Editing Entropy (section)

== Definitions and descriptions ==
{{quote box|width=30em|quote=Any method involving the notion of entropy, the very existence of which depends on the second law of thermodynamics, will doubtless seem to many far-fetched, and may repel beginners as obscure and difficult of comprehension.|source=[[Willard Gibbs]], ''Graphical Methods in the Thermodynamics of Fluids''<ref>{{cite book|title=The scientific papers of J. Willard Gibbs in Two Volumes|url=https://archive.org/stream/scientificpapers01gibbuoft#page/11/mode/1up |volume=1 |year=1906 |publisher=Longmans, Green, and Co. |page=11 |access-date=2011-02-26}}</ref>}}

The concept of entropy is described by two principal approaches, the macroscopic perspective of [[classical thermodynamics]], and the microscopic description central to [[statistical mechanics]]. 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 — modelled at first classically, e.g. Newtonian particles constituting a gas, and later quantum-mechanically (photons, [[phonons]], 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 properties]] are defined by physical variables that define a state of [[thermodynamic equilibrium]], which essentially are [[state variable]]s. State variables depend only on the equilibrium condition, not on the path evolution to that state. State variables can be functions of state, also called [[state function]]s, in a sense that one state variable is a [[Function (mathematics)|mathematical function]] 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]]. A system composed of a pure substance of a single [[Phase (matter)|phase]] at a particular uniform temperature and pressure is determined, and is thus a particular state, and has a particular volume.  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]] of any state function, such as entropy, over this reversible cycle is zero.

=== Reversible process ===
The entropy change ''<math display="inline">\mathrm{d} S</math>'' of a system can be well-defined as a small portion of [[heat]] ''<math display="inline">\delta Q_{\mathsf{rev}}</math>'' transferred from the surroundings to the system during a reversible process divided by the [[temperature]] ''<math display="inline">T</math>'' of the system during this [[heat transfer]]:<math display="block">\mathrm{d} S = \frac{\delta Q_\mathsf{rev}}{T}</math>The reversible process is [[Quasistatic process|quasistatic]] (i.e., it occurs without any dissipation, deviating only infinitesimally from the thermodynamic equilibrium), and it may conserve total entropy. For example, in the [[Carnot cycle]], while the heat flow from a hot reservoir to a cold reservoir represents the increase in the entropy in a cold reservoir, the work output, if reversibly and perfectly stored, represents the decrease in the entropy which could be used to operate the heat engine in reverse, returning to the initial state; thus the total entropy change may still be zero at all times if the entire process is reversible.

In contrast, an irreversible process increases the total entropy of the system and surroundings.<ref>{{cite web|last1=Lower|first1=Stephen|title=What is entropy?|url=http://www.chem1.com/acad/webtext/thermeq/TE2.html|website=chem1.com|access-date=21 May 2016}}</ref> Any process that happens quickly enough to deviate from the thermal equilibrium cannot be reversible; the total entropy increases, and the potential for maximum work to be done during the process is lost.<ref>{{cite web|title=6.5 Irreversibility, Entropy Changes, and ''Lost Work''|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node48.html|website=web.mit.edu|access-date=21 May 2016}}</ref>

=== Carnot cycle ===
The concept of entropy arose from [[Rudolf Clausius]]'s study of the [[Carnot cycle]] which is a [[thermodynamic cycle]] performed by a Carnot heat engine as a reversible heat engine.<ref>{{cite book| last1=Lavenda| first1=Bernard H.| title=A new perspective on thermodynamics| date=2010|publisher=Springer|location=New York|isbn=978-1-4419-1430-9|edition=Online-Ausg.|chapter=2.3.4}}</ref> In a Carnot cycle, the heat <math display="inline">Q_\mathsf{H}</math> is transferred from a hot reservoir to a working gas at the constant temperature <math display="inline">T_\mathsf{H}</math> during [[Isothermal process|isothermal]] expansion stage and the heat <math display="inline">Q_\mathsf{C}</math> is transferred from a working gas to a cold reservoir at the constant temperature <math display="inline">T_\mathsf{C}</math> during [[Isothermal process|isothermal]] compression stage. According to [[Carnot's theorem (thermodynamics)|Carnot's theorem]], a heat engine with two thermal reservoirs can produce a [[work (physics)|work]] <math display="inline">W</math> if and only if there is a temperature difference between reservoirs. Originally, Carnot did not distinguish between heats <math display="inline">Q_\mathsf{H}</math> and <math display="inline">Q_\mathsf{C}</math>, as he assumed [[caloric theory]] to be valid and hence that the total heat in the system was conserved. But in fact, the magnitude of heat <math display="inline">Q_\mathsf{H}</math> is greater than the magnitude of heat <math display="inline">Q_\mathsf{C}</math>.<ref>{{cite book|last1=Carnot|first1=Sadi Carnot|editor1-last=Fox|editor1-first=Robert |title=Reflexions on the motive power of fire|url=https://archive.org/details/reflexionsonmoti0000carn|url-access=registration| date=1986|publisher=Lilian Barber Press| location=New York|isbn=978-0-936508-16-0|pages=[https://archive.org/details/reflexionsonmoti0000carn/page/26 26]}}</ref><ref>{{cite book|last1=Truesdell|first1=C.|title=The tragicomical history of thermodynamics 1822–1854|url=https://archive.org/details/tragicomicalhist18221854iiic|url-access=limited|date=1980|publisher=Springer|location=New York|isbn=978-0-387-90403-0|pages=[https://archive.org/details/tragicomicalhist18221854iiic/page/n85 78]–85}}</ref> Through the efforts of [[Rudolf Clausius|Clausius]] and [[Lord Kelvin|Kelvin]], the work <math display="inline">W</math> done by a reversible heat engine was found to be the product of the Carnot efficiency (i.e., the efficiency of all reversible heat engines with the same pair of thermal reservoirs) and the heat <math display="inline">Q_\mathsf{H}</math> absorbed by a working body of the engine during isothermal expansion:<math display="block">W = \frac{ T_\mathsf{H} - T_\mathsf{C} }{ T_\mathsf{H} } \cdot Q_\mathsf{H} = \left( 1 - \frac{ T_\mathsf{C} }{ T_\mathsf{H} } \right) Q_\mathsf{H}</math>To derive the Carnot efficiency 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 [[James Prescott Joule|Joule]] in a letter to Kelvin. This allowed Kelvin to establish his absolute temperature scale.<ref>{{cite book |last1=Clerk Maxwell|first1=James|editor1-last=Pesic|editor1-first=Peter|title=Theory of heat|date=2001|publisher=Dover Publications |location=Mineola|isbn=978-0-486-41735-6|pages=115–158}}</ref>

It is known that a work <math display="inline">W > 0</math> produced by an engine over a cycle equals to a net heat <math display="inline"> Q_\Sigma = \left\vert Q_\mathsf{H} \right\vert - \left\vert Q_\mathsf{C} \right\vert </math> absorbed over a cycle.<ref name="PlanckBook">{{cite book |last=Planck |first=M. |title=Treatise on Thermodynamics |page=§90 & §137|quote=eqs.(39), (40), & (65) |publisher=Dover Publications |year=1945}}.</ref> Thus, with the sign convention for a heat <math display="inline"> Q</math> transferred in a thermodynamic process (<math display="inline"> Q > 0</math> for an absorption and <math display="inline"> Q < 0</math> for a dissipation) we get:<math display="block">W - Q_\Sigma = W - \left\vert Q_\mathsf{H} \right\vert + \left\vert Q_\mathsf{C} \right\vert = W - Q_\mathsf{H} - Q_\mathsf{C} = 0</math>Since this equality holds over an entire Carnot cycle, it gave Clausius the hint that at each stage of the cycle the difference between a work and a net heat would be conserved, rather than a net heat itself. Which means there exists a [[state function]] <math display="inline">U</math> with a change of <math display="inline">\mathrm{d} U = \delta Q - \mathrm{d} W</math>. It is called an [[internal energy]] and forms a central concept for the [[first law of thermodynamics]].<ref name="Clausius1867">{{cite book |author=Rudolf Clausius |title=The Mechanical Theory of Heat: With Its Applications to the Steam-engine and to the Physical Properties of Bodies|url=https://books.google.com/books?id=8LIEAAAAYAAJ |year=1867 |publisher=J. Van Voorst |isbn=978-1-4981-6733-8|page=28}}</ref>

Finally, comparison for both the representations of a work output in a Carnot cycle gives us:<ref name="PlanckBook" /><ref name="FermiBook">{{cite book |last=Fermi |first=E. |title=Thermodynamics |page=48 |quote=eq.(64) |publisher=Dover Publications (still in print) |year=1956}}.</ref><math display="block">\frac{\left\vert Q_\mathsf{H} \right\vert}{T_\mathsf{H}} - \frac{\left\vert Q_\mathsf{C} \right\vert}{T_\mathsf{C}} = \frac{Q_\mathsf{H}}{T_\mathsf{H}} + \frac{Q_\mathsf{C}}{T_\mathsf{C}} = 0</math>Similarly to the derivation of internal energy, this equality implies existence of a [[state function]] <math display="inline">S</math> with a change of <math display="inline">\mathrm{d} S = \delta Q / T</math> and which is conserved over an entire cycle. Clausius called this state function ''entropy''.

In addition, the total change of entropy in both thermal reservoirs over Carnot cycle is zero too, since the inversion of a heat transfer direction means a sign inversion for the heat transferred during isothermal stages:<math display="block">- \frac{ Q_\mathsf{H} }{ T_\mathsf{H} } - \frac{ Q_\mathsf{C} }{ T_\mathsf{C} } = \Delta S_\mathsf{r, H} +  \Delta S_\mathsf{r, C} = 0</math>Here we denote the entropy change for a thermal reservoir by <math display="inline">\Delta S_{\mathsf{r}, i} = - Q_i / T_i</math>, where <math display="inline">i</math> is either <math display="inline">\mathsf{H}</math> for a hot reservoir or <math display="inline">\mathsf{C}</math> for a cold one.

If we consider a heat engine which is less effective than Carnot cycle (i.e., the work <math display="inline"> W</math> produced by this engine is less than the maximum predicted by Carnot's theorem), its work output is capped by Carnot efficiency as:<math display="block"> W < \left( 1 - \frac{T_\mathsf{C}}{T_\mathsf{H}} \right) Q_\mathsf{H} </math>Substitution of the work <math display="inline">W</math> as the net heat into the inequality above gives us:<math display="block">\frac{Q_\mathsf{H}}{T_\mathsf{H}} + \frac{Q_\mathsf{C}}{T_\mathsf{C}} < 0</math>or in terms of the entropy change <math display="inline">\Delta S_{\mathsf{r}, i}</math>:<math display="block">\Delta S_\mathsf{r, H} + \Delta S_\mathsf{r, C} > 0</math>A [[Carnot cycle]] and an entropy as shown above prove to be useful in the study of any classical thermodynamic heat engine: other cycles, such as an [[Otto cycle|Otto]], [[Diesel cycle|Diesel]] or [[Brayton cycle]], could be analysed from the same standpoint. Notably, any machine or cyclic process converting heat into work (i.e., heat engine) that is claimed to produce an efficiency greater than the one of Carnot is not viable — due to violation of [[Second law of thermodynamics|the second law of thermodynamics]].

For further analysis of sufficiently discrete systems, such as an assembly of particles, [[Statistical mechanics|statistical thermodynamics]] must be used. Additionally, descriptions of devices operating near the limit of [[Matter wave|de Broglie waves]], e.g. [[Solar cell|photovoltaic cells]], have to be consistent with [[Quantum statistical mechanics|quantum statistics]].

=== Classical thermodynamics ===
{{Main|Entropy (classical thermodynamics)}} 
{{Conjugate variables (thermodynamics)}}

The thermodynamic definition of entropy was developed in the early 1850s by [[Rudolf Clausius]] and essentially describes how to measure the entropy of an [[isolated system]] in [[thermodynamic equilibrium]] with its parts. Clausius created the term entropy as an [[Intensive and extensive properties|extensive]] thermodynamic variable that was shown to be useful in characterizing the [[Carnot cycle]]. Heat transfer in the isotherm steps (isothermal expansion and isothermal compression) of the Carnot cycle was found to be proportional to the temperature of a system (known as its [[absolute temperature]]). This relationship was expressed in an increment of entropy that is equal to incremental heat transfer divided by temperature. Entropy was found to vary in the thermodynamic cycle but eventually returned to the same value at the end of every cycle. Thus it was found to be a [[function of state]], 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]], entropy of an isolated [[Thermodynamic system|system]] 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 theorem|Clausius equality]], for a reversible cyclic thermodynamic process: <math display="block">\oint{\frac{\delta Q_\mathsf{rev}}{T}} = 0</math>which means the line integral <math display="inline">\int_L{\delta Q_\mathsf{rev} / T}</math> is [[State function|path-independent]]. Thus we can define a state function <math display="inline">S</math>, called ''entropy'':<math display="block">\mathrm{d} S = \frac{\delta Q_\mathsf{rev}}{T}</math>Therefore, thermodynamic entropy has the dimension of energy divided by temperature, and the unit [[joule]] per [[kelvin]] (J/K) in the International System of Units (SI).

To find the entropy difference between any two states of the system, the integral must be evaluated for some reversible path between the initial and final states.<ref>{{Cite book|last=Atkins|first=Peter|author2=Julio De Paula|title=Physical Chemistry, 8th ed.|publisher=Oxford University Press|year=2006|page=79|isbn=978-0-19-870072-2}}</ref> Since an 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.<ref>{{Cite book|last=Engel|first=Thomas|author2=Philip Reid|title=Physical Chemistry|publisher=Pearson Benjamin Cummings|year=2006|page=86|isbn=978-0-8053-3842-3}}</ref> However, the heat transferred to or from the surroundings is different as well as its entropy change.

We can calculate the change of entropy only by integrating the above formula. To obtain the absolute value of the entropy, we consider the [[third law of thermodynamics]]: perfect crystals at the [[absolute zero]] have an entropy <math display="inline">S = 0</math>.

From a macroscopic perspective, in [[classical thermodynamics]] the entropy is interpreted as a [[state function]] of a [[thermodynamic system]]: that is, a property depending only on the current state of the system, independent of how that state came to be achieved. In any process, where the system gives up <math>\Delta E</math> of energy to the surrounding at the temperature <math display="inline">T</math>, its entropy falls by <math display="inline">\Delta S</math> and at least <math display="inline">T \cdot \Delta S</math> of that energy must be given up to the system's surroundings as a heat. Otherwise, this process cannot go forward. In classical thermodynamics, the entropy of a system is defined if and only if it is in a [[thermodynamic equilibrium]] (though a [[chemical equilibrium]] is not required: for example, the entropy of a mixture of two moles of hydrogen and one mole of oxygen in [[Standard temperature and pressure|standard conditions]] is well-defined).

=== Statistical mechanics ===
{{main|Entropy (statistical thermodynamics)}}
The statistical definition was developed by [[Ludwig Boltzmann]] in the 1870s by analysing the statistical behaviour 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 the [[Boltzmann constant]]. In short, the thermodynamic definition of entropy provides the experimental verification of entropy, while the statistical definition of entropy extends the concept, providing an explanation and a deeper understanding of its nature.

The [[Entropy (statistical thermodynamics)|interpretation of entropy in statistical mechanics]] is the measure of uncertainty, disorder, or ''mixedupness'' in the phrase of [[Josiah Willard Gibbs|Gibbs]], 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 [[Microstate (statistical mechanics)|microstates]]. In contrast to the macrostate, which characterizes plainly observable average quantities, a microstate specifies all molecular details about the system including the position and momentum 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).<ref name=McH>{{cite book |last1=Licker |first1=Mark D. |title=McGraw-Hill concise encyclopedia of chemistry |date=2004 |publisher=McGraw-Hill Professional |location=New York |isbn=978-0-07-143953-4}}</ref><ref name="Sethna78" /><ref>{{cite book |last1=Clark |first1=John O. E. |title=The essential dictionary of science |date=2004 |publisher=Barnes & Noble |location=New York |isbn=978-0-7607-4616-5}}</ref> 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 ([[microstate (statistical mechanics)|microstates]]) that could cause the observed macroscopic state ([[macrostate]]) of the system. The constant of proportionality is the [[Boltzmann constant]].

The Boltzmann constant, and therefore entropy, have [[dimension (physics)|dimensions]] of energy divided by temperature, which has a unit of [[joule]]s per [[kelvin]] (J⋅K<sup>−1</sup>) in the [[International System of Units]] (or kg⋅m<sup>2</sup>⋅s<sup>−2</sup>⋅K<sup>−1</sup> in terms of base units). The entropy of a substance is usually given as an [[Intensive and extensive properties#Intensive properties|intensive property]] — either entropy per unit [[mass]] (SI unit: J⋅K<sup>−1</sup>⋅kg<sup>−1</sup>) or entropy per unit [[amount of substance]] (SI unit: J⋅K<sup>−1</sup>⋅mol<sup>−1</sup>).

Specifically, entropy is a [[logarithmic scale|logarithmic]] measure for the system with a number of states, each with a probability <math display="inline">p_i</math> of being occupied (usually given by the [[Boltzmann distribution]]):<math display="block">S = - k_\mathsf{B} \sum_i{p_i \ln{p_i}}</math>where <math display="inline">k_\mathsf{B}</math> is the [[Boltzmann constant]] and the summation is performed over all possible microstates of the system.<ref name="Perplexed">[http://charlottewerndl.net/Entropy_Guide.pdf Frigg, R. and Werndl, C. "Entropy – A Guide for the Perplexed"] {{Webarchive|url=https://web.archive.org/web/20110813112247/http://charlottewerndl.net/Entropy_Guide.pdf |date=13 August 2011 }}. In ''Probabilities in Physics''; Beisbart C. and Hartmann, S. (eds.); Oxford University Press, Oxford, 2010.</ref>

In case states are defined in a continuous manner, the summation is replaced by an [[integral]] over all possible states, or equivalently we can consider the [[expected value]] of [[Entropy (information theory)#Rationale|the logarithm of the probability]] that a microstate is occupied:<math display="block">S = - k_\mathsf{B} \left\langle \ln{p} \right\rangle</math>This definition assumes the basis states to be picked in a way that there is no information on their relative phases. In a general case the expression is:<math display="block">S = - k_\mathsf{B}\ \mathrm{tr}{\left( \hat{\rho} \times \ln{\hat{\rho}} \right)}</math>where <math display="inline">\hat{\rho}</math> is a [[density matrix]], <math>\mathrm{tr}</math> is a [[Trace class|trace operator]] and <math>\ln</math> is a [[matrix logarithm]]. The density matrix formalism is not required if the system is in thermal equilibrium so long as the basis states are chosen to be [[Quantum state|eigenstates]] of the [[Hamiltonian (quantum mechanics)|Hamiltonian]]. For most practical purposes it can be taken as the fundamental definition of entropy since all other formulae for <math display="inline">S</math> can be derived from it, but not vice versa.

In what has been called ''[[Fundamental postulate of statistical mechanics|the fundamental postulate in statistical mechanics]]'', among system microstates of the same energy (i.e., [[Degenerate energy levels|degenerate microstates]]) each microstate is assumed to be populated with equal probability <math display="inline">p_i = 1 / \Omega</math>, where <math display="inline">\Omega</math> is the number of microstates whose energy equals that of the system. Usually, this assumption is justified for an isolated system in a thermodynamic equilibrium.<ref>{{cite book|last1=Schroeder|first1=Daniel V.|title=An introduction to thermal physics|url=https://archive.org/details/introductiontoth00schr_817|url-access=limited|date=2000|publisher=Addison Wesley|location=San Francisco, CA |isbn=978-0-201-38027-9|page=[https://archive.org/details/introductiontoth00schr_817/page/n68 57]}}</ref> Then in case of an isolated system the previous formula reduces to:<math display="block">S = k_\mathsf{B} \ln{\Omega}</math>In thermodynamics, such a system is one with a fixed volume, number of molecules, and internal energy, called a [[microcanonical ensemble]].

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.<ref>{{Cite journal|last=Jaynes|first=E. T.|date=1 May 1965|title=Gibbs vs Boltzmann Entropies|url=https://aapt.scitation.org/doi/10.1119/1.1971557|journal=American Journal of Physics|volume=33|issue=5|pages=391–398|doi=10.1119/1.1971557|bibcode=1965AmJPh..33..391J|issn=0002-9505}}</ref>

The interpretative model has a central role in determining entropy. The qualifier "for a given set of macroscopic variables" above has deep implications when two observers use different sets of macroscopic variables. For example, consider observer A using variables <math display="inline">U</math>, <math display="inline">V</math>, <math display="inline">W</math> and observer B using variables <math display="inline">U</math>, <math display="inline">V</math>, <math display="inline">W</math>, <math display="inline">X</math>. If observer B changes variable <math display="inline">X</math>, then observer A will see a violation of the second law of thermodynamics, since he does not possess information about variable <math display="inline">X</math> and its influence on the system. In other words, one must choose a complete set of macroscopic variables to describe the system, i.e. every independent parameter that may change during experiment.<ref>{{cite book |url=http://www.mdpi.org/lin/entropy/cgibbs.pdf |author=Jaynes, E. T. |chapter=The Gibbs Paradox |title=Maximum Entropy and Bayesian Methods |editor1=Smith, C. R. |editor2=Erickson, G. J. |editor3=Neudorfer, P. O. |publisher=Kluwer Academic: Dordrecht |year=1992 |pages=1–22 |access-date=17 August 2012}}</ref>

Entropy can also be defined for any [[Markov process]]es 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.

=== Entropy of a system ===
[[File:system boundary.svg|thumb|A [[thermodynamic system]]]]

[[File:Temperature-entropy chart for steam, imperial units.svg|thumb|A [[temperature–entropy diagram]] for steam. The vertical axis represents uniform temperature, and the horizontal axis represents specific entropy. Each dark line on the graph represents constant pressure, and these form a mesh with light grey lines of constant volume. (Dark-blue is liquid water, light-blue is liquid-steam mixture, and faint-blue is steam. Grey-blue represents supercritical liquid water.)]]

In a [[thermodynamic system]], pressure and temperature tend to become uniform over time because the [[equilibrium state]] has higher [[probability]] (more possible [[combination]]s of [[microstate (statistical mechanics)|microstates]]) than any other state. As an example, for a glass of ice water in air at [[room temperature]], the difference in temperature between the warm room (the surroundings) and the cold glass of ice and water (the system and not part of the room) decreases 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]] such as the room and ice water taken together, the dispersal of energy from warmer to cooler always results in a net increase in entropy. Thus, when the "universe" of the room and ice water system has reached a temperature equilibrium, the entropy change from the initial state is at a maximum. The entropy of the [[thermodynamic system]] is a measure of how far the equalisation has progressed.

Thermodynamic entropy is a non-conserved [[state function]] that is of great importance in the sciences of [[physics]] and [[chemistry]].<ref name="McH" /><ref name="Wiley91">{{cite book|last1=Sandler|first1=Stanley I.|title=Chemical, biochemical, and engineering thermodynamics|url=https://archive.org/details/chemicalbiochemi00sand|url-access=limited|date=2006|publisher=John Wiley & Sons|location=New York|isbn=978-0-471-66174-0|page=[https://archive.org/details/chemicalbiochemi00sand/page/n104 91]|edition=4th}}</ref> 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.<ref name="McQuarrie817">{{cite book|last1=Simon|first1= John D. |first2=Donald A. |last2=McQuarrie |title=Physical chemistry : a molecular approach|date=1997|publisher=Univ. Science Books|location=Sausalito, Calif.|isbn=978-0-935702-99-6|page=817|edition=Rev.}}</ref><ref>{{Cite book|last=Haynie|first=Donald T.|title=Biological Thermodynamics|publisher=[[Cambridge University Press]]|year=2001|isbn=978-0-521-79165-6}}</ref> For [[isolated system]]s, entropy never decreases.<ref name="Wiley91" /> 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.<ref name="McH" /><ref name="Sethna78">{{cite book|last1=Sethna|first1=James P.|title=Statistical mechanics : entropy, order parameters, and complexity.|url=https://archive.org/details/statisticalmecha00seth_912|url-access=limited|date=2006|publisher=Oxford University Press|location=Oxford|isbn=978-0-19-856677-9|page=[https://archive.org/details/statisticalmecha00seth_912/page/n97 78]|edition=[Online-Ausg.]}}</ref><ref name="OxSci">{{cite book|last1=Daintith|first1=John|title=A dictionary of science|date=2005|publisher=Oxford University Press|location=Oxford|isbn=978-0-19-280641-3|edition=5th}}</ref><ref>{{Cite book|last=de Rosnay|first=Joel|title=The Macroscope&nbsp;– a New World View (written by an M.I.T.-trained biochemist)|publisher=Harper & Row, Publishers|year=1979|isbn=978-0-06-011029-1|title-link=M.I.T.}}</ref>

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]] states that the entropy of an [[isolated system]] 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.<ref>{{cite web|last=McGovern|first=J. A.|title=Heat Capacities|url=http://theory.phy.umist.ac.uk/~judith/stat_therm/node50.html|url-status=dead|archive-url=https://web.archive.org/web/20120819175243/http://theory.phy.umist.ac.uk/~judith/stat_therm/node50.html|archive-date=19 August 2012|access-date=27 January 2013}}</ref> 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.<ref>{{cite journal|last1=Ben-Naim|first1=Arieh|title=On the So-Called Gibbs Paradox, and on the Real Paradox|journal=Entropy|date=21 September 2007|volume=9|issue=3|pages=132–136|doi=10.3390/e9030133|url=http://www.mdpi.org/entropy/papers/e9030132.pdf|bibcode=2007Entrp...9..132B|doi-access=free}}</ref>

=== Equivalence of definitions ===
Proofs of equivalence between the entropy in statistical mechanics — the [[Entropy (statistical thermodynamics)#Gibbs entropy formula|Gibbs entropy formula]]:<math display="block">S = - k_\mathsf{B} \sum_i{p_i \ln{p_i}}</math>and the entropy in classical thermodynamics:<math display="block">\mathrm{d} S = \frac{\delta Q_\mathsf{rev}}{T}</math>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 generalised [[Boltzmann distribution]] and the identification of the thermodynamic internal energy as the ensemble average <math display="inline">U = \left\langle E_i \right\rangle </math>.<ref>{{cite book |last= Callen|first= Herbert|date= 2001|title= Thermodynamics and an Introduction to Thermostatistics (2nd ed.)|publisher= John Wiley and Sons|isbn= 978-0-471-86256-7}}</ref> 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 [[Boltzmann distribution#Generalized Boltzmann distribution|generalized Boltzmann distribution]].<ref>{{cite journal |last1= Gao |first1= Xiang |last2= Gallicchio |first2= Emilio |first3= Adrian |last3= Roitberg |year= 2019 |title= The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy |journal= The Journal of Chemical Physics|volume= 151|issue= 3|pages= 034113|doi= 10.1063/1.5111333|pmid= 31325924 |arxiv= 1903.02121 |bibcode= 2019JChPh.151c4113G |s2cid= 118981017 }}</ref>

Furthermore, it has been shown that the definitions of entropy in statistical mechanics is the only entropy that is equivalent to the classical thermodynamics entropy under the following postulates:<ref name="Gao2022">{{cite journal |last1= Gao |first1= Xiang |date= March 2022 |title= The Mathematics of the Ensemble Theory |journal= Results in Physics|volume= 34|pages= 105230|doi= 10.1016/j.rinp.2022.105230 |bibcode= 2022ResPh..3405230G |s2cid= 221978379 |doi-access= free |arxiv= 2006.00485 }}</ref>
{{ordered list
| The probability density function is proportional to some function of the ensemble parameters and random variables.
| Thermodynamic state functions are described by ensemble averages of random variables.
| At infinite temperature, all the microstates have the same probability.
}}