TheInfoList

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 Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, where it was first recognized, to the microscopic description of nature in
statistical physics Statistical physics is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), ...
, and to the principles of
information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamentally established by the ...
. It has found far-ranging applications in
chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in the real world. T ...

and
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, in biological systems and their relation to life, in
cosmology Cosmology (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...
,
economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interact ...

,
sociology Sociology is a social science Social science is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the scie ...
, weather science,
climate change Contemporary climate change includes both the global warming caused by humans, and its impacts on Earth's weather patterns. There have been previous periods of climate change, but the current changes are more rapid than any known even ...
, and
information system An information system (IS) is a formal, sociotechnical, organizational system designed to collect, process, information storage, store, and information distribution, distribute information. From a sociotechnical perspective, information systems ar ...
s including the transmission of information in
telecommunication Telecommunication is the transmission of information by various types of technologies over wire A wire is a single usually cylindrical A cylinder (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Gr ...
. The thermodynamic concept was referred to by Scottish scientist and engineer
Macquorn Rankine William John Macquorn Rankine (; 5 July 1820 – 24 December 1872) was a Scottish mechanical engineer who also contributed to civil engineering Civil engineering is a Regulation and licensure in engineering, professional engineering disciplin ...
in 1850 with the names ''thermodynamic function'' and ''heat-potential''. In 1865, German physicist
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citize ...
, 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 Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

. 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 In the history of thermodynamics, disgregation was defined in 1862 by Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist A physicist is a scientist A scientist is a person who co ...
. 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, , pp. 576–577. A consequence of entropy is that certain processes are irreversible or impossible, aside from the requirement of not violating the
conservation of energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
, the latter being expressed in the
first law of thermodynamics The first law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter tran ...
. Entropy is central to the
second law of thermodynamics The second law of thermodynamics establishes the concept of entropy 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 an ...
, 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 Thermodynamic equilibrium is an axiomatic An axiom, postulate or assumption is a statement that is taken to be true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online ...
, where the entropy is highest. Austrian physicist
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austria Austria, officially the Republic of Austria, is a landlocked country in the southern part of Central Europe, located on the Eastern Alps. It is compo ...
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 distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
s into a new field of thermodynamics, called
statistical mechanics In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
, and found the link between the microscopic interactions, which fluctuate about an average configuration, to the macroscopically observable behavior, in form of a simple
logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

ic law, with a
proportionality constant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor In mathematics, two varying quantities are said to be in a Binary relation, relation of proportionality, Multiplication, multiplicatively connected to a Constant (mathematics), c ...
, that has become one of the defining universal constants for the modern
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...
(SI). In 1948,
Bell Labs Nokia Bell Labs (formerly named Bell Labs Innovations (1996–2007), AT&T Bell Laboratories (1984–1996) and Bell Telephone Laboratories (1925–1984)) is an American industrial research and scientific development company A company, ab ...
scientist
Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbe ...
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 John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian Americans, Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. Von Neumann was generally rega ...

'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 Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamentally established by the ...
. This description has been identified as a universal definition of the concept of entropy.Arieh Ben-Naim, ''A Farewell to Entropy: Statistical Thermodynamics Based on Information'', World-Scientific Publishing Co., Singapore, 2008, ISBN 978-981-270-706-2

# History

In his 1803 paper, ''Fundamental Principles of Equilibrium and Movement'', the French mathematician
Lazare Carnot Lazare Nicolas Marguerite, Count Carnot (13 May 1753 – 2 August 1823) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of s ...

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 ''Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power'' is a book published in 1824 by French people, French physicist Nicolas Léonard Sadi Carnot, Sadi Carnot.full text of 1897 ed. ( s:Reflections on the Motive P ...
'', which posited that in all heat-engines, whenever " caloric" (what is now known as heat) falls through a temperature difference, work or
motive power ''Motive Power'' is a bi-monthly railway related magazine that focuses on diesel locomotives in Rail transport in Australia, Australia. The first issue was published on 23 August 1998. Its headquarters is in Sydney. The content includes photograp ...

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 The reversible water wheel powering a mine hoist in ''De re metallica'' (Georgius Agricola">De_re_metallica.html" ;"title="mine hoist in ''De re metallica">mine hoist in ''De re metallica'' (Georgius Agricola, 1566) A water wheel is a machi ...

. That was an early insight into the
second law of thermodynamics The second law of thermodynamics establishes the concept of entropy 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 an ...
. 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 Sir Benjamin Thompson, Count Rumford, (german: von Rumford; March 26, 1753August 21, 1814) was an American-born British and whose challenges to established were part of the 19th-century revolution in . He served as lieutenant-colonel of th ...

, who showed in 1789 that heat could be created by friction, as when cannon bores are machined. Carnot reasoned that if the body of the working substance, such as a body of steam, is returned to its original state at the end of a complete engine cycle, "no change occurs in the condition of the working body". The
first law of thermodynamics The first law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter tran ...
, deduced from the heat-friction experiments of
James Joule James Prescott Joule (; 24 December 1818 11 October 1889) was an English physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical m ...

in 1843, expresses the concept of energy, and its
conservation Conservation is the preservation or efficient use of resources, or the conservation of various quantities under physical laws. Conservation may also refer to: Environment and natural resources * Nature conservation, the protection and manageme ...
in all processes; the first law, however, is unsuitable to separately quantify the effects of
friction Friction is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related en ...

and
dissipation In , dissipation is the result of an that takes place in homogeneous . In a dissipative process, (, bulk flow , or system ) from an initial form to a final form, where the capacity of the final form to do is less than that of the initial form. ...
. In the 1850s and 1860s, German physicist
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citize ...
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. n the Motive Power of Heat, and on the Laws which can be deduced from it for the Theory of Heat: Poggendorff's ''Annalen der Physik und Chemie'' He described his observations as a dissipative use of energy, resulting in a ''transformation-content'' (''Verwandlungsinhalt'' in German), of a
thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, whic ...
or working body of
chemical species A chemical species is a chemical substance A chemical substance is a form of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touc ...
during a change of
state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * The State (newspaper), ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, Un ...
. That was in contrast to earlier views, based on the theories of
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

, 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'' ropē which is translated in an established lexicon as ''turning'' or ''change'' 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''. "Sucht man für ''S'' einen bezeichnenden Namen, so könnte man, ähnlich wie von der Gröſse ''U'' gesagt ist, sie sey der ''Wärme- und Werkinhalt'' des Körpers, von der Gröſse ''S'' sagen, sie sey der ''Verwandlungsinhalt'' des Körpers. Da ich es aber für besser halte, die Namen derartiger für die Wissenschaft wichtiger Gröſsen aus den alten Sprachen zu entnehmen, damit sie unverändert in allen neuen Sprachen angewandt werden können, so schlage ich vor, die Gröſse ''S'' nach dem griechischen Worte ἡ τροπή, die Verwandlung, die ''Entropie'' des Körpers zu nennen. Das Wort ''Entropie'' habei ich absichtlich dem Worte ''Energie'' möglichst ähnlich gebildet, denn die beiden Gröſsen, welche durch diese Worte benannt werden sollen, sind ihren physikalischen Bedeutungen nach einander so nahe verwandt, daſs eine gewisse Gleichartigkeit in der Benennung mir zweckmäſsig zu seyn scheint." (p. 390). The word was adopted into the English language in 1868. Later, scientists such as
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austria Austria, officially the Republic of Austria, is a landlocked country in the southern part of Central Europe, located on the Eastern Alps. It is compo ...
,
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in tr ...

, and
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as num ...

gave entropy a statistical basis. In 1877, Boltzmann visualized a probabilistic way to measure the entropy of an ensemble of
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
particles, in which he defined entropy as proportional to the
natural logarithm The natural logarithm of a number is its logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
of the number of microstates such a gas could occupy. The
proportionality constant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
in this definition, called the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor In mathematics, two varying quantities are said to be in a Binary relation, relation of proportionality, Multiplication, multiplicatively connected to a Constant (mathematics), c ...
, has become one of the defining universal constants for the modern
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...
(SI). Henceforth, the essential problem in
statistical thermodynamics In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior throug ...
has been to determine the distribution of a given amount of energy ''E'' over ''N'' identical systems.
Constantin Carathéodory Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, ...
, a Greek mathematician, linked entropy with a mathematical definition of irreversibility, in terms of trajectories and integrability.

# Etymology

In 1865, Clausius named the concept of "the differential of a quantity which depends on the configuration of the system," ''
entropy 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 thermodynamic ...
'' () after the Greek word for 'transformation'. "Clausius coined the word entropy for $S$: ″I prefer going to the ancient languages for the names of important scientific quantities, so that they may mean the same thing in all living tongues. I propose, accordingly, to call $S$ the entropy of a body, after the Greek word 'transformation.' I have designedly coined the word entropy to be similar to 'energy,' for these two quantities are so analogous in their physical significance, that an analogy of denomination seemed to me helpful.″" He gave "transformational content" () as a synonym, paralleling his "thermal and ergonal content" () as the name of $U$, but preferring the term ''entropy'' as a close parallel of the word ''energy'', as he found the concepts nearly "analogous in their physical significance." This term was formed by replacing the root of ('work') by that of ('transformation').

# Definitions and descriptions

The concept of entropy is described by two principal approaches, the macroscopic perspective of
classical thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, and the microscopic description central to
statistical mechanics In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
. The classical approach defines entropy in terms of macroscopically measurable physical properties, such as bulk mass, volume, pressure, and temperature. 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. Newtonian particles constituting a gas, and later quantum-mechanically (photons,
phonons In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...
, spins, etc.). The two approaches form a consistent, unified view of the same phenomenon as expressed in the second law of thermodynamics, which has found universal applicability to physical processes.

## State variables and functions of state

Many
thermodynamic propertiesIn thermodynamics, a physical property is any property that is measurable, and whose value describes a state of a physical system. Thermodynamic properties are defined as characteristic features of a system, capable of specifying the system's state. ...
are defined by physical variables that define a state of
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic An axiom, postulate or assumption is a statement that is taken to be true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online ...
; these are ''state variables''. State variables depend only on the equilibrium condition, not on the path evolution to that state. State variables can be
functions of state In the thermodynamics of equilibrium, a state function, function of state, or point function is a function defined for a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form ...
, also called ''state functions'', in a sense that one state variable is a
mathematical function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
of other state variables. Often, if some properties of a system are determined, they are sufficient to determine the state of the system and thus other properties' values. For example, temperature and pressure of a given quantity of gas determine its state, and thus also its volume via the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the na ...

. A system composed of a pure substance of a single
phase Phase or phases may refer to: Science * State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter) In the physical sciences, a phase is a region of space (a thermodynamic system A thermodynamic system is a ...
at a particular uniform temperature and pressure is determined, and is thus a particular state, and has not only a particular volume but also a specific entropy. The fact that entropy is a function of state makes it useful. In the Carnot cycle, the working fluid returns to the same state that it had at the start of the cycle, hence the change or
line integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of any state function, such as entropy, over this reversible cycle is zero.

## Reversible process

Total entropy may be conserved during a reversible process. The entropy change of the system (not including the surroundings) is well-defined as heat divided by temperature, $d S = \frac$. A reversible process is a quasistatic one that deviates only infinitesimally from thermodynamic equilibrium and avoids friction or other dissipation. Any process that happens quickly enough to deviate from thermal equilibrium cannot be reversible, total entropy increases, and the potential for maximum work to be done in the process is also lost. For example, in the , 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 may still be zero at all times if the entire process is reversible. An irreversible process increases the total entropy of system and surroundings.

## Carnot cycle

The concept of entropy arose from
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citize ...
's study of the . In a Carnot cycle, heat is absorbed isothermally at temperature from a 'hot' reservoir and given up isothermally as heat to a 'cold' reservoir at . According to Carnot's principle,
work Work may refer to: * Work (human activity) Work or labor is intentional activity people perform to support themselves, others, or the needs and wants of a wider community. Alternatively, work can be viewed as the human activity that cont ...

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 (). Carnot did not distinguish between and , since he was using the incorrect hypothesis that
caloric theory The caloric theory is an obsolete scientific theory In Science#History, science, a theory is superseded when a scientific consensus once widely accepted it, but current science considers it inadequate, incomplete, or debunked (i.e., wrong). Such ...
was valid, and hence heat was conserved (the incorrect assumption that and were equal in magnitude) when, in fact, is greater than the magnitude of . Through the efforts of Clausius and
Kelvin The kelvin is the base unit of temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal en ...

, 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 (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. The possibility that the Carnot function could be the temperature as measured from a zero point of temperature was suggested by
Joule The joule ( ; symbol: J) is a derived unit of energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates ...

in a letter to Kelvin. This allowed Kelvin to establish his absolute temperature scale. It is also known that the net work produced by the system in one cycle is the net heat absorbed, which is the sum (or difference of the magnitudes) of the
heat In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ...

> 0 absorbed from the hot reservoir and the waste heat < 0 given off to the cold reservoir:. Since the latter is valid over the entire cycle, this gave Clausius the hint that at each stage of the cycle, work and heat would not be equal, but rather their difference would be the change of a state function that would vanish upon completion of the cycle. The state function was called the
internal energy The internal energy of a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that ca ...
central to the
first law of thermodynamics The first law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter tran ...
. Now equating () and () gives. $\frac +\frac = 0$ This implies that there is a function of state whose change is and that is conserved over a complete cycle of the Carnot cycle. Clausius called this state function ''entropy''. One can see that entropy was discovered through mathematics rather than through laboratory results. It is a mathematical construct and has no easy physical analogy. This makes the concept somewhat obscure or abstract, akin to how the concept of energy arose. Clausius then asked what would happen if there should be less work produced by the system than that predicted by Carnot's principle. The right-hand side of the first equation would be the upper bound of the work output by the system, which would now be converted into an inequality $W < \left(1 - \frac\right) Q_\text$ When the second equation is used to express the work as a net or total heat exchanged in a cycle, we get $Q_\text+Q_\text<\left(1-\frac\right)Q_\text$ or $, Q_\text, >\fracQ_\text$ So more heat is given up to the cold reservoir than in the Carnot cycle. If we denote the entropy changes by for the two stages of the process, then the above inequality can be written as a decrease in the entropy $\Delta S_\text+ \Delta S_\text<0$ or $\Delta S_\text < - \Delta S_\text = , \Delta S_\text,$ The magnitude of the entropy that leaves the system is greater than the entropy that enters the system, implying that some irreversible process prevents the cycle from producing the maximum amount of work predicted by the Carnot equation. The Carnot cycle and efficiency are useful because they define the upper bound of the possible work output and the efficiency of any classical thermodynamic heat engine. Other cycles, such as the
Otto cycle An Otto cycle is an idealized thermodynamic cycle that describes the functioning of a typical spark ignition engine, spark ignition piston engine. It is the thermodynamic cycle most commonly found in automobile engines. The Otto cycle is a d ...

,
Diesel cycle The Diesel cycle is a combustion process of a reciprocating internal combustion engine An internal combustion engine (ICE or IC engine) is a heat engine In thermodynamics Thermodynamics is a branch of physics that deals with heat, ...
and
Brayton cycle The Brayton cycle is a thermodynamic cycle named after George Brayton that describes the workings of a constant-pressure heat engine. The original Brayton engines used a piston compressor and piston expander, but more modern gas turbine engines ...

, can be analyzed from the standpoint of the Carnot cycle. Any machine or cyclic process that converts heat to work and is claimed to produce an efficiency greater than the Carnot efficiency is not viable because it violates the second law of thermodynamics. For very small numbers of particles in the system, statistical thermodynamics must be used. The efficiency of devices such as photovoltaic cells requires an analysis from the standpoint of quantum mechanics.

## Classical thermodynamics

The thermodynamic definition of entropy was developed in the early 1850s by
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citize ...
and essentially describes how to measure the entropy of an
isolated system In physical science, an isolated system is either of the following: # a physical system In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , i ...
in
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic An axiom, postulate or assumption is a statement that is taken to be true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online ...
with its parts. Clausius created the term entropy as an extensive thermodynamic variable that was shown to be useful in characterizing the . Heat transfer along the isotherm steps of the Carnot cycle was found to be proportional to the temperature of a system (known as its
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of ...
). This relationship was expressed in increments of entropy equal to the ratio of incremental heat transfer divided by temperature, which was found to vary in the thermodynamic cycle but eventually return to the same value at the end of every cycle. Thus it was found to be a
function of state In the Thermodynamics#Equilibrium_thermodynamics, thermodynamics of equilibrium, a state function, function of state, or point function is a function defined for a system relating several state variables or state quantities that depends only on the ...
, specifically a thermodynamic state of the system. While Clausius based his definition on a reversible process, there are also irreversible processes that change entropy. Following the
second law of thermodynamics The second law of thermodynamics establishes the concept of entropy 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 an ...
, entropy of an isolated
system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purp ...
always increases for irreversible processes. The difference between an isolated system and closed system is that energy may ''not'' flow to and from an isolated system, but energy flow to and from a closed system is possible. Nevertheless, for both closed and isolated systems, and indeed, also in open systems, irreversible thermodynamics processes may occur. According to the Clausius equality, for a reversible cyclic process: $\oint \frac = 0$. This means the line integral $\int_L \frac$ is path-independent. So we can define a state function called entropy, which satisfies $d S = \frac$. 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. Since entropy is a state function, the entropy change of the system for an irreversible path is the same as for a reversible path between the same two states. However, the heat transferred to or from, and the entropy change of, the surroundings is different. We can only obtain the change of entropy by integrating the above formula. To obtain the absolute value of the entropy, we need the
third law of thermodynamics The third law of thermodynamics states as follows, regarding the properties of closed systems in thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics Thermodynamics is a branch of physics that deals wit ...
, which states that ''S'' = 0 at
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature Thermodynamic temperature is the measure of ''absolute temperature'' and is one of the principal parameters of thermodynamics. A thermodynamic temperature reading of zero deno ...
for perfect crystals. From a macroscopic perspective, in
classical thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
the entropy is interpreted as a
state function In the thermodynamics of equilibrium, a state function, function of state, or point function is a function defined for a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form ...
of a
thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, whic ...
: that is, a property depending only on the current state of the system, independent of how that state came to be achieved. In any process where the system gives up energy Δ''E'', and its entropy falls by Δ''S'', a quantity at least ''T''R Δ''S'' of that energy must be given up to the system's surroundings as heat (''T''R is the temperature of the system's external surroundings). Otherwise the process cannot go forward. In classical thermodynamics, the entropy of a system is defined only if it is in physical
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic An axiom, postulate or assumption is a statement that is taken to be true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online ...
(but chemical equilibrium is not required: the entropy of a mixture of two moles of hydrogen and one mole of oxygen at 1 bar pressure and 298 K is well-defined).

## Statistical mechanics

The statistical definition was developed by
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austria Austria, officially the Republic of Austria, is a landlocked country in the southern part of Central Europe, located on the Eastern Alps. It is compo ...
in the 1870s by analyzing the statistical behavior of the microscopic components of the system. Boltzmann showed that this definition of entropy was equivalent to the thermodynamic entropy to within a constant factor—known as
Boltzmann's constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
. In summary, the thermodynamic definition of entropy provides the experimental definition of entropy, while the statistical definition of entropy extends the concept, providing an explanation and a deeper understanding of its nature. The interpretation of entropy in statistical mechanics is the measure of uncertainty, disorder, or ''mixedupness'' in the phrase of , which remains about a system after its observable macroscopic properties, such as temperature, pressure and volume, have been taken into account. For a given set of macroscopic variables, the entropy measures the degree to which the probability of the system is spread out over different possible
microstates Image:BlankMap-World-v6 small states.png, upright=1.4, Map of the smallest states in the world by land area. Note many of these are not considered microstates A microstate or ministate is a sovereign state having a very small population or very ...
. In contrast to the macrostate, which characterizes plainly observable average quantities, a microstate specifies all molecular details about the system including the position and velocity of every molecule. The more such states are available to the system with appreciable probability, the greater the entropy. In statistical mechanics, entropy is a measure of the number of ways a system can be arranged, often taken to be a measure of "disorder" (the higher the entropy, the higher the disorder). 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 Image:BlankMap-World-v6 small states.png, upright=1.4, Map of the smallest states in the world by land area. Note many of these are not considered microstates A microstate or ministate is a sovereign state having a very small population or very ...
) that could cause the observed macroscopic state ( macrostate) of the system. The constant of proportionality is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor In mathematics, two varying quantities are said to be in a Binary relation, relation of proportionality, Multiplication, multiplicatively connected to a Constant (mathematics), c ...
. Boltzmann's constant, and therefore entropy, have
dimensions thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature ...
of energy divided by temperature, which has a unit of
joule The joule ( ; symbol: J) is a derived unit of energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates ...

s per
kelvin The kelvin is the base unit of temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal en ...

(J⋅K−1) in the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...
(or kg⋅m2⋅s−2⋅K−1 in terms of base units). The entropy of a substance is usually given as an
intensive propertyIn grammar, an intensive word form is one which denotes stronger, more forceful, or more concentrated action relative to the root on which the intensive is built. Intensives are usually lexical formations, but there may be a regular process for formi ...
either entropy per unit
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
(SI unit: J⋅K−1⋅kg−1) or entropy per unit
amount of substance In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a ...
(SI unit: J⋅K−1⋅mol−1). Specifically, entropy is a measure of the number of system states with significant probability of being occupied: :$S = -k_\sum_i p_i \log p_i,$ ($p_i$ is the probability that the system is in $i$th state, usually given by the
Boltzmann distribution In statistical mechanics In physics, statistical mechanics is a mathematical framework that applies Statistics, statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natu ...

; if states are defined in a continuous manner, the summation is replaced by an
integral In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

over all possible states) or, equivalently, the expected value of that a microstate is occupied :$S = -k_ \langle\log p\rangle$ where ''k''B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor In mathematics, two varying quantities are said to be in a Binary relation, relation of proportionality, Multiplication, multiplicatively connected to a Constant (mathematics), c ...
, equal to . 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.Frigg, R. and Werndl, C. "Entropy – A Guide for the Perplexed"
In ''Probabilities in Physics''; Beisbart C. and Hartmann, S. Eds; Oxford University Press, Oxford, 2010
This definition assumes that the basis set of states has been picked so that there is no information on their relative phases. In a different basis set, the more general expression is :$S = -k_ \operatorname\left(\widehat \log\left(\widehat\right)\right),$ where $\widehat$ is the density matrix, $\operatorname$ is trace (linear algebra), trace and $\log$ is the matrix logarithm. This density matrix formulation is not needed in cases of thermal equilibrium so long as the basis states are chosen to be energy eigenstates. For most practical purposes, this can be taken as the fundamental definition of entropy since all other formulas for ''S'' can be mathematically derived from it, but not vice versa. In what has been called ''the fundamental assumption of statistical thermodynamics'' or ''Fundamental postulate of statistical mechanics, the fundamental postulate in statistical mechanics'', among system microstates of the same energy (Degenerate energy levels, degenerate microstates) each microstate is assumed to be populated with equal probability; this assumption is usually justified for an isolated system in equilibrium. Then for an isolated system ''p''''i'' = 1/Ω, where Ω is the number of microstates whose energy equals the system's energy, and the previous equation reduces to :$S = k_ \log \Omega.$ In thermodynamics, such a system is one in which the volume, number of molecules, and internal energy are fixed (the microcanonical ensemble). For a given thermodynamic system, the ''excess entropy'' is defined as the entropy minus that of an ideal gas at the same density and temperature, a quantity that is always negative because an ideal gas is maximally disordered. This concept plays an important role in liquid-state theory. For instance, Rosenfeld's excess-entropy scaling principle states that reduced transport coefficients throughout the two-dimensional phase diagram are functions uniquely determined by the excess entropy. The most general interpretation of entropy is as a measure of the extent of uncertainty about a system. The equilibrium state of a system maximizes the entropy because it does not reflect all information about the initial conditions, except for the conserved variables. This uncertainty is not of the everyday subjective kind, but rather the uncertainty inherent to the experimental method and interpretative model. The interpretative model has a central role in determining entropy. The qualifier "for a given set of macroscopic variables" above has deep implications: if two observers use different sets of macroscopic variables, they see different entropies. For example, if observer A uses the variables ''U'', ''V'' and ''W'', and observer B uses ''U'', ''V'', ''W'', ''X'', then, by changing ''X'', observer B can cause an effect that looks like a violation of the second law of thermodynamics to observer A. In other words: the set of macroscopic variables one chooses must include everything that may change in the experiment, otherwise one might see decreasing entropy. Entropy can be defined for any Markov processes with reversible dynamics and the detailed balance property. 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. A definition of entropy based on geometric considerations (quantitative geometrical thermodynamics (QGT)) obeys an entropic version of Liouville's theorem (Hamiltonian), Liouville's theorem so that entropy is defined in terms of degrees of freedom, which may also apply to small systems.

## Entropy of a system

Entropy arises directly from the . It can also be described as the reversible heat divided by temperature. Entropy is a fundamental function of state. In a
thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, whic ...
, pressure, density, and temperature tend to become uniform over time because the equilibrium state has higher probability (more possible combinations of
microstates Image:BlankMap-World-v6 small states.png, upright=1.4, Map of the smallest states in the world by land area. Note many of these are not considered microstates A microstate or ministate is a sovereign state having a very small population or very ...
) than any other state. As an example, for a glass of ice water in air at room temperature, 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 equalize as portions of the thermal energy from the warm surroundings spread 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. In other words, the entropy of the room has decreased as some of its energy has been dispersed to the ice and water, of which the entropy has increased. 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 In physical science, an isolated system is either of the following: # a physical system In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , i ...
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 A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, whic ...
is a measure of how far the equalization has progressed. Thermodynamic entropy is a non-conserved
state function In the thermodynamics of equilibrium, a state function, function of state, or point function is a function defined for a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form ...
that is of great importance in the sciences of
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

and
chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in the real world. T ...

. Historically, the concept of entropy evolved to explain why some processes (permitted by conservation laws) occur spontaneously while their T-symmetry, time reversals (also permitted by conservation laws) do not; systems tend to progress in the direction of increasing entropy. For
isolated system In physical science, an isolated system is either of the following: # a physical system In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , i ...
s, entropy never decreases. This fact has several important consequences in science: first, it prohibits "perpetual motion" machines; and second, it implies the Entropy (arrow of time), arrow of entropy has the same direction as the arrow of time. Increases in the total entropy of system and surroundings correspond to irreversible changes, because some energy is expended as waste heat, limiting the amount of work a system can do. Unlike many other functions of state, entropy cannot be directly observed but must be calculated. Absolute standard molar entropy of a substance can be calculated from the measured temperature dependence of its heat capacity. The molar entropy of ions is obtained as a difference in entropy from a reference state defined as zero entropy. The
second law of thermodynamics The second law of thermodynamics establishes the concept of entropy 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 an ...
states that the entropy of an
isolated system In physical science, an isolated system is either of the following: # a physical system In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , i ...
must increase or remain constant. Therefore, entropy is not a conserved quantity: for example, in an isolated system with non-uniform temperature, heat might irreversibly flow and the temperature become more uniform such that entropy increases. Chemical reactions cause changes in entropy and system entropy, in conjunction with enthalpy, plays an important role in determining in which direction a chemical reaction spontaneously proceeds. One dictionary definition of entropy is that it is "a measure of thermal energy per unit temperature that is not available for useful work" in a cyclic process. For instance, a substance at uniform temperature is at maximum entropy and cannot drive a heat engine. 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. 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 is no net exchange of heat or work – the entropy change is entirely due to the mixing of the different substances. At a statistical mechanical level, this results due to the change in available volume per particle with mixing.

## Equivalence of definitions

Proofs of equivalence between the definition of entropy in statistical mechanics (the Entropy (statistical thermodynamics)#Gibbs entropy formula, Gibbs entropy formula $S = -k_\sum_i p_i \log p_i$) and in classical thermodynamics ($d S = \frac$ together with the fundamental thermodynamic relation) are known for the microcanonical ensemble, the canonical ensemble, the grand canonical ensemble, and the isothermal–isobaric ensemble. These proofs are based on the probability density of microstates of the generalized
Boltzmann distribution In statistical mechanics In physics, statistical mechanics is a mathematical framework that applies Statistics, statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natu ...

and the identification of the thermodynamic internal energy as the ensemble average $U=\left\langle E_\right\rangle$. Thermodynamic relations are then employed to derive the well-known Gibbs entropy formula. 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 In statistical mechanics In physics, statistical mechanics is a mathematical framework that applies Statistics, statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natu ...

.

# Second law of thermodynamics

The
second law of thermodynamics The second law of thermodynamics establishes the concept of entropy 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 an ...
requires that, in general, the total entropy of any system does not decrease other than by increasing the entropy of some other system. Hence, in a system isolated from its environment, the entropy of that system tends not to decrease. It follows that heat cannot flow from a colder body to a hotter body without the application of work to the colder body. Secondly, it is impossible for any device operating on a cycle to produce net work from a single temperature reservoir; the production of net work requires flow of heat from a hotter reservoir to a colder reservoir, or a single expanding reservoir undergoing adiabatic cooling, which performs adiabatic process, adiabatic work. As a result, there is no possibility of a perpetual motion machine. It follows that a reduction in the increase of entropy in a specified process, such as a chemical reaction, means that it is energetically more efficient. It follows from the second law of thermodynamics that the entropy of a system that is not isolated may decrease. An air conditioner, for example, may cool the air in a room, thus reducing the entropy of the air of that system. The heat expelled from the room (the system), which the air conditioner transports and discharges to the outside air, always makes a bigger contribution to the entropy of the environment than the decrease of the entropy of the air of that system. Thus, the total of entropy of the room plus the entropy of the environment increases, in agreement with the second law of thermodynamics. In mechanics, the second law in conjunction with the fundamental thermodynamic relation places limits on a system's ability to do work (thermodynamics), useful work. The entropy change of a system at temperature $T$ absorbing an infinitesimal amount of heat $\delta q$ in a reversible way, is given by $\delta q / T$. More explicitly, an energy $T_R S$ is not available to do useful work, where $T_R$ is the temperature of the coldest accessible reservoir or heat sink external to the system. For further discussion, see ''Exergy''. Statistical mechanics demonstrates that entropy is governed by probability, thus allowing for a decrease in disorder even in an isolated system. Although this is possible, such an event has a small probability of occurring, making it unlikely. The applicability of a second law of thermodynamics is limited to systems in or sufficiently near thermodynamic equilibrium, equilibrium state, so that they have defined entropy. Some inhomogeneous systems out of thermodynamic equilibrium still satisfy the hypothesis of Thermodynamic equilibrium # local and global equilibrium , local thermodynamic equilibrium, so that entropy density is locally defined as an intensive quantity. For such systems, there may apply a principle of maximum time rate of entropy production. It states that such a system may evolve to a steady state that maximizes its time rate of entropy production. This does not mean that such a system is necessarily always in a condition of maximum time rate of entropy production; it means that it may evolve to such a steady state.

# Applications

## The fundamental thermodynamic relation

The entropy of a system depends on its internal energy and its external parameters, such as its volume. 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. This relation is known as the ''fundamental thermodynamic relation''. If external pressure $p$ bears on the volume $V$ as the only external parameter, this relation is: : $dU = T \, dS - p \, dV$ Since both internal energy and entropy are monotonic functions of temperature $T$, implying that the internal energy is fixed when one specifies the entropy and the volume, this relation is valid even if the change from one state of thermal equilibrium to another with infinitesimally larger entropy and volume happens in a non-quasistatic way (so during this change the system may be very far out of thermal equilibrium and then the whole-system entropy, pressure, and temperature may not exist). The fundamental thermodynamic relation implies many thermodynamic identities that are valid in general, independent of the microscopic details of the system. Important examples are the Maxwell relations and the relations between heat capacities.

## Entropy in chemical thermodynamics

Thermodynamic entropy is central in chemical thermodynamics, enabling changes to be quantified and the outcome of reactions predicted. The
second law of thermodynamics The second law of thermodynamics establishes the concept of entropy 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 an ...
states that entropy in an
isolated system In physical science, an isolated system is either of the following: # a physical system In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , i ...
– the combination of a subsystem under study and its surroundings – increases during all spontaneous chemical and physical processes. The Clausius equation of $\delta q_ / T = \Delta S$ introduces the measurement of entropy change, $\Delta S$. Entropy change describes the direction and quantifies the magnitude of simple changes such as heat transfer between systems – always from hotter to cooler spontaneously. The thermodynamic entropy therefore has the dimension of energy divided by temperature, and the unit
joule The joule ( ; symbol: J) is a derived unit of energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates ...

per
kelvin The kelvin is the base unit of temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal en ...

(J/K) in the International System of Units (SI). Thermodynamic entropy is an Intensive and extensive properties, extensive property, meaning that it scales with the size or extent of a system. In many processes it is useful to specify the entropy as an Intensive and extensive properties, intensive property independent of the size, as a specific entropy characteristic of the type of system studied. Specific entropy may be expressed relative to a unit of mass, typically the kilogram (unit: J⋅kg−1⋅K−1). Alternatively, in chemistry, it is also referred to one Mole (unit), mole of substance, in which case it is called the ''molar entropy'' with a unit of J⋅mol−1⋅K−1. Thus, when one mole of substance at about is warmed by its surroundings to , the sum of the incremental values of $q_ / T$ constitute each element's or compound's standard molar entropy, an indicator of the amount of energy stored by a substance at . Entropy change also measures the mixing of substances as a summation of their relative quantities in the final mixture. Entropy is equally essential in predicting the extent and direction of complex chemical reactions. For such applications, $\Delta S$ must be incorporated in an expression that includes both the system and its surroundings, $\Delta S_ = \Delta S_ + \Delta S_$. This expression becomes, via some steps, the Gibbs free energy equation for reactants and products in the system: $\Delta G$ [the Gibbs free energy change of the system] $= \Delta H$ [the enthalpy change] $- T\,\Delta S$ [the entropy change].

## World's technological capacity to store and communicate entropic information

A 2011 study in Science (journal) estimated the world's technological capacity to store and communicate optimally compressed information normalized on the most effective compression algorithms available in the year 2007, therefore estimating the entropy of the technologically available sources."The World’s Technological Capacity to Store, Communicate, and Compute Information"
Martin Hilbert and Priscila López (2011), Science (journal), 332(6025), 60–65; free access to the article through here: martinhilbert.net/WorldInfoCapacity.html
The author's estimate that human kind's technological capacity to store information grew from 2.6 (entropically compressed) exabytes in 1986 to 295 (entropically compressed) exabytes in 2007. The world's technological capacity to receive information through one-way broadcast networks was 432 exabytes of (entropically compressed) information in 1986, to 1.9 zettabytes in 2007. The world's effective capacity to exchange information through two-way telecommunication networks was 281 petabytes of (entropically compressed) information in 1986, to 65 (entropically compressed) exabytes in 2007.

## Entropy balance equation for open systems

In chemical engineering, the principles of thermodynamics are commonly applied to "Open system (systems theory), open systems", i.e. those in which heat, work (thermodynamics), work, and
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
flow across the system boundary. Flows of both heat ($\dot$) and work, i.e. $\dot_\text$ (Work (thermodynamics), shaft work) and $P\left(dV/dt\right)$ (pressure-volume work), across the system boundaries, in general cause changes in the entropy of the system. Transfer as heat entails entropy transfer $\dot/T$, where $T$ is the absolute thermodynamic temperature of the system at the point of the heat flow. If there are mass flows across the system boundaries, they also influence the total entropy of the system. This account, in terms of heat and work, is valid only for cases in which the work and heat transfers are by paths physically distinct from the paths of entry and exit of matter from the system. To derive a generalized entropy balanced equation, we start with the general balance equation for the change in any extensive quantity $\theta$ in a
thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, whic ...
, a quantity that may be either conserved, such as energy, or non-conserved, such as entropy. The basic generic balance expression states that $d\theta/dt$, i.e. the rate of change of $\theta$ in the system, equals the rate at which $\theta$ enters the system at the boundaries, minus the rate at which $\theta$ leaves the system across the system boundaries, plus the rate at which $\theta$ is generated within the system. For an open thermodynamic system in which heat and work are transferred by paths separate from the paths for transfer of matter, using this generic balance equation, with respect to the rate of change with time $t$ of the extensive quantity entropy $S$, the entropy balance equation is:The overdots represent derivatives of the quantities with respect to time. :$\frac = \sum_^K \dot_k \hat_k + \frac + \dot_\text$ where *$\sum_^K \dot_k \hat_k$ is the net rate of entropy flow due to the flows of mass into and out of the system (where $\hat$ is entropy per unit mass). *$\frac$ is the rate of entropy flow due to the flow of heat across the system boundary. *$\dot_\text$ is the rate of entropy production within the system. This entropy production arises from processes within the system, including chemical reactions, internal matter diffusion, internal heat transfer, and frictional effects such as viscosity occurring within the system from mechanical work transfer to or from the system. If there are multiple heat flows, the term $\dot/T$ is replaced by $\sum \dot_j/T_j,$ where $\dot_j$ is the heat flow and $T_j$ is the temperature at the $j$th heat flow port into the system. Note that the nomenclature "entropy balance" is misleading and often deemed inappropriate because entropy is not a conserved quantity. In other words, the term $\dot_\text$ is never a known quantity but always a derived one based on the expression above. Therefore, the open system version of the second law is more appropriately described as the "entropy generation equation" since it specifies that $\dot_\text \ge 0$, with zero for reversible processes or greater than zero for irreversible ones.

# Entropy change formulas for simple processes

For certain simple transformations in systems of constant composition, the entropy changes are given by simple formulas.

## Isothermal expansion or compression of an ideal gas

For the expansion (or compression) of an
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
from an initial volume $V_0$ and pressure $P_0$ to a final volume $V$ and pressure $P$ at any constant temperature, the change in entropy is given by: :$\Delta S = n R \ln \frac = - n R \ln \frac .$ Here $n$ is the amount of gas (in Mole (unit), moles) and $R$ is the ideal gas constant. These equations also apply for expansion into a finite vacuum or a throttling process (thermodynamics), throttling process, where the temperature, internal energy and enthalpy for an ideal gas remain constant.

## Cooling and heating

For pure heating or cooling of any system (gas, liquid or solid) at constant pressure from an initial temperature $T_0$ to a final temperature $T$, the entropy change is :$\Delta S = n C_P \ln \frac.$ provided that the constant-pressure molar heat capacity (or specific heat) ''C''''P'' is constant and that no phase transition occurs in this temperature interval. Similarly at constant volume, the entropy change is :$\Delta S = n C_v \ln \frac,$ where the constant-volume molar heat capacity ''C''v is constant and there is no phase change. At low temperatures near absolute zero, Debye T3 law, heat capacities of solids quickly drop off to near zero, so the assumption of constant heat capacity does not apply. Since entropy is a Functions of state, state function, the entropy change of any process in which temperature and volume both vary is the same as for a path divided into two steps – heating at constant volume and expansion at constant temperature. For an ideal gas, the total entropy change is :$\Delta S = nC_v \ln \frac + nR \ln \frac.$ Similarly if the temperature and pressure of an ideal gas both vary, :$\Delta S = nC_P \ln \frac - nR \ln \frac.$

## Phase transitions

Reversible phase transitions occur at constant temperature and pressure. The reversible heat is the enthalpy change for the transition, and the entropy change is the enthalpy change divided by the thermodynamic temperature. For fusion (melting) of a solid to a liquid at the melting point ''T''m, the entropy of fusion is :$\Delta S_\text = \frac.$ Similarly, for vaporization of a liquid to a gas at the boiling point ''T''b, the entropy of vaporization is :$\Delta S_\text = \frac.$

# Approaches to understanding entropy

As a fundamental aspect of thermodynamics and physics, several different approaches to entropy beyond that of Clausius and Boltzmann are valid.

## Standard textbook definitions

The following is a list of additional definitions of entropy from a collection of textbooks: * a measure of energy dispersal at a specific temperature. * a measure of disorder in the universe or of the availability of the energy in a system to do work. * a measure of a system's thermal energy per unit temperature that is unavailable for doing useful work (thermodynamics), work. In Boltzmann's definition, entropy is a measure of the number of possible microscopic states (or microstates) of a system in thermodynamic equilibrium. Consistent with the Boltzmann definition, the second law of thermodynamics needs to be re-worded as such that entropy increases over time, though the underlying principle remains the same.

## Order and disorder

Entropy is often loosely associated with the amount of wikt:order, order or Randomness, disorder, or of Chaos theory, chaos, in a
thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, whic ...
. The traditional qualitative description of entropy is that it refers to changes in the status quo of the system and is a measure of "molecular disorder" and the amount of wasted energy in a dynamical energy transformation from one state or form to another. In this direction, several recent authors have derived exact entropy formulas to account for and measure disorder and order in atomic and molecular assemblies. One of the simpler entropy order/disorder formulas is that derived in 1984 by thermodynamic physicist Peter Landsberg, based on a combination of thermodynamics and
information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamentally established by the ...
arguments. He argues that when constraints operate on a system, such that it is prevented from entering one or more of its possible or permitted states, as contrasted with its forbidden states, the measure of the total amount of "disorder" in the system is given by: :$\text=.\,$ Similarly, the total amount of "order" in the system is given by: :$\text=1-.\,$ 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.

## Energy dispersal

The concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature. Similar terms have been in use from early in the history of
classical thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, and with the development of
statistical thermodynamics In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior throug ...
and quantum mechanics, 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. As the
second law of thermodynamics The second law of thermodynamics establishes the concept of entropy 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 an ...
shows, in an
isolated system In physical science, an isolated system is either of the following: # a physical system In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , i ...
internal portions at different temperatures 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 The first law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter tran ...
(compare discussion in next section). 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 or matter and often both".

## Relating entropy to energy ''usefulness''

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. This is because energy supplied at a higher temperature (i.e. with low entropy) tends to be more useful than the same amount of energy available at a lower temperature. Mixing a hot parcel of a fluid with a cold one produces a parcel of intermediate temperature, in which the overall increase in entropy represents a "loss" that can never be replaced. Thus, the fact that the entropy of the universe is steadily increasing, means that its total energy is becoming less useful: eventually, this leads to the "heat death of the Universe."

A definition of entropy based entirely on the relation of adiabatic accessibility between equilibrium states was given by Elliott H. Lieb, E.H.Lieb and Jakob Yngvason, J. Yngvason in 1999. This approach has several predecessors, including the pioneering work of
Constantin Carathéodory Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, ...
from 1909 and the monograph by R. Giles. In the setting of Lieb and Yngvason one starts by picking, for a unit amount of the substance under consideration, two reference states $X_0$ and $X_1$ such that the latter is adiabatically accessible from the former but not vice versa. Defining the entropies of the reference states to be 0 and 1 respectively the entropy of a state $X$ is defined as the largest number $\lambda$ such that $X$ is adiabatically accessible from a composite state consisting of an amount $\lambda$ in the state $X_1$ and a complementary amount, $\left(1-\lambda\right)$, in the state $X_0$. A simple but important result within this setting is that entropy is uniquely determined, apart from a choice of unit and an additive constant for each chemical element, by the following properties: It is monotonic with respect to the relation of adiabatic accessibility, additive on composite systems, and extensive under scaling.

## Entropy in quantum mechanics

In quantum statistical mechanics, the concept of entropy was developed by
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian Americans, Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. Von Neumann was generally rega ...

and is generally referred to as "von Neumann entropy", : $S = - k_\mathrm\operatorname \left( \rho \log \rho \right) ,$ where ''ρ'' is the density matrix and Tr is the trace (linear algebra), trace operator. 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, : $S = - k_\mathrm\sum_i p_i \, \log \, p_i,$ i.e. in such a basis the density matrix is diagonal. Von Neumann established a rigorous 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 function 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.

## Information theory

When viewed in terms of
information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamentally established by the ...
, the entropy state function is the amount of information in the system that is needed to fully specify the microstate of the system. Entropy is the measure of the amount of missing information before reception. Often called ''Shannon entropy'', it was originally devised by
Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbe ...
in 1948 to study the size of information of a transmitted message. The definition of information entropy is expressed in terms of a discrete set of probabilities $p_i$ so that :$H\left(X\right) = -\sum_^n p\left(x_i\right) \log p\left(x_i\right).$ 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 the average size of information of a message. For the case of equal probabilities (i.e. each message is equally probable), the Shannon entropy (in bits) is just the number of binary questions needed to determine the content of the message. Most researchers consider information entropy and thermodynamic entropy directly linked to the same concept, while others argue that they are distinct. Both expressions are mathematically similar. If $W$ is the number of microstates that can yield a given macrostate, and each microstate has the same ''A priori knowledge, a priori'' probability, then that probability is $1 = p = 1/W$. The Shannon entropy (in Nat (unit), nats) is: :$H = -\sum_^W p \log \left(p\right)= \log \left(W\right)$ and if entropy is measured in units of $k$ per nat, then the entropy is given by: :$H = k \log \left(W\right)$ which is the Boltzmann's entropy formula, Boltzmann entropy formula, where $k$ is Boltzmann's constant, which may be interpreted as the thermodynamic entropy per nat. Some authors argue for dropping the word entropy for the $H$ function of information theory and using Shannon's other term, "uncertainty", instead.

## Measurement

The entropy of a substance can be measured, although in an indirect way. The measurement, known as entropymetry, is done on a closed system (with particle number N and volume V being constants) and uses the definition of temperature in terms of entropy, while limiting energy exchange to heat ($dU \rightarrow dQ$). : $T := \left\left(\frac\right\right)_ \Rightarrow \cdots \Rightarrow \; dS=dQ/T$ The resulting relation describes how entropy changes $dS$ when a small amount of energy $dQ$ is introduced into the system at a certain temperature $T$. The process of measurement goes as follows. First, a sample of the substance is cooled as close to absolute zero as possible. At such temperatures, the entropy approaches zerodue to the definition of temperature. Then, small amounts of heat are introduced into the sample and the change in temperature is recorded, until the temperature reaches a desired value (usually 25 °C). The obtained data allows the user to integrate the equation above, yielding the absolute value of entropy of the substance at the final temperature. This value of entropy is called calorimetric entropy.

# Interdisciplinary applications

Although the concept of entropy was originally a thermodynamic concept, it has been adapted in other fields of study, including
information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamentally established by the ...
, psychodynamics, thermoeconomics/ecological economics, and evolution.

## Philosophy and theoretical physics

Entropy is the only quantity in the physical sciences that seems to imply a particular direction of progress, sometimes called an arrow of time. As time progresses, the second law of thermodynamics states that the entropy of an
isolated system In physical science, an isolated system is either of the following: # a physical system In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , i ...
never decreases in large systems over significant periods of time. Hence, from this perspective, entropy measurement is thought of as a clock in these conditions.

## Biology

Chiavazzo ''et al.'' proposed that where cave spiders choose to lay their eggs can be explained through entropy minimization. Entropy has been proven useful in the analysis of base pair sequences in DNA. Many entropy-based measures have been shown to distinguish between different structural regions of the genome, differentiate between coding and non-coding regions of DNA, and can also be applied for the recreation of evolutionary trees by determining the evolutionary distance between different species.

## Cosmology

Assuming that a finite universe is an isolated system, the second law of thermodynamics states that its total entropy is continually increasing. It has been speculated, since the 19th century, that the universe is fated to a heat death of the universe, heat death in which all the energy ends up as a homogeneous distribution of thermal energy so that no more work can be extracted from any source. 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. Black hole entropy, The entropy of a black hole is proportional to the surface area of the black hole's event horizon. 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 entropy-increasing processes, if they are totally effective matter and energy traps. However, the escape of energy from black holes might be possible due to quantum activity (see Hawking radiation). The role of entropy in cosmology remains a controversial subject since the time of
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austria Austria, officially the Republic of Austria, is a landlocked country in the southern part of Central Europe, located on the Eastern Alps. It is compo ...
. Recent work has cast some 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, moving the universe further from the heat death with time, not closer. This results in an "entropy gap" pushing the system further away from the posited heat death equilibrium. Other complicating factors, such as the energy density of the vacuum and macroscopic quantum mechanics, quantum effects, are difficult to reconcile with thermodynamical models, making any predictions of large-scale thermodynamics extremely difficult. Current theories suggest the entropy gap to have been originally opened up by inflation (cosmology), the early rapid exponential expansion of the universe. (in honor of John Wheeler's 90th birthday)

## Economics

Romanian American economist Nicholas Georgescu-Roegen, a List of people considered father or mother of a scientific field, progenitor in economics and a Paradigm shift#Kuhnian paradigm shifts, paradigm founder of ecological economics, made extensive use of the entropy concept in his Nicholas Georgescu-Roegen#Magnum opus on The Entropy Law and the Economic Process, magnum opus on ''The Entropy Law and the Economic Process''. Due to Georgescu-Roegen's work, the laws of thermodynamics now form an Ecological economics#Methodology, integral part of the ecological economics school. Although his work was Nicholas Georgescu-Roegen#Mistakes and controversies, blemished somewhat by mistakes, a full chapter on the economics of Georgescu-Roegen has approvingly been included in one elementary physics textbook on the historical development of thermodynamics. In economics, Georgescu-Roegen's work has generated the term Pessimism#Entropy pessimism, 'entropy pessimism'. Since the 1990s, leading ecological economist and Steady-state economy#Herman Daly's concept of a steady-state economy, steady-state theorist Herman Daly – a student of Georgescu-Roegen – has been the economics profession's most influential proponent of the entropy pessimism position.

* Autocatalytic reactions and order creation * Boltzmann entropy – a type of Gibbs entropy, which neglects internal statistical correlations in the overall particle distribution * Brownian ratchet * Clausius–Duhem inequality * Configuration entropy * Conformational entropy – associated with the physical arrangement of a polymer chain that assumes a compact or globular protein, globular state in solution * Departure function * Enthalpy * Entropic explosion – an explosion in which the reactants expand without releasing much heat * Entropic force * Entropy unit * Entropic value at risk * Entropy (information theory) * Entropy (computing) * Entropy (statistical thermodynamics) * Entropy and life * Entropy (order and disorder) * Entropy of mixing – the change in the entropy when two different chemical substances or component (thermodynamics), components are mixed * Entropy rate * Entropy production * Extropianism#Extropy, Extropy * Free entropy – a thermodynamic potential analogous to free energy * Geometrical frustration * Gibbs entropy – a precise definition of entropy * Harmonic entropy * Heat death of the universe * Info-metrics * Laws of thermodynamics * Loop entropy – is the entropy lost upon bringing together two residues of a polymer within a prescribed distance * Multiplicity function * Negentropy (negative entropy) * Orders of magnitude (entropy) * Phase space#Thermodynamics and statistical mechanics, Phase space * Principle of maximum entropy * Residual entropy – the entropy present after a substance is cooled arbitrarily close to
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature Thermodynamic temperature is the measure of ''absolute temperature'' and is one of the principal parameters of thermodynamics. A thermodynamic temperature reading of zero deno ...
* Sackur–Tetrode equation, Sackur–Tetrode entropy – the entropy of a monatomic classical ideal gas determined via quantum considerations * Standard molar entropy – the entropy in one mole of substance under STP * Stirling's formula * Thermodynamic databases for pure substances * Thermodynamic potential * Thermodynamic equilibrium * Tsallis entropy – a generalization of the Boltzmann and Gibbs definitions

# References

* * * * * * * * * * * * * * * Lambert, Frank L.
entropysite.oxy.edu
* * * * * * Sharp, Kim (2019). ''Entropy and the Tao of Counting: A Brief Introduction to Statistical Mechanics and the Second Law of Thermodynamics'' (SpringerBriefs in Physics). Springer Nature. . * Spirax-Sarco Limited
Entropy – A Basic Understanding
A primer on entropy tables for steam engineering *

Entropy and the Second Law of Thermodynamics
– an A-level physics lecture with 'derivation' of entropy based on Carnot cycle * Khan Academy: entropy lectures, part o
Chemistry playlist
*
Proof: S (or Entropy) is a valid state variable
*
Thermodynamic Entropy Definition Clarification
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Reconciling Thermodynamic and State Definitions of Entropy
*
Entropy Intuition
*
More on Entropy

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at ''Scholarpedia'' {{Authority control Entropy, Physical quantities Philosophy of thermal and statistical physics State functions Asymmetry