Ice melting in a warm room is a common example of increasing entropy,^{[note 1]} described in 1862 by Rudolf Clausius as an increase in the disgregation of the water molecules in ice.^[1]

Entropy is a thermodynamic property that can be used to determine the energy not available for work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when converting energy to work. During this work, entropy accumulates in the system, which then dissipates in the form of waste heat.

In classical thermodynamics, the concept of entropy is defined phenomenologically by the second law of thermodynamics, which states that the entropy of an isolated system always increases or remains constant. Thus, entropy is also a measure of the tendency of a process, such as a chemical reaction, to be entropically favored, or to proceed in a particular direction. It determines that thermal energy always flows spontaneously from regions of higher temperature to regions of lower temperature, in the form of heat. These processes reduce the state of order of the initial systems, and therefore entropy is an expression of disorder or randomness. This is the basis of the modern microscopic interpretation of entropy in statistical mechanics, where entropy is defined as the amount of additional information needed to specify the exact physical state of a system, given its thermodynamic specification. The second law is then a consequence of this definition and the fundamental postulate of statistical mechanics.

Thermodynamic entropy has the dimension of energy divided by temperature, which has a unit of joules per kelvin (J/K) in the International System of Units.

The term entropy was coined in 1865 by Rudolf Clausius based on the Greek εντροπία [entropía], a turning toward, from εν- [en-] (in) and τροπή [tropē] (turn, conversion).^{[2][note 2]}

Thermodynamical and statistical descriptions[link]

There are two related definitions of entropy: the thermodynamic definition and the statistical mechanics definition. The thermodynamic definition was developed in the early 1850s by Rudolf Clausius and essentially describes how to measure the entropy of an isolated system in thermodynamic equilibrium. Importantly, it makes no reference to the microscopic nature of matter. The statistical definition was developed by Ludwig Boltzmann 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 number which has since been known as Boltzmann's constant. 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.

Thermodynamic entropy is a non-conserved state function that is of great importance in the sciences of physics and chemistry.^[4][5] Historically, the concept of entropy evolved in order to explain why some processes (permitted by conservation laws) occur spontaneously while their time reversals (also permitted by conservation laws) do not; systems tend to progress in the direction of increasing entropy.^[6][7] For isolated systems, entropy never decreases.^[5] This fact has several important consequences in science: first, it prohibits "perpetual motion" machines; and second, it implies the arrow of entropy has the same directionality as the arrow of time. Increases in entropy correspond to irreversible changes in a system, because some energy is expended as waste heat, limiting the amount of work a system can do.^{[4][8][8][9][10][11]}

In statistical mechanics, entropy is a measure of the number of ways in which a system may be arranged, often taken to be a measure of "disorder" (the higher the entropy, the higher the disorder).^{[4][9][10][12][13][14][15]} 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) which could give rise to the observed macroscopic state (macrostate) of the system. The constant of proportionality is the Boltzmann constant.

Second law of thermodynamics[link]

The second law of thermodynamics states that in general the total entropy of any system will 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 will tend not to decrease. It follows that heat will not flow from a colder body to a hotter body without the application of work (the imposition of order) 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. As a result, there is no possibility of a perpetual motion system. 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, will always make a bigger contribution to the entropy of the environment than will 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 useful work.^[16] The entropy change of a system at temperature T absorbing an infinitesimal amount of heat Failed to parse (Missing texvc executable; please see math/README to configure.): \delta q

in a reversible way, is given by Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\delta q}{T}

. More explicitly, an energy T_RS 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 a closed system. Although this is possible, such an event has a small probability of occurring, making it unlikely. Even if such an event were to occur, it would result in a transient decrease that would affect only a limited number of particles in the system.^[17]

Definitions and descriptions[link]

Thermodynamic entropy is more generally defined from a statistical thermodynamics viewpoint, in which the molecular nature of matter is explicitly considered. Alternatively entropy can be defined from a classical thermodynamics viewpoint, in which the molecular interactions are not considered and instead the system is viewed from perspective of the gross motion of very large masses of molecules and the behavior of individual molecules is averaged and obscured. Historically, the classical thermodynamics definition developed first, and it has more recently been extended in the area of non-equilibrium thermodynamics.

Carnot cycle[link]

The concept of entropy arose from Rudolf Clausius's study of the Carnot cycle.^[18] In a Carnot cycle, heat (Failed to parse (Missing texvc executable; please see math/README to configure.): Q_1 ) is absorbed from a 'hot' reservoir, isothermally at the higher temperature Failed to parse (Missing texvc executable; please see math/README to configure.): T_1 , and given up isothermally to a 'cold' reservoir, Failed to parse (Missing texvc executable; please see math/README to configure.): Q_2 , at a lower temperature, Failed to parse (Missing texvc executable; please see math/README to configure.): T_2 . According to Carnot's principle, work can only be done 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 Failed to parse (Missing texvc executable; please see math/README to configure.): Q_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): Q_2

, since he was working under the hypothesis that caloric theory was valid, and hence heat was conserved.^[19] Through the efforts of Clausius and Kelvin, it is now known that the maximum work that can be done is the product of the Carnot efficiency and the heat absorbed at the hot reservoir: Failed to parse (Missing texvc executable; please see math/README to configure.): W=\left(1-\frac{T_2}{T_1}\right)Q_1.

In order to derive the Carnot efficiency, Failed to parse (Missing texvc executable; please see math/README to configure.): 1-\frac{T_2}{T_1} , Kelvin had to evaluate the ratio of the work done to the heat absorbed in the isothermal expansion with the help of the Carnot-Clapeyron equation which contained an unknown function, known as the Carnot function. The fact that the Carnot function could be the temperature, measured from zero, was suggested by Joule in a letter to Kelvin, and this allowed Kelvin to establish his absolute temperature scale.^[20] It is also known that the work is the difference in the heat absorbed at the hot reservoir and rejected at the cold one: Failed to parse (Missing texvc executable; please see math/README to configure.): W=Q_1-Q_2

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 a state function that would vanish upon completion of the cycle. The state function was called the internal energy and it became the first law of thermodynamics.^[21]

Now equating the two expressions gives Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{Q_1}{T_1}-\frac{Q_2}{T_2}=0.

If we allow Failed to parse (Missing texvc executable; please see math/README to configure.): Q_2

to incorporate the algebraic sign, this becomes a sum and implies that there is a function of state which is conserved over a complete cycle. Clausius called this state function entropy. This is the second law of thermodynamics.

Then Clausius asked what would happen if there would be less work done than that predicted by Carnot's principle. The right-hand side of the first equation would be the upper bound of the work, which would now be converted into an inequality Failed to parse (Missing texvc executable; please see math/README to configure.): W<\left(1-\frac{T_2}{T_1}\right)Q_1

When the second equation is used to express the work as a difference in heats, we get Failed to parse (Missing texvc executable; please see math/README to configure.): Q_1-Q_2<\left(1-\frac{T_2}{T_1}\right)Q_1

or Failed to parse (Missing texvc executable; please see math/README to configure.): Q_2>\frac{T_2}{T_1}Q_1

So more heat is given off to the cold reservoir than in the Carnot cycle. If we denote the entropies by Failed to parse (Missing texvc executable; please see math/README to configure.): S_i=Q_i/T_i

for the two states, then the above inequality can be written as a decrease in the entropy

Failed to parse (Missing texvc executable; please see math/README to configure.): S_1-S_2<0

The wasted heat implies that irreversible processes must have prevented the cycle from carrying out maximum work.

Statistical thermodynamics[link]

The interpretation of entropy in statistical mechanics is the measure of uncertainty, or mixedupness in the phrase of 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 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 velocity of every molecule. The more such states available to the system with appreciable probability, the greater the entropy.

Specifically, entropy is a logarithmic measure of the density of states:

Failed to parse (Missing texvc executable; please see math/README to configure.): S = - k_{\mathrm{B}}\sum_i P_i \ln P_i \, ,

where k_B is the Boltzmann constant, equal to 1.38065×10⁻²³ J K⁻¹. The summation is over all the possible microstates of the system, and P_i is the probability that the system is in the ith microstate.^[22] 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 some rare and recondite situations, a generalization of this formula may be needed to account for quantum coherence effects, but in any situation where a classical notion of probability makes sense, the above equation accurately describes the entropy of the system.)

In what has been called the fundamental assumption of statistical thermodynamics or the fundamental postulate in statistical mechanics, the occupation of any microstate is assumed to be equally probable (i.e. P_i=1/Ω since Ω is the number of microstates); this assumption is usually justified for an isolated system in equilibrium.^[23] Then the previous equation reduces to:

Failed to parse (Missing texvc executable; please see math/README to configure.): S = k_{\mathrm{B}} \ln \Omega \, ,

In thermodynamics, such a system is one in which the volume, number of molecules, and internal energy are fixed (the microcanonical ensemble).

The most general interpretation of entropy is as a measure of our uncertainty about a system. The equilibrium state of a system maximizes the entropy because we have lost all information about the initial conditions except for the conserved variables; maximizing the entropy maximizes our ignorance about the details of the system.^[24] 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 will observe 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!^[25]

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.

Classical thermodynamics[link]

According to the Clausius equality, for a reversible process: Failed to parse (Missing texvc executable; please see math/README to configure.): \oint \frac{\delta Q}{T}=0 . This means the line integral Failed to parse (Missing texvc executable; please see math/README to configure.): \int_L \frac{\delta Q}{T}

is path independent.

So we can define a state function S called entropy, which satisfies: Failed to parse (Missing texvc executable; please see math/README to configure.): dS = \frac{\delta Q}{T} \!

With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the Third Law of Thermodynamics, which states that S=0 at absolute zero for perfect crystals.

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. The state function has the important property that, when multiplied by a reference temperature, it can be understood as a measure of the amount of energy in a physical system that cannot be used to do thermodynamic work; i.e., work mediated by thermal energy.^{[citation needed]} More precisely, in any process where the system gives up energy ΔE, and its entropy falls by ΔS, a quantity at least T_R ΔS of that energy must be given up to the system's surroundings as unusable heat (T_R is the temperature of the system's external surroundings). Otherwise the process will not go forward. In classical thermodynamics, the entropy of a system is defined only if it is in thermodynamic equilibrium.

In a thermodynamic system, pressure, density, and temperature tend to become uniform over time because this equilibrium state has higher probability (more possible combinations of microstates) than any other; see statistical mechanics. In the ice melting example, the difference in temperature between a warm room (the surroundings) and cold glass of ice and water (the system and not part of the room), begins to be equalized as portions of the thermal energy from the warm surroundings spread to the cooler system of ice and water.

A thermodynamic system

Over time the temperature of the glass and its contents and the temperature of the room become equal. The entropy of the room has decreased as some of its energy has been dispersed to the ice and water. However, as calculated in the example, the entropy of the system of ice and water has increased more than the entropy of the surrounding room has decreased. In an isolated system such as the room and ice water taken together, the dispersal of energy from warmer to cooler always results in a net increase in entropy. Thus, when the "universe" of the room and ice water system has reached a temperature equilibrium, the entropy change from the initial state is at a maximum. The entropy of the thermodynamic system is a measure of how far the equalization has progressed.

A special case of entropy increase, the entropy of mixing, occurs when two or more different substances are mixed. If the substances are at the same temperature and pressure, there will be no net exchange of heat or work – the entropy change will be 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.^[26]

History[link]

Rudolf Clausius, originator of the concept of entropy

The first law of thermodynamics, formalized based on the heat-friction experiments of James Joule in 1843, deals with the concept of energy, which is conserved in all processes; the first law, however, is unable to quantify the effects of friction and dissipation.

The analysis which led to the concept of entropy began with the work of French mathematician Lazare Carnot who in his 1803 paper Fundamental Principles of Equilibrium and Movement proposed that in any machine the accelerations and shocks of the moving parts represent losses of moment of activity. In other words, in any natural process there exists an inherent tendency towards the dissipation of useful energy. Building on this work, in 1824 Lazare's son Sadi Carnot published Reflections on the Motive Power of Fire which posited that in all heat-engines whenever "caloric", or what is now known as heat, falls through a temperature difference, work or motive power can be produced from the actions of the "fall of caloric" between a hot and cold body. This was an early insight into the second law of thermodynamics.^[27]

Carnot based his views of heat partially on the early 18th century "Newtonian hypothesis" that both heat and light were types of indestructible forms of matter, which are attracted and repelled by other matter, and partially on the contemporary views of Count Rumford who showed (1789) that heat could be created by friction as when cannon bores are machined.^[28] Carnot reasoned that if the body of the working substance, such as a body of steam, is returned to its original state (temperature and pressure) at the end of a complete engine cycle, that "no change occurs in the condition of the working body". This latter comment was amended in his foot notes, and it was this comment that led to the development of entropy.^{[citation needed]}

In the 1850s and 1860s, German physicist Rudolf Clausius objected to this supposition, i.e. that no change occurs in the working body, and gave this "change" a mathematical interpretation by questioning the nature of the inherent loss of usable heat when work is done, e.g. heat produced by friction.^[29] Clausius described entropy as the transformation-content, i.e. dissipative energy use, of a thermodynamic system or working body of chemical species during a change of state.^[29] This was in contrast to earlier views, based on the theories of Isaac Newton, that heat was an indestructible particle that had mass.

Later, scientists such as Ludwig Boltzmann, Josiah Willard Gibbs, and James Clerk Maxwell gave entropy a statistical basis. In 1877 Boltzmann visualized a probabilistic way to measure the entropy of an ensemble of ideal gas particles, in which he defined entropy to be proportional to the logarithm of the number of microstates such a gas could occupy. Henceforth, the essential problem in statistical thermodynamics, i.e. according to Erwin Schrödinger, has been to determine the distribution of a given amount of energy E over N identical systems. Carathéodory linked entropy with a mathematical definition of irreversibility, in terms of trajectories and integrability.

Consequences and applications[link]

The arrow of time[link]

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 never decreases. Hence, from this perspective, entropy measurement is thought of as a kind of clock.

The fundamental thermodynamic relation[link]

The entropy of a system depends on its internal energy and the external parameters, such as the volume. In the thermodynamic limit this fact leads to an equation relating the change in the internal energy to changes in the entropy and the external parameters. This relation is known as the fundamental thermodynamic relation. If the volume is the only external parameter, this relation is: Failed to parse (Missing texvc executable; please see math/README to configure.): dU = T dS - P dV

Since 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 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[link]

Thermodynamic entropy is central in chemical thermodynamics, enabling changes to be quantified and the outcome of reactions predicted. The second law of thermodynamics states that entropy in an isolated system —the combination of a subsystem under study and its surroundings— increases during all spontaneous chemical and physical processes. The Clausius equation of δq_rev/T = ΔS introduces the measurement of entropy change, ΔS. Entropy change describes the direction and quantifies the magnitude of simple changes such as heat transfer between systems —always from hotter to cooler spontaneously.^[30]

The thermodynamic entropy therefore has the dimension of energy divided by temperature, and the unit joule per kelvin (J/K) in the International System of Units (SI).

Thermodynamic entropy is an 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 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: Jkg⁻¹K⁻¹). Alternatively, in chemistry, it is also referred to one mole of substance, in which case it is called the molar entropy with a unit of Jmol⁻¹K⁻¹.

Thus, when one mole of substance at 0K is warmed by its surroundings to 298K, the sum of the incremental values of q_rev/T constitute each element's or compound's standard molar entropy, a fundamental physical property and an indicator of the amount of energy stored by a substance at 298K.^[31][32] Entropy change also measures the mixing of substances as a summation of their relative quantities in the final mixture.^[33]

Entropy is equally essential in predicting the extent and direction of complex chemical reactions. For such applications, ΔS must be incorporated in an expression that includes both the system and its surroundings, ΔS_universe = ΔS_surroundings + ΔS _system. This expression becomes, via some steps, the Gibbs free energy equation for reactants and products in the system: ΔG [the Gibbs free energy change of the system] = ΔH [the enthalpy change] −T ΔS [the entropy change].^[31]

Entropy change[link]

When an ideal gas undergoes a change, its entropy may also change. For cases where the specific heat does not change and either volume, pressure or temperature is also constant, the change in entropy can be easily calculated.^[34]

When specific heat and volume are constant, the change in entropy is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta S = n C_v \ln \frac{T}{T_0}

.

When specific heat and pressure are constant, the change in entropy is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta S = n C_p \ln \frac{T}{T_0}

.

When specific heat and temperature are constant, the change in entropy is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta S = n R \ln \frac{V}{V_0}

.

In these equations Failed to parse (Missing texvc executable; please see math/README to configure.): C_v

is the specific heat at constant volume, Failed to parse (Missing texvc executable; please see math/README to configure.): C_p
is the specific heat at constant pressure, Failed to parse (Missing texvc executable; please see math/README to configure.): R
is the ideal gas constant, and Failed to parse (Missing texvc executable; please see math/README to configure.): n
is the number of moles of gas.

For some other transformations, not all of these properties (specific heat, volume, pressure or temperature) are constant. In these cases, for only 1 mole of an ideal gas, the change in entropy can be given by^[35] either:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta S = C_v \ln \frac{T}{T_0} + R \ln \frac{V}{V_0}

or

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta S = C_p \ln \frac{T}{T_0} - R \ln \frac{P}{P_0}

.

Entropy balance equation for open systems[link]

In chemical engineering, the principles of thermodynamics are commonly applied to "open systems", i.e. those in which heat, work, and mass flow across the system boundary. In a system in which there are flows of both heat (Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{Q} ) and work, i.e. Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{W}_S

(shaft work) and P(dV/dt) (pressure-volume work), across the system boundaries, the heat flow, but not the work flow, causes a change in the entropy of the system. This rate of entropy change is Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{Q}/T,
where T is the absolute thermodynamic temperature of the system at the point of the heat flow. If, in addition, there are mass flows across the system boundaries, the total entropy of the system will also change due to this convected flow.

During steady-state continuous operation, an entropy balance applied to an open system accounts for system entropy changes related to heat flow and mass flow across the system boundary.

To derive a generalized entropy balanced equation, we start with the general balance equation for the change in any extensive quantity Θ in a thermodynamic system, a quantity that may be either conserved, such as energy, or non-conserved, such as entropy. The basic generic balance expression states that dΘ/dt, i.e. the rate of change of Θ in the system, equals the rate at which Θ enters the system at the boundaries, minus the rate at which Θ leaves the system across the system boundaries, plus the rate at which Θ is generated within the system. Using this generic balance equation, with respect to the rate of change with time of the extensive quantity entropy S, the entropy balance equation for an open thermodynamic system is:^[36]

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{dS}{dt} = \sum_{k=1}^K \dot{M}_k \hat{S}_k + \frac{\dot{Q}}{T} + \dot{S}_{gen}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{k=1}^K \dot{M}_k \hat{S}_k

= the net rate of entropy flow due to the flows of mass into and out of the system (where Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{S}
= entropy per unit mass).

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\dot{Q}}{T}

= the rate of entropy flow due to the flow of heat across the system boundary.

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{S}_{gen}

= the rate of entropy production within the system.

Note, also, that if there are multiple heat flows, the term Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{Q}/T

is to be replaced by Failed to parse (Missing texvc executable; please see math/README to configure.): \sum \dot{Q}_j/T_j,
where Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{Q}_j
is the heat flow and Failed to parse (Missing texvc executable; please see math/README to configure.): T_j
is the temperature at the jth heat flow port into the system.

Entropy in quantum mechanics (von Neumann entropy)[link]

In quantum statistical mechanics, the concept of entropy was developed by John von Neumann and is generally referred to as "von Neumann entropy", Failed to parse (Missing texvc executable; please see math/README to configure.): S = - k_\mathrm{B}\mathrm{Tr} ( \rho \log \rho ) \!

where Failed to parse (Missing texvc executable; please see math/README to configure.): \rho

is the density matrix and Tr is the trace operator.

This upholds the correspondence principle, because in the classical limit, i.e. whenever the classical notion of probability applies, this expression is equivalent to the familiar classical definition of entropy, Failed to parse (Missing texvc executable; please see math/README to configure.): S = - k_\mathrm{B}\sum_i p_i \, \log \, p_i

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.

Approaches to understanding entropy[link]

Order and disorder[link]

Entropy has often been loosely associated with the amount of order, disorder, and/or chaos in a thermodynamic system. 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.^[38] In this direction, several recent authors have derived exact entropy formulas to account for and measure disorder and order in atomic and molecular assemblies.^{[39][40][41][42]} 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 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:^[41][42]

Failed to parse (Missing texvc executable; please see math/README to configure.): \mbox{Disorder}={C_D\over C_I}.\,

Similarly, the total amount of "order" in the system is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mbox{Order}=1-{C_O\over C_I}.\,

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.^[40]

Energy dispersal[link]

The concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature.^[43] Similar terms have been in use from early in the history of classical thermodynamics, and with the development of statistical thermodynamics and quantum theory, entropy changes have been described in terms of the mixing or "spreading" of the total energy of each constituent of a system over its particular quantized energy levels.

Ambiguities in the terms disorder and chaos, which usually have meanings directly opposed to equilibrium, contribute to widespread confusion and hamper comprehension of entropy for most students.^[44] As the second law of thermodynamics shows, in an isolated system internal portions at different temperatures will tend to adjust to a single uniform temperature and thus produce equilibrium. A recently developed educational approach avoids ambiguous terms and describes such spreading out of energy as dispersal, which leads to loss of the differentials required for work even though the total energy remains constant in accordance with the first law of thermodynamics^[45] (compare discussion in next section). Physical chemist Peter Atkins, for example, who previously wrote of dispersal leading to a disordered state, now writes that "spontaneous changes are always accompanied by a dispersal of energy".^[30][46]

[edit] Relating entropy to energy usefulness

Following on from the above, it is possible (in a thermal context) to regard entropy as an indicator or measure of the effectiveness or usefulness of a particular quantity of energy.^[47] This is because energy supplied at a high temperature (i.e. with low entropy) tends to be more useful than the same amount of energy available at room 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” which 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 will lead to the "heat death of the Universe".

Entropy and molecule complexity[link]

In a situation where a reaction involves equal moles hydrogen gas (H₂) on the reactant side and water vapor (H₂O) on the product side, would the reaction be spontaneous? Due to the complexity of the shapes of the molecules, water vapor would be favored and the forward reaction would be spontaneous. Since water vapor's has a bent shape compared to hydrogen gas's linear shape, it has a larger array of possible positions that each molecule can be situated as at some particular point in time. The second law of thermodynamics says that an increase in entropy is favored, which is why the statement before is valid.

Ice melting example[link]

The illustration for this article is a classic example in which entropy increases in a small "universe", a thermodynamic system consisting of the "surroundings" (the warm room) and "system" (glass, ice, cold water). In this universe, some thermal energy δQ from the warmer room surroundings (at 298 K or 25 °C) will spread out to the cooler system of ice and water at its constant temperature T of 273 K (0 °C), the melting temperature of ice. The entropy of the system will change by the amount dS = δQ/T, in this example δQ/273 K. (The thermal energy δQ for this process is the energy required to change water from the solid state to the liquid state, and is called the enthalpy of fusion, i.e. the ΔH for ice fusion.) The entropy of the surroundings will change by an amount dS = −δQ/298 K. So in this example, the entropy of the system increases, whereas the entropy of the surroundings decreases.

It is important to realize that the decrease in the entropy of the surrounding room is less than the increase in the entropy of the ice and water: the room temperature of 298 K is larger than 273 K and therefore the ratio, (entropy change), of δQ/298 K for the surroundings is smaller than the ratio (entropy change), of δQ/273 K for the ice+water system. To find the entropy change of our "universe", we add up the entropy changes for its constituents: the surrounding room and the ice+water. The total entropy change is positive; this is always true in spontaneous events in a thermodynamic system and it shows the predictive importance of entropy: the final net entropy after such an event is always greater than was the initial entropy.

As the temperature of the cool water rises to that of the room and the room further cools imperceptibly, the sum of the δQ/T over the continuous range, at many increments, in the initially cool to finally warm water can be found by calculus. The entire miniature "universe", i.e. this thermodynamic system, has increased in entropy. Energy has spontaneously become more dispersed and spread out in that "universe" than when the glass of ice water was introduced and became a "system" within it.

Notice that the system will reach a point where the room, the glass and the contents of the glass will be at the same temperature. In this situation, nothing else can happen: although thermal energy does exist in the room (in fact, the amount of thermal energy is the same as in the beginning, since it is a closed system), it is now unable to do useful work, as there is no longer a temperature gradient. Unless an external event intervenes (thus breaking the definition of a closed system), the room is destined to remain in the same condition for all eternity. Therefore, following the same reasoning but considering the whole universe as our "room", we reach a similar conclusion: that, at a certain point in the distant future, the whole universe will be a uniform, isothermic and inert body of matter, in which there will be no available energy to do work. This condition is known as the "heat death of the Universe".

Entropy and adiabatic accessibility[link]

A definition of entropy based entirely on the relation of adiabatic accessibility between equilibrium states was given by E.H.Lieb and J. Yngvason in 1999.^[48] This approach has several predecessors, including the pioneering work of Constantin Carathéodory from 1909 ^[49] and the monograph by R. Giles from 1964.^[50] In the setting of Lieb and Yngvason one starts by picking, for a unit amount of the substance under consideration, two reference states Failed to parse (Missing texvc executable; please see math/README to configure.): X_0

 and Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): X
is defined as the largest number Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda
 such that Failed to parse (Missing texvc executable; please see math/README to configure.): X
is adiabatically accessible from a composite state consisting of an amount Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda
in the state Failed to parse (Missing texvc executable; please see math/README to configure.): X_1
and a complementary amount, Failed to parse (Missing texvc executable; please see math/README to configure.): (1-\lambda)

, in the state Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

Standard textbook definitions[link]

The following is a list of additional definitions of entropy from a collection of textbooks:

• a measure of energy dispersal at a specific temperature.^[30]
• a measure of disorder in the universe or of the availability of the energy in a system to do work.^[51]

Interdisciplinary applications of entropy[link]

Although the concept of entropy was originally a thermodynamic construct, it has been adapted in other fields of study, including information theory, psychodynamics, thermoeconomics, and evolution.^[40][52][53]

Thermodynamic and statistical mechanics concepts[link]

• Entropy unit – a non-S.I. unit of thermodynamic entropy, usually denoted "e.u." and equal to one calorie per Kelvin per mole, or 4.184 Joules per Kelvin per mole.^[54]
• Gibbs entropy – the usual statistical mechanical entropy of a thermodynamic system.
• Boltzmann entropy – a type of Gibbs entropy, which neglects internal statistical correlations in the overall particle distribution.
• Tsallis entropy – a generalization of the standard Boltzmann-Gibbs entropy.
• Standard molar entropy – is the entropy content of one mole of substance, under conditions of standard temperature and pressure.
• Residual entropy – the entropy present after a substance is cooled arbitrarily close to absolute zero.
• Entropy of mixing – the change in the entropy when two different chemical substances or components are mixed.
• Loop entropy – is the entropy lost upon bringing together two residues of a polymer within a prescribed distance.
• Conformational entropy – is the entropy associated with the physical arrangement of a polymer chain that assumes a compact or globular state in solution.
• Entropic force – a microscopic force or reaction tendency related to system organization changes, molecular frictional considerations, and statistical variations.
• Free entropy – an entropic thermodynamic potential analogous to the free energy.
• Entropic explosion – an explosion in which the reactants undergo a large change in volume without releasing a large amount of heat.
• Entropy change – a change in entropy dS between two equilibrium states is given by the heat transferred dQ_rev divided by the absolute temperature T of the system in this interval.^[55]
• Sackur-Tetrode entropy – the entropy of a monatomic classical ideal gas determined via quantum considerations.

Entropy and life[link]

For nearly a century and a half, beginning with Clausius' 1863 memoir "On the Concentration of Rays of Heat and Light, and on the Limits of its Action", much writing and research has been devoted to the relationship between thermodynamic entropy and the evolution of life. The argument that life feeds on negative entropy or negentropy was asserted by physicist Erwin Schrödinger in a 1944 book What is Life?. He posed, "How does the living organism avoid decay?" The obvious answer is: "By eating, drinking, breathing and (in the case of plants) assimilating." Recent writings have used the concept of Gibbs free energy to elaborate on this issue.^[56] While energy from nutrients is necessary to sustain an organism's order, there is also the Schrödinger prescience: "An organism's astonishing gift of concentrating a stream of order on itself and thus escaping the decay into atomic chaos – of drinking orderliness from a suitable environment – seems to be connected with the presence of the aperiodic solids..." We now know that the 'aperiodic' crystal is DNA and that the irregular arrangement is a form of information. "The DNA in the cell nucleus contains the master copy of the software, in duplicate. This software seems to control by "specifying an algorithm, or set of instructions, for creating and maintaining the entire organism containing the cell."^[57] DNA and other macromolecules determine an organism's life cycle: birth, growth, maturity, decline, and death. Nutrition is necessary but not sufficient to account for growth in size as genetics is the governing factor. At some point, organisms normally decline and die even while remaining in environments that contain sufficient nutrients to sustain life. The controlling factor must be internal and not nutrients or sunlight acting as causal exogenous variables. Organisms inherit the ability to create unique and complex biological structures; it is unlikely for those capabilities to be reinvented or be taught each generation. Therefore DNA must be operative as the prime cause in this characteristic as well. Applying Boltzmann's perspective of the second law, the change of state from a more probable, less ordered and high entropy arrangement to one of less probability, more order, and lower entropy seen in biological ordering calls for a function like that known of DNA. DNA's apparent information processing function provides a resolution of the paradox posed by life and the entropy requirement of the second law.^[58]

In 1982, American biochemist Albert Lehninger argued that the "order" produced within cells as they grow and divide is more than compensated for by the "disorder" they create in their surroundings in the course of growth and division. "Living organisms preserve their internal order by taking from their surroundings free energy, in the form of nutrients or sunlight, and returning to their surroundings an equal amount of energy as heat and entropy."^[59]

Evolution-related concepts:

• Negentropy – a shorthand colloquial phrase for negative entropy.^[60]
• Ectropy – a measure of the tendency of a dynamical system to do useful work and grow more organized.^[38]
• Extropy – a metaphorical term defining the extent of a living or organizational system's intelligence, functional order, vitality, energy, life, experience, and capacity and drive for improvement and growth.
• Ecological entropy – a measure of biodiversity in the study of biological ecology.

In a study titled "Natural selection for least action" published in the Proceedings of The Royal Society A., Ville Kaila and Arto Annila of the University of Helsinki describe how the second law of thermodynamics can be written as an equation of motion to describe evolution, showing how natural selection and the principle of least action can be connected by expressing natural selection in terms of chemical thermodynamics. In this view, evolution explores possible paths to level differences in energy densities and so increase entropy most rapidly. Thus, an organism serves as an energy transfer mechanism, and beneficial mutations allow successive organisms to transfer more energy within their environment.^[61]

Cosmology[link]

Since a finite universe is an isolated system, the Second Law of Thermodynamics states that its total entropy is constantly increasing. It has been speculated, since the 19th century, that the universe is fated to a 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 Sir 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. The entropy of a black hole is proportional to the surface area of the black hole's event horizon.^[62] 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. Hawking has, however, recently changed his stance on this aspect.^{[citation needed]}

The role of entropy in cosmology remains a controversial subject. 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.^[63] Other complicating factors, such as the energy density of the vacuum and macroscopic quantum effects, are difficult to reconcile with thermodynamical models, making any predictions of large-scale thermodynamics extremely difficult.^[64]

The entropy gap is widely believed to have been originally opened up by the early rapid exponential expansion of the universe.

Information theory[link]

In information theory, entropy is the measure of the amount of information that is missing before reception and is sometimes referred to as Shannon entropy.^[65] Shannon entropy is a broad and general concept which finds applications in information theory as well as thermodynamics. It was originally devised by Claude Shannon in 1948 to study the amount of information in a transmitted message. The definition of the information entropy is, however, quite general, and is expressed in terms of a discrete set of probabilities Failed to parse (Missing texvc executable; please see math/README to configure.): p_i

Failed to parse (Missing texvc executable; please see math/README to configure.): H(X) = -\sum_{i=1}^n {p(x_i) \log p(x_i)}.

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 amount of information in 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 yes/no questions needed to determine the content of the message.^[22]

The question of the link between information entropy and thermodynamic entropy is a debated topic. While most authors argue that there is a link between the two,^[66][67][68] a few argue that they have nothing to do with each other.^[22][69]

The expressions for the two entropies are similar. The information entropy H for equal probabilities Failed to parse (Missing texvc executable; please see math/README to configure.): p_i = p = 1/n

is

Failed to parse (Missing texvc executable; please see math/README to configure.): H = k\, \log(1/p),

where k is a constant which determines the units of entropy. For example, if the units are bits, then k = 1/ln(2). The thermodynamic entropy S, from a statistical mechanical point of view, was first expressed by Boltzmann:

Failed to parse (Missing texvc executable; please see math/README to configure.): S = k_\mathrm{B} \log(1/p),

where p is the probability of a system's being in a particular microstate, given that it is in a particular macrostate, and Failed to parse (Missing texvc executable; please see math/README to configure.): k_\mathrm{B}

is Boltzmann's constant. It can be seen that one may think of the thermodynamic entropy as Boltzmann's constant, divided by log(2), times the number of yes/no questions that must be asked in order to determine the microstate of the system, given that we know the macrostate. The link between thermodynamic and information entropy was developed in a series of papers by Edwin Jaynes beginning in 1957.[70]

There are many ways of demonstrating the equivalence of "information entropy" and "physics entropy", that is, the equivalence of "Shannon entropy" and "Boltzmann entropy". Nevertheless, some authors argue for dropping the word entropy for the H function of information theory and using Shannon's other term "uncertainty" instead.^[71]

Mathematics[link]

• Kolmogorov-Sinai entropy – a mathematical type of entropy in dynamical systems related to measures of partitions.^[22]
• Relative entropy – is a natural distance measure from a "true" probability distribution P to an arbitrary probability distribution Q.
• Rényi entropy – a generalized entropy measure for fractal systems.
• Topological entropy – a way of defining entropy in an iterated function map in ergodic theory.
• Volume entropy – a Riemannian invariant measuring the exponential rate of volume growth.

Sociology[link]

The concept of entropy has also entered the domain of sociology, generally as a metaphor for chaos, disorder or dissipation of energy, rather than as a direct measure of thermodynamic or information entropy:

• Corporate entropy – energy waste as red tape and business team inefficiency, i.e. energy lost to waste.^[72] (This definition is comparable to von Clausewitz's concept of friction in war.)
• Economic entropy – a semi-quantitative measure of the irrevocable dissipation and degradation of natural materials and available energy with respect to economic activity.^[67][73]
• Entropology – the study or discussion of entropy or the name sometimes given to thermodynamics without differential equations.^[74][75]
• Psychological entropy – the distribution of energy in the psyche, which tends to seek equilibrium or balance among all the structures of the psyche.^[76]
• Social entropy – a measure of social system structure, having both theoretical and statistical interpretations, i.e. society (macrosocietal variables) measured in terms of how the individual functions in society (microsocietal variables); also related to social equilibrium.^[77]

See also[link]

Thermodynamics
Branches Classical · Statistical · Chemical Equilibrium / Non-equilibrium Thermofluids
Laws Zeroth · First · Second · Third
Systems State: Equation of state Ideal gas · Real gas Phase of matter · Equilibrium Control volume · Instruments Processes: Isobaric · Isochoric · Isothermal Adiabatic · Isentropic · Isenthalpic Quasistatic · Polytropic Free expansion Reversibility · Irreversibility Endoreversibility Cycles: Heat engines · Heat pumps Thermal efficiency
System properties Property diagrams Intensive and extensive properties State functions: Temperature / Entropy (intro.) † Pressure / Volume † Chemical potential / Particle no. † († Conjugate variables) Vapor quality Reduced properties Process functions: Work · Heat
Material properties Specific heat capacity  Failed to parse (Missing texvc executable; please see math/README to configure.): c= Failed to parse (Missing texvc executable; please see math/README to configure.): T Failed to parse (Missing texvc executable; please see math/README to configure.): \partial S Failed to parse (Missing texvc executable; please see math/README to configure.): N Failed to parse (Missing texvc executable; please see math/README to configure.): \partial T Compressibility  Failed to parse (Missing texvc executable; please see math/README to configure.): \beta=- Failed to parse (Missing texvc executable; please see math/README to configure.): 1 Failed to parse (Missing texvc executable; please see math/README to configure.): \partial V Failed to parse (Missing texvc executable; please see math/README to configure.): V Failed to parse (Missing texvc executable; please see math/README to configure.): \partial p Thermal expansion  Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha= Failed to parse (Missing texvc executable; please see math/README to configure.): 1 Failed to parse (Missing texvc executable; please see math/README to configure.): \partial V Failed to parse (Missing texvc executable; please see math/README to configure.): V Failed to parse (Missing texvc executable; please see math/README to configure.): \partial T Property database
Equations Carnot's theorem Clausius theorem Fundamental relation Ideal gas law Maxwell relations Table of thermodynamic equations
Potentials Free energy · Free entropy Internal energy Failed to parse (Missing texvc executable; please see math/README to configure.): U(S,V) Enthalpy Failed to parse (Missing texvc executable; please see math/README to configure.): H(S,p)=U+pV Helmholtz free energy Failed to parse (Missing texvc executable; please see math/README to configure.): A(T,V)=U-TS Gibbs free energy Failed to parse (Missing texvc executable; please see math/README to configure.): G(T,p)=H-TS
History and culture Philosophy: Entropy and time · Entropy and life Brownian ratchet Maxwell's demon Heat death paradox Loschmidt's paradox Synergetics History: General · Heat · Entropy · Gas laws Perpetual motion Theories: Caloric theory · Vis viva Theory of heat Mechanical equivalent of heat Motive power Publications: "An Experimental Enquiry Concerning ... Heat" "On the Equilibrium of Heterogeneous Substances" "Reflections on the Motive Power of Fire" Timelines of: Thermodynamics · Heat engines Art: Maxwell's thermodynamic surface Education: Entropy as energy dispersal
Scientists Daniel Bernoulli Sadi Carnot Benoît Paul Émile Clapeyron Rudolf Clausius Hermann von Helmholtz Constantin Carathéodory Pierre Duhem Josiah Willard Gibbs James Prescott Joule James Clerk Maxwell Julius Robert von Mayer William Rankine John Smeaton Georg Ernst Stahl Benjamin Thompson William Thomson, 1st Baron Kelvin John James Waterston
v t e

Notes[link]

References[link]