of functional energy released from combustion reactions is always lost to dissipation or friction and is thus not transformed into useful work .^{[citation needed]} Early heatpowered engines such as Thomas Savery's (1698), the Newcomen engine (1712) and the Cugnot steam tricycle (1769) were inefficient, converting less than two percent of the input energy into useful work output; a great deal of useful energy was dissipated or lost into what seemed like a state of immeasurable randomness.^{[original research?]} Over the next two centuries, physicists investigated this puzzle of lost energy; the result was the concept of entropy.In the early 1850s, Rudolf Clausius set forth the concept of the thermodynamic system and posited the argument that in any irreversible process a small amount of heat energy δQ is incrementally dissipated across the system boundary. Clausius continued to develop his ideas of lost energy, and coined the term entropy.Since the mid20th century the concept of entropy has found application in the analogous field of data loss in information transmission systems.^{[citation needed]}
Contents
[hide]
[edit]Classical thermodynamic views
Main article: classical thermodynamicsIn 1803, mathematician Lazare Carnot published a work entitled Fundamental Principles of Equilibrium and Movement. This work includes a discussion on the efficiency of fundamental machines, i.e. pulleys and inclined planes. Lazare Carnot saw through all the details of the mechanisms to develop a general discussion on the conservation of mechanical energy. Over the next three decades, Lazare Carnot’s theorem was taken as a statement that in any machine the accelerations and shocks of the moving parts all represent losses of moment of activity, i.e. the useful work done. From this Lazare drew the inference that perpetual motion was impossible. This loss of moment of activity was the firstever rudimentary statement of the second law of thermodynamics and the concept of 'transformationenergy' or entropy, i.e. energy lost to dissipation and friction.^{[1]}Lazare Carnot died in exile in 1823. During the following year Lazare’s son Sadi Carnot, having graduated from the École Polytechnique training school for engineers, but now living on halfpay with his brother Hippolyte in a small apartment in Paris, wrote the Reflections on the Motive Power of Fire. In this paper, Sadi visualized an ideal engine in which the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Building on his father's work, Sadi postulated the concept that “some caloric is always lost”, not being converted to mechanical work. Hence any real heat engine could not realize the Carnot cycle's reversibility and was condemned to be less efficient. This lost caloric was a precursory form of entropy loss as we now know it. Though formulated in terms of caloric, rather than entropy, this was an early insight into the second law of thermodynamics.[edit]1854 definition
In his 1854 memoir, Clausius first develops the concepts of interior work, i.e. that "which the atoms of the body exert upon each other", and exterior work, i.e. that "which arise from foreign influences [to] which the body may be exposed", which may act on a working body of fluid or gas, typically functioning to work a piston. He then discusses the three categories into which heat Q may be divided: Heat employed in increasing the heat actually existing in the body.
 Heat employed in producing the interior work.
 Heat employed in producing the exterior work.
Building on this logic, and following a mathematical presentation of the first fundamental theorem, Clausius then presented the firstever mathematical formulation of entropy, although at this point in the development of his theories he called it "equivalencevalue", perhaps referring to the concept of themechanical equivalent of heat which was developing at the time rather than entropy, a term which was to come into use later.^{[2]} He stated:^{[3]}the second fundamental theorem in the mechanical theory of heat may thus be enunciated: If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generations of the quantity of heat Q from work at the temperature T , has the equivalencevalue:
and the passage of the quantity of heat Q from the temperature T_{1} to the temperature T_{2}, has the equivalencevalue:

wherein T is a function of the temperature, independent of the nature of the process by which the transformation is effected.
In modern terminology, we think of this equivalencevalue as "entropy", symbolized by S. Thus, using the above description, we can calculate the entropy change ΔS for the passage of the quantity ofheat Q from the temperature T_{1}, through the "working body" of fluid (see heat engine), which was typically a body of steam, to the temperature T_{2} as shown below:If we make the assignment:
Then, the entropy change or "equivalencevalue" for this transformation is:
which equals:
and by factoring out Q, we have the following form, as was derived by Clausius:
[edit]1856 definition
In 1856, Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:
where N is the "equivalencevalue" of all uncompensated transformations involved in a cyclical process. This equivalencevalue was a precursory formulation of entropy.^{[4]}[edit]1862 definition
Main article: disgregationIn 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: The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing.
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 was an early formulation of the second law and one of the original forms of the concept of entropy.[edit]1865 definition
In 1865, Clausius gave irreversible heat loss, or what he had previously been calling "equivalencevalue", a name:^{[5]}^{[6]}
“ I propose to name the quantity S the entropy of the system, after the Greek word [τροπη trope], the transformation. I have deliberately chosen the word entropy to be as similar as possible to the word energy: the two quantities to be named by these words are so closely related in physical significance that a certain similarity in their names appears to be appropriate. ”
Although Clausius did not specify why he chose the symbol "S" to represent entropy, it is arguable that Clausius chose "S" in honor of Sadi Carnot, to whose 1824 article Clausius devoted over 15 years of work and research. On the first page of his original 1850 article "On the Motive Power of Heat, and on the Laws which can be Deduced from it for the Theory of Heat", Clausius calls Carnot the most important of the researchers in the theory of heat.^{[7]}[edit]Later developments
In 1876, physicist J. Willard Gibbs, building on the work of Clausius, Hermann von Helmholtz and others, proposed that the measurement of "available energy" ΔG in a thermodynamic system could be mathematically accounted for by subtracting the "energy loss" TΔS from total energy change of the system ΔH. These concepts were further developed by James Clerk Maxwell [1871] and Max Planck[1903].[edit]Statistical thermodynamic views
Main article: statistical thermodynamicsIn 1877, Ludwig Boltzmann formulated the alternative definition of entropy S defined as:
where k_{B} is Boltzmann's constant and
 Ω is the number of microstates consistent with the given macrostate.
Boltzmann saw entropy as a measure of statistical "mixedupness" or disorder. This concept was soon refined by J. Willard Gibbs, and is now regarded as one of the cornerstones of the theory ofstatistical mechanics.[edit]Information theory
An analog to thermodynamic entropy is information entropy. In 1948, while working at Bell Telephone Laboratories electrical engineer Claude Shannon set out to mathematically quantify the statistical nature of “lost information” in phoneline signals. To do this, Shannon developed the very general concept of information entropy, a fundamental cornerstone of information theory. Although the story varies, initially it seems that Shannon was not particularly aware of the close similarity between his new quantity and earlier work in thermodynamics. In 1949, however, when Shannon had been working on his equations for some time, he happened to visit the mathematician John von Neumann. During their discussions, regarding what Shannon should call the “measure of uncertainty” or attenuation in phoneline signals with reference to his new information theory, according to one source:^{[8]}
“ My greatest concern was what to call it. I thought of calling it ‘information’, but the word was overly used, so I decided to call it ‘uncertainty’. When I discussed it with John von Neumann, he had a better idea. Von Neumann told me, ‘You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage. ”
According to another source, when von Neumann asked him how he was getting on with his information theory, Shannon replied:^{[9]}
“ The theory was in excellent shape, except that he needed a good name for “missing information”. “Why don’t you call it entropy”, von Neumann suggested. “In the first place, a mathematical development very much like yours already exists in Boltzmann’s statistical mechanics, and in the second place, no one understands entropy very well, so in any discussion you will be in a position of advantage. ”
In 1948 Shannon published his famous paper A Mathematical Theory of Communication, in which he devoted a section to what he calls Choice, Uncertainty, and Entropy.^{[10]} In this section, Shannon introduces an H function of the following form:
where K is a positive constant. Shannon then states that “any quantity of this form, where K merely amounts to a choice of a unit of measurement, plays a central role in information theory as measures of information, choice, and uncertainty.” Then, as an example of how this expression applies in a number of different fields, he references R.C. Tolman’s 1938 Principles of Statistical Mechanics, stating that “the form of H will be recognized as that of entropy as defined in certain formulations of statistical mechanics where p_{i} is the probability of a system being in cell i of its phase space… H is then, for example, the H in Boltzmann’s famous H theorem.” As such, over the last fifty years, ever since this statement was made, people have been overlapping the two concepts or even stating that they are exactly the same.Shannon's information entropy is a much more general concept than statistical thermodynamic entropy. Information entropy is present whenever there are unknown quantities that can be described only by a probability distribution. In a series of papers by E. T. Jaynes starting in 1957,^{[11]}^{[12]} the statistical thermodynamic entropy can be seen as just a particular application of Shannon's information entropy to the probabilities of particular microstates of a system occurring in order to produce a particular macrostate.[edit]Popular use
The term entropy is often used in popular language to denote a variety of unrelated phenomena. One example is the concept of corporate entropy as put forward somewhat humorously by authors Tom DeMarco and Timothy Lister in their 1987 classic publication Peopleware, a book on growing and managing productive teams and successful software projects. Here, they view energy waste as red tape and business team inefficiency as a form of entropy, i.e. energy lost to waste. This concept has caught on and is now common jargon in business schools.[edit]Terminology overlap
When necessary, to disambiguate between the statistical thermodynamic concept of entropy, and entropylike formulae put forward by different researchers, the statistical thermodynamic entropy is most properly referred to as the Gibbs entropy. The terms BoltzmannGibbs entropy or BG entropy, and BoltzmannGibbsShannon entropy or BGS entropy are also seen in the literature.[edit]See also
[edit]References
 ^ Mendoza, E. (1988). Reflections on the Motive Power of Fire – and other Papers on the Second Law of Thermodynamics by E. Clapeyron and R. Clausius. New York: Dover Publications. ISBN 0486446417.
 ^ Mechanical Theory of Heat, by Rudolf Clausius, 18501865
 ^ Published in Poggendoff’s Annalen, December 1854, vol. xciii. p. 481; translated in the Journal de Mathematiques, vol. xx. Paris, 1855, and in the Philosophical Magazine, August 1856, s. 4. vol. xii, p. 81
 ^ Clausius, Rudolf. (1856). "On the Application of the Mechanical theory of Heat to the SteamEngine." as found in: Clausius, R. (1865). The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies. London: John van Voorst, 1 Paternoster Row. MDCCCLXVII.
 ^ Laidler, Keith J. (1995). The Physical World of Chemistry. Oxford University Press. pp. 104–105. ISBN 0198559194.
 ^ OED, Second Edition, 1989, "Clausius (Pogg. Ann. CXXV. 390), assuming (unhistorically) the etymological sense of energy to be ‘workcontents’ (werkinhalt), devised the term entropy as a corresponding designation for the ‘transformationcontents’ (verwandlungsinhalt) of a system"
 ^ Clausius, Rudolf (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). ISBN 0486590658.
 ^ M. Tribus, E.C. McIrvine, “Energy and information”, Scientific American, 224 (September 1971).
 ^ Avery, John (2003). Information Theory and Evolution. World Scientific. ISBN 9812384006.
 ^ C.E. Shannon, "A Mathematical Theory of Communication", Bell System Technical Journal, vol. 27, pp. 379423, 623656, July, October, 1948, Eprint, PDF
 ^ E. T. Jaynes (1957) Information theory and statistical mechanics, Physical Review 106:620
 ^ E. T. Jaynes (1957) Information theory and statistical mechanics II, Physical Review 108:171
[edit]External links
History of entrop
The concept of entropy developed in response to the observation that a certain amount of functional energy released from combustion reactions is always lost to dissipation or friction and is thus not transformed into useful work .^{[citation needed]} Early heatpowered engines such as Thomas Savery's (1698), the Newcomen engine (1712) and the Cugnot steam tricycle (1769) were inefficient, converting less than two percent of the input energy into useful work output; a great deal of useful energy was dissipated or lost into what seemed like a state of immeasurable randomness.^{[original research?]} Over the next two centuries, physicists investigated this puzzle of lost energy; the result was the concept of entropy.
In the early 1850s, Rudolf Clausius set forth the concept of the thermodynamic system and posited the argument that in any irreversible process a small amount of heat energy δQ is incrementally dissipated across the system boundary. Clausius continued to develop his ideas of lost energy, and coined the term entropy.
Since the mid20th century the concept of entropy has found application in the analogous field of data loss in information transmission systems.^{[citation needed]}
Contents[hide] 
[edit]Classical thermodynamic views
Main article: classical thermodynamics
In 1803, mathematician Lazare Carnot published a work entitled Fundamental Principles of Equilibrium and Movement. This work includes a discussion on the efficiency of fundamental machines, i.e. pulleys and inclined planes. Lazare Carnot saw through all the details of the mechanisms to develop a general discussion on the conservation of mechanical energy. Over the next three decades, Lazare Carnot’s theorem was taken as a statement that in any machine the accelerations and shocks of the moving parts all represent losses of moment of activity, i.e. the useful work done. From this Lazare drew the inference that perpetual motion was impossible. This loss of moment of activity was the firstever rudimentary statement of the second law of thermodynamics and the concept of 'transformationenergy' or entropy, i.e. energy lost to dissipation and friction.^{[1]}
Lazare Carnot died in exile in 1823. During the following year Lazare’s son Sadi Carnot, having graduated from the École Polytechnique training school for engineers, but now living on halfpay with his brother Hippolyte in a small apartment in Paris, wrote the Reflections on the Motive Power of Fire. In this paper, Sadi visualized an ideal engine in which the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Building on his father's work, Sadi postulated the concept that “some caloric is always lost”, not being converted to mechanical work. Hence any real heat engine could not realize the Carnot cycle's reversibility and was condemned to be less efficient. This lost caloric was a precursory form of entropy loss as we now know it. Though formulated in terms of caloric, rather than entropy, this was an early insight into the second law of thermodynamics.
[edit]1854 definition
In his 1854 memoir, Clausius first develops the concepts of interior work, i.e. that "which the atoms of the body exert upon each other", and exterior work, i.e. that "which arise from foreign influences [to] which the body may be exposed", which may act on a working body of fluid or gas, typically functioning to work a piston. He then discusses the three categories into which heat Q may be divided:
 Heat employed in increasing the heat actually existing in the body.
 Heat employed in producing the interior work.
 Heat employed in producing the exterior work.
Building on this logic, and following a mathematical presentation of the first fundamental theorem, Clausius then presented the firstever mathematical formulation of entropy, although at this point in the development of his theories he called it "equivalencevalue", perhaps referring to the concept of themechanical equivalent of heat which was developing at the time rather than entropy, a term which was to come into use later.^{[2]} He stated:^{[3]}
the second fundamental theorem in the mechanical theory of heat may thus be enunciated: If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generations of the quantity of heat Q from work at the temperature T , has the equivalencevalue:and the passage of the quantity of heat Q from the temperature T_{1} to the temperature T_{2}, has the equivalencevalue:
wherein T is a function of the temperature, independent of the nature of the process by which the transformation is effected.
In modern terminology, we think of this equivalencevalue as "entropy", symbolized by S. Thus, using the above description, we can calculate the entropy change ΔS for the passage of the quantity ofheat Q from the temperature T_{1}, through the "working body" of fluid (see heat engine), which was typically a body of steam, to the temperature T_{2} as shown below:
If we make the assignment:
Then, the entropy change or "equivalencevalue" for this transformation is:
which equals:
and by factoring out Q, we have the following form, as was derived by Clausius:
[edit]1856 definition
In 1856, Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:
where N is the "equivalencevalue" of all uncompensated transformations involved in a cyclical process. This equivalencevalue was a precursory formulation of entropy.^{[4]}
[edit]1862 definition
Main article: disgregation
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:
 The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing.
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 was an early formulation of the second law and one of the original forms of the concept of entropy.
[edit]1865 definition
In 1865, Clausius gave irreversible heat loss, or what he had previously been calling "equivalencevalue", a name:^{[5]}^{[6]}
“  I propose to name the quantity S the entropy of the system, after the Greek word [τροπη trope], the transformation. I have deliberately chosen the word entropy to be as similar as possible to the word energy: the two quantities to be named by these words are so closely related in physical significance that a certain similarity in their names appears to be appropriate.  ” 
Although Clausius did not specify why he chose the symbol "S" to represent entropy, it is arguable that Clausius chose "S" in honor of Sadi Carnot, to whose 1824 article Clausius devoted over 15 years of work and research. On the first page of his original 1850 article "On the Motive Power of Heat, and on the Laws which can be Deduced from it for the Theory of Heat", Clausius calls Carnot the most important of the researchers in the theory of heat.^{[7]}
[edit]Later developments
In 1876, physicist J. Willard Gibbs, building on the work of Clausius, Hermann von Helmholtz and others, proposed that the measurement of "available energy" ΔG in a thermodynamic system could be mathematically accounted for by subtracting the "energy loss" TΔS from total energy change of the system ΔH. These concepts were further developed by James Clerk Maxwell [1871] and Max Planck[1903].
[edit]Statistical thermodynamic views
Main article: statistical thermodynamics
In 1877, Ludwig Boltzmann formulated the alternative definition of entropy S defined as:
where
 k_{B} is Boltzmann's constant and
 Ω is the number of microstates consistent with the given macrostate.
Boltzmann saw entropy as a measure of statistical "mixedupness" or disorder. This concept was soon refined by J. Willard Gibbs, and is now regarded as one of the cornerstones of the theory ofstatistical mechanics.
[edit]Information theory
An analog to thermodynamic entropy is information entropy. In 1948, while working at Bell Telephone Laboratories electrical engineer Claude Shannon set out to mathematically quantify the statistical nature of “lost information” in phoneline signals. To do this, Shannon developed the very general concept of information entropy, a fundamental cornerstone of information theory. Although the story varies, initially it seems that Shannon was not particularly aware of the close similarity between his new quantity and earlier work in thermodynamics. In 1949, however, when Shannon had been working on his equations for some time, he happened to visit the mathematician John von Neumann. During their discussions, regarding what Shannon should call the “measure of uncertainty” or attenuation in phoneline signals with reference to his new information theory, according to one source:^{[8]}
“  My greatest concern was what to call it. I thought of calling it ‘information’, but the word was overly used, so I decided to call it ‘uncertainty’. When I discussed it with John von Neumann, he had a better idea. Von Neumann told me, ‘You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage.  ” 
According to another source, when von Neumann asked him how he was getting on with his information theory, Shannon replied:^{[9]}
“  The theory was in excellent shape, except that he needed a good name for “missing information”. “Why don’t you call it entropy”, von Neumann suggested. “In the first place, a mathematical development very much like yours already exists in Boltzmann’s statistical mechanics, and in the second place, no one understands entropy very well, so in any discussion you will be in a position of advantage.  ” 
In 1948 Shannon published his famous paper A Mathematical Theory of Communication, in which he devoted a section to what he calls Choice, Uncertainty, and Entropy.^{[10]} In this section, Shannon introduces an H function of the following form:
where K is a positive constant. Shannon then states that “any quantity of this form, where K merely amounts to a choice of a unit of measurement, plays a central role in information theory as measures of information, choice, and uncertainty.” Then, as an example of how this expression applies in a number of different fields, he references R.C. Tolman’s 1938 Principles of Statistical Mechanics, stating that “the form of H will be recognized as that of entropy as defined in certain formulations of statistical mechanics where p_{i} is the probability of a system being in cell i of its phase space… H is then, for example, the H in Boltzmann’s famous H theorem.” As such, over the last fifty years, ever since this statement was made, people have been overlapping the two concepts or even stating that they are exactly the same.
Shannon's information entropy is a much more general concept than statistical thermodynamic entropy. Information entropy is present whenever there are unknown quantities that can be described only by a probability distribution. In a series of papers by E. T. Jaynes starting in 1957,^{[11]}^{[12]} the statistical thermodynamic entropy can be seen as just a particular application of Shannon's information entropy to the probabilities of particular microstates of a system occurring in order to produce a particular macrostate.
[edit]Popular use
The term entropy is often used in popular language to denote a variety of unrelated phenomena. One example is the concept of corporate entropy as put forward somewhat humorously by authors Tom DeMarco and Timothy Lister in their 1987 classic publication Peopleware, a book on growing and managing productive teams and successful software projects. Here, they view energy waste as red tape and business team inefficiency as a form of entropy, i.e. energy lost to waste. This concept has caught on and is now common jargon in business schools.
[edit]Terminology overlap
When necessary, to disambiguate between the statistical thermodynamic concept of entropy, and entropylike formulae put forward by different researchers, the statistical thermodynamic entropy is most properly referred to as the Gibbs entropy. The terms BoltzmannGibbs entropy or BG entropy, and BoltzmannGibbsShannon entropy or BGS entropy are also seen in the literature.
[edit]See also
[edit]References
 ^ Mendoza, E. (1988). Reflections on the Motive Power of Fire – and other Papers on the Second Law of Thermodynamics by E. Clapeyron and R. Clausius. New York: Dover Publications. ISBN 0486446417.
 ^ Mechanical Theory of Heat, by Rudolf Clausius, 18501865
 ^ Published in Poggendoff’s Annalen, December 1854, vol. xciii. p. 481; translated in the Journal de Mathematiques, vol. xx. Paris, 1855, and in the Philosophical Magazine, August 1856, s. 4. vol. xii, p. 81
 ^ Clausius, Rudolf. (1856). "On the Application of the Mechanical theory of Heat to the SteamEngine." as found in: Clausius, R. (1865). The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies. London: John van Voorst, 1 Paternoster Row. MDCCCLXVII.
 ^ Laidler, Keith J. (1995). The Physical World of Chemistry. Oxford University Press. pp. 104–105. ISBN 0198559194.
 ^ OED, Second Edition, 1989, "Clausius (Pogg. Ann. CXXV. 390), assuming (unhistorically) the etymological sense of energy to be ‘workcontents’ (werkinhalt), devised the term entropy as a corresponding designation for the ‘transformationcontents’ (verwandlungsinhalt) of a system"
 ^ Clausius, Rudolf (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). ISBN 0486590658.
 ^ M. Tribus, E.C. McIrvine, “Energy and information”, Scientific American, 224 (September 1971).
 ^ Avery, John (2003). Information Theory and Evolution. World Scientific. ISBN 9812384006.
 ^ C.E. Shannon, "A Mathematical Theory of Communication", Bell System Technical Journal, vol. 27, pp. 379423, 623656, July, October, 1948, Eprint, PDF
 ^ E. T. Jaynes (1957) Information theory and statistical mechanics, Physical Review 106:620
 ^ E. T. Jaynes (1957) Information theory and statistical mechanics II, Physical Review 108:171
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