Entropy is a
thermodynamic property that can be used to determine the energy available for useful 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
dissipatesIn physics, dissipation embodies the concept of a dynamical system where important mechanical models, such as waves or oscillations, lose energy over time, typically from friction or turbulence. The lost energy converts into heat, which raises the temperature of the system. Such systems are called...
in the form of waste
heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
.
In classical thermodynamics, the concept of entropy is defined
phenomenologicallyThe term phenomenology in science is used to describe a body of knowledge that relates empirical observations of phenomena to each other, in a way that is consistent with fundamental theory, but is not directly derived from theory. For example, we find the following definition in the Concise...
by the
second law of thermodynamicsThe second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
, 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 energyThermal energy is the part of the total internal energy of a thermodynamic system or sample of matter that results in the system's temperature....
always flows spontaneously from regions of higher temperature to regions of lower temperature, in the form of
heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
. These processes reduce the state of order of the initial systems, and therefore entropy is an expression of disorder or randomness. This picture is the basis of the modern microscopic interpretation of entropy in
statistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, 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
energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
divided by
temperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
, and a unit of
jouleThe joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...
s per
kelvinThe kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
(J/K) in the
International System of UnitsThe International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...
.
The term
entropy was coined in 1865 by
Rudolf ClausiusRudolf Julius Emanuel Clausius , was a German physicist and mathematician and is considered one of the central founders of the science of thermodynamics. By his restatement of Sadi Carnot's principle known as the Carnot cycle, he put the theory of heat on a truer and sounder basis...
based on the Greek εντροπία [entropía],
a turning toward, from εν [en] (in) and τροπή [tropē] (turn, conversion).
[A machine in this context includes engineered devices as well as biological organisms.]
Thermodynamical and statistical descriptions
There are two related definitions of entropy: the
thermodynamicThermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
definition and the
statistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
definition. The thermodynamic definition was developed in the early 1850s by
Rudolf ClausiusRudolf Julius Emanuel Clausius , was a German physicist and mathematician and is considered one of the central founders of the science of thermodynamics. By his restatement of Sadi Carnot's principle known as the Carnot cycle, he put the theory of heat on a truer and sounder basis...
and essentially describes how to measure the entropy of an
isolated systemIn the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy  mass stays constant...
in
thermodynamic equilibriumIn thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...
. Importantly, it makes no reference to the microscopic nature of matter. The statistical definition was developed by
Ludwig BoltzmannLudwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...
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 nonconserved
state functionIn thermodynamics, a state function, function of state, state quantity, or state variable is a property of a system that depends only on the current state of the system, not on the way in which the system acquired that state . A state function describes the equilibrium state of a system...
that is of great importance in the sciences of
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and
chemistryChemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
. Historically, the concept of entropy evolved in order to explain why some processes (permitted by conservation laws) occur spontaneously while their
time reversalTime reversal may refer to:* In physics, Tsymmetry  the study of thermodynamics and the symmetry of certain physical laws where the concept of time is reversed — ie...
s (also permitted by conservation laws) do not; systems tend to progress in the direction of increasing entropy. For
isolated systemIn the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy  mass stays constant...
s, entropy never decreases. This fact has several important consequences in
scienceScience is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...
: first, it prohibits "
perpetual motionPerpetual motion describes hypothetical machines that operate or produce useful work indefinitely and, more generally, hypothetical machines that produce more work or energy than they consume, whether they might operate indefinitely or not....
" machines; and second, it implies the
arrow of entropyEntropy is the only quantity in the physical sciences that requires a particular direction for time, sometimes called an arrow of time. As one goes "forward" in time, the second law of thermodynamics says, the entropy of an isolated system will increase...
has the same directionality as the
arrow of timeThe arrow of time, or time’s arrow, is a term coined in 1927 by the British astronomer Arthur Eddington to describe the "oneway direction" or "asymmetry" 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.
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). This definition describes the entropy as being proportional to the logarithm of the number of possible microscopic configurations of the individual atoms and molecules of the system (
microstatesIn statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations...
) which could give rise to the observed macroscopic state (macrostate) of the system. The constant of proportionality is the
Boltzmann constant.
The second law of thermodynamics
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 motionPerpetual motion describes hypothetical machines that operate or produce useful work indefinitely and, more generally, hypothetical machines that produce more work or energy than they consume, whether they might operate indefinitely or not....
system. It follows that a reduction in the increase of entropy in a specified process, such as a
chemical reactionA chemical reaction is a process that leads to the transformation of one set of chemical substances to another. Chemical reactions can be either spontaneous, requiring no input of energy, or nonspontaneous, typically following the input of some type of energy, such as heat, light or electricity...
, 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 workIn thermodynamics, work performed by a system is the energy transferred to another system that is measured by the external generalized mechanical constraints on the system. As such, thermodynamic work is a generalization of the concept of mechanical work in mechanics. Thermodynamic work encompasses...
. The entropy change of a system at temperature T absorbing an infinitesimal amount of heat
in a reversible way, is given by
. More explicitly, an energy
T_{R}S is not available to do useful work, where
T_{R} is the temperature of the coldest accessible reservoir or heat sink external to the system. For further discussion, see
ExergyIn thermodynamics, the exergy of a system is the maximum useful work possible during a process that brings the system into equilibrium with a heat reservoir. When the surroundings are the reservoir, exergy is the potential of a system to cause a change as it achieves equilibrium with its...
.
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.
Definitions and descriptions
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
nonequilibrium thermodynamicsNonequilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium; for they are changing or can be triggered to change over time, and are continuously and discontinuously...
.
Carnot cycle
The concept of entropy arose from
Rudolf ClausiusRudolf Julius Emanuel Clausius , was a German physicist and mathematician and is considered one of the central founders of the science of thermodynamics. By his restatement of Sadi Carnot's principle known as the Carnot cycle, he put the theory of heat on a truer and sounder basis...
's study of the
Carnot cycleThe Carnot cycle is a theoretical thermodynamic cycle proposed by Nicolas Léonard Sadi Carnot in 1824 and expanded by Benoit Paul Émile Clapeyron in the 1830s and 40s. It can be shown that it is the most efficient cycle for converting a given amount of thermal energy into work, or conversely,...
. In a Carnot cycle, heat (
) is absorbed from a 'hot' reservoir, isothermally at the higher temperature
, and given up isothermally to a 'cold' reservoir,
, at a lower temperature,
. According to Carnot's principle, work can only be done when there is a drop in the temperature, and the work should be some function of the difference in temperature and the heat absorbed. Carnot did not distinguish between
and
, since he was working under the hypothesis that
caloric theoryThe caloric theory is an obsolete scientific theory that heat consists of a selfrepellent fluid called caloric that flows from hotter bodies to colder bodies. Caloric was also thought of as a weightless gas that could pass in and out of pores in solids and liquids...
was valid, and hence heat was conserved. 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:
In order to derive the Carnot efficiency,
Kelvin had to evaluate the ratio of the work done to the heat absorbed in the isothermal expansion with the help of the CarnotClapeyron 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
JouleJames Prescott Joule FRS was an English physicist and brewer, born in Salford, Lancashire. Joule studied the nature of heat, and discovered its relationship to mechanical work . This led to the theory of conservation of energy, which led to the development of the first law of thermodynamics. The...
in a letter to Kelvin, and this allowed Kelvin to establish his absolute temperature scale.
It is also known that the work is the difference in the heat absorbed at the hot reservoir and rejected at the cold one:
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 thermodynamicsThe first law of thermodynamics is an expression of the principle of conservation of work.The law states that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed...
.
Now equating the two expressions gives
If we allow
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 thermodynamicsThe second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
.
Then Clausius asked what would happen if there would be less work done than that predicted by Carnot's principle. The righthand side of the first equation would be the upper bound of the work, which would now be converted into an inequality
When the second equation is used to express the work as a difference in heats, we get
or
So more heat is given off to the cold reservoir than in the Carnot cycle. If we denote the entropies by
for the two states, then the above inequality can be written as a decrease in the entropy
The wasted heat implies that irreversible processes must have prevented the cycle from carrying out maximum work.
Statistical thermodynamics
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
microstatesIn statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations...
. 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
logarithmicA logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...
measure of the
density of statesIn solidstate and condensed matter physics, the density of states of a system describes the number of states per interval of energy at each energy level that are available to be occupied by electrons. Unlike isolated systems, like atoms or molecules in gas phase, the density distributions are not...
:

where
k_{B} is the
Boltzmann constant, equal to .
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.
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. Then the previous equation reduces to:

In thermodynamics, such a system is one in which the volume, number of molecules, and internal energy are fixed (the
microcanonical ensembleIn statistical physics, the microcanonical ensemble is a theoretical tool used to describe the thermodynamic properties of an isolated system. In such a system, the possible macrostates of the system all have the same energy and the probability for the system to be in any given microstate is the same...
).
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. 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!
Entropy can be defined for any
Markov processIn probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a timevarying random phenomenon for which a specific property holds...
es with
reversible dynamics Mathematics :In mathematics, a dynamical system is invertible if the forward evolution is onetoone, not manytoone; so that for every state there exists a welldefined reversetime evolution operator....
and the
detailed balanceThe principle of detailed balance is formulated for kinetic systems which are decomposed into elementary processes : At equilibrium, each elementary process should be equilibrated by its reverse process....
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
According to the
Clausius equality, for a reversible process:
That means the line integral
is path independent.
So we can define a state function S called entropy, which satisfies:
With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need
Third Law of ThermodynamicsThe third law of thermodynamics is a statistical law of nature regarding entropy:For other materials, the residual entropy is not necessarily zero, although it is always zero for a perfect crystal in which there is only one possible ground state.History:...
, which states that S=0 at
absolute zeroAbsolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....
for perfect crystals.
From a macroscopic perspective, in classical thermodynamics the entropy is interpreted as a
state functionIn thermodynamics, a state function, function of state, state quantity, or state variable is a property of a system that depends only on the current state of the system, not on the way in which the system acquired that state . A state function describes the equilibrium state of a system...
of a
thermodynamic systemA thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....
: 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
energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
in a physical system that cannot be used to do
thermodynamic workIn thermodynamics, work performed by a system is the energy transferred to another system that is measured by the external generalized mechanical constraints on the system. As such, thermodynamic work is a generalization of the concept of mechanical work in mechanics. Thermodynamic work encompasses...
; i.e., work mediated by thermal energy. More precisely, in any process where the system gives up energy Δ
E, and its entropy falls by Δ
S, a quantity at least
T_{R} Δ
S of that energy must be given up to the system's surroundings as unusable
heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
(
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 equilibriumIn thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...
.
In a
thermodynamic systemA thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....
, pressure, density, and temperature tend to become uniform over time because this equilibrium state has higher
probabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
(more possible
combinationIn mathematics a combination is a way of selecting several things out of a larger group, where order does not matter. In smaller cases it is possible to count the number of combinations...
s of
microstatesIn statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations...
) than any other; see
statistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
. 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.
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 systemIn the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy  mass stays constant...
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 systemA thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....
is a measure of how far the equalization has progressed.
A special case of entropy increase, the
entropy of mixingIn thermodynamics the entropy of mixing is the increase in the total entropy of a compound system, when different and chemically nonreacting chemical substances or material components are mixed by removing partition between the system's initially separate volumes...
, 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.
History
The
first law of thermodynamicsThe first law of thermodynamics is an expression of the principle of conservation of work.The law states that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed...
, formalized based on the heatfriction experiments of James Joule in 1843, deals with the concept of energy, which is
conservedThe nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
in all processes; the first law, however, is unable to quantify the effects of
frictionFriction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
and
dissipationIn physics, dissipation embodies the concept of a dynamical system where important mechanical models, such as waves or oscillations, lose energy over time, typically from friction or turbulence. The lost energy converts into heat, which raises the temperature of the system. Such systems are called...
.
The analysis which led to the concept of entropy began with the work of French mathematician
Lazare CarnotLazare Nicolas Marguerite, Comte Carnot , the Organizer of Victory in the French Revolutionary Wars, was a French politician, engineer, and mathematician.Education and early life:...
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 CarnotNicolas Léonard Sadi Carnot was a French military engineer who, in his 1824 Reflections on the Motive Power of Fire, gave the first successful theoretical account of heat engines, now known as the Carnot cycle, thereby laying the foundations of the second law of thermodynamics...
published
Reflections on the Motive Power of Fire which posited that in all heatengines whenever "
caloricThe caloric theory is an obsolete scientific theory that heat consists of a selfrepellent fluid called caloric that flows from hotter bodies to colder bodies. Caloric was also thought of as a weightless gas that could pass in and out of pores in solids and liquids...
", or what is now known as
heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
, falls through a temperature difference, work or
motive powerIn thermodynamics, motive power is an agency, as water or steam, used to impart motion. Generally, motive power is defined as a natural agent, as water, steam, wind, electricity, etc., used to impart motion to machinery; a motor; a mover. The term may also define something, as a locomotive or a...
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 thermodynamicsThe second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
.
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. 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.
In the 1850s and 1860s, German physicist
Rudolf ClausiusRudolf Julius Emanuel Clausius , was a German physicist and mathematician and is considered one of the central founders of the science of thermodynamics. By his restatement of Sadi Carnot's principle known as the Carnot cycle, he put the theory of heat on a truer and sounder basis...
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. Clausius described entropy as the
transformationcontent, i.e. dissipative
energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
use, of a
thermodynamic systemA thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....
or working body of
chemical speciesChemical species are atoms, molecules, molecular fragments, ions, etc., being subjected to a chemical process or to a measurement. Generally, a chemical species can be defined as an ensemble of chemically identical molecular entities that can explore the same set of molecular energy levels on a...
during a change of
stateA thermodynamic state is a set of values of properties of a thermodynamic system that must be specified to reproduce the system. The individual parameters are known as state variables, state parameters or thermodynamic variables. Once a sufficient set of thermodynamic variables have been...
. This was in contrast to earlier views, based on the theories of
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
, that heat was an indestructible particle that had mass.
Later, scientists such as
Ludwig BoltzmannLudwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...
,
Josiah Willard GibbsJosiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...
, and
James Clerk MaxwellJames Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
gave entropy a statistical basis. In 1877 Boltzmann visualized a probabilistic way to measure the entropy of an ensemble of
ideal gasAn ideal gas is a theoretical gas composed of a set of randomlymoving, noninteracting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
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 SchrödingerErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...
, 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.
The arrow of time
Entropy is the only quantity in the physical sciences that seems to imply a particular direction of progress, sometimes called an
arrow of timeThe arrow of time, or time’s arrow, is a term coined in 1927 by the British astronomer Arthur Eddington to describe the "oneway direction" or "asymmetry" of time...
. As time progresses, the second law of thermodynamics states that the entropy of an
isolated systemIn the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy  mass stays constant...
never decreases. Hence, from this perspective, entropy measurement is thought of as a kind of clock.
The fundamental thermodynamic relation
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:
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 nonquasistatic 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 relationsMaxwell's relations are a set of equations in thermodynamics which are derivable from the definitions of the thermodynamic potentials. The Maxwell relations are statements of equality among the second derivatives of the thermodynamic potentials. They follow directly from the fact that the order of...
and the
relations between heat capacities.
Entropy in chemical thermodynamics
Thermodynamic entropy is central in
chemical thermodynamicsChemical thermodynamics is the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics...
, enabling changes to be quantified and the outcome of reactions predicted. The
second law of thermodynamicsThe second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
states that entropy in an
isolated systemIn the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy  mass stays constant...
—the combination of a subsystem under study and its surroundings— increases during all spontaneous chemical and physical processes. The Clausius equation of δ
q_{rev}/
T = Δ
S introduces the measurement of entropy change, Δ
S. Entropy change describes the direction and quantifies the magnitude of simple changes such as
heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
transfer between systems —always from hotter to cooler spontaneously.
The thermodynamic entropy therefore has the dimension of energy divided by temperature, and the unit
jouleThe joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...
per
kelvinThe kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
(J/K) in the International System of Units (SI).
Thermodynamic entropy is an
extensiveIn the physical sciences, an intensive property , is a physical property of a system that does not depend on the system size or the amount of material in the system: it is scale invariant.By contrast, an extensive property In the physical sciences, an intensive property (also called a bulk...
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 propertyIn the physical sciences, an intensive property , is a physical property of a system that does not depend on the system size or the amount of material in the system: it is scale invariant.By contrast, an extensive property In the physical sciences, an intensive property (also called a bulk...
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
kilogramThe kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water...
(unit: ). Alternatively, in chemistry, it is also referred to one
moleThe mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon12 , the isotope of carbon with atomic weight 12. This corresponds to a value...
of substance, in which case it is called the
molar entropy with a unit of .
Thus, when one mole of substance at is warmed by its surroundings to , 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 . Entropy change also measures the mixing of substances as a summation of their relative quantities in the final mixture.
Entropy is equally essential in predicting the extent and direction of complex chemical reactions. For such applications, Δ
S must be incorporated in an expression that includes both the system and its surroundings, Δ
S_{universe} = Δ
S_{surroundings} + Δ
S _{system}. This expression becomes, via some steps, the
Gibbs free energyIn thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or processinitiating work obtainable from a thermodynamic system at a constant temperature and pressure...
equation for reactants and products in the system: Δ
G [the Gibbs free energy change of the system] = Δ
H [the enthalpy change] −
T Δ
S [the entropy change].
Entropy change
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.
When specific heat and volume are constant, the change in entropy is given by:
.
When specific heat and pressure are constant, the change in entropy is given by:
.
When specific heat and temperature are constant, the change in entropy is given by:
.
In these equations
is the specific heat at constant volume,
is the specific heat at constant pressure,
is the ideal gas constant, and
is the number of
molesThe mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon12 , the isotope of carbon with atomic weight 12. This corresponds to a value...
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 either:
or
.
Entropy balance equation for open systems
In
chemical engineeringChemical engineering is the branch of engineering that deals with physical science , and life sciences with mathematics and economics, to the process of converting raw materials or chemicals into more useful or valuable forms...
, the principles of thermodynamics are commonly applied to "open systems", i.e. those in which
heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
,
workIn thermodynamics, work performed by a system is the energy transferred to another system that is measured by the external generalized mechanical constraints on the system. As such, thermodynamic work is a generalization of the concept of mechanical work in mechanics. Thermodynamic work encompasses...
, and
massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
flow across the system boundary. In a system in which there are flows of both heat (
) and work, i.e.
(shaft work) and
P(dV/dt) (pressurevolume work), across the system boundaries, the heat flow, but not the work flow, causes a change in the entropy of the system. This rate of entropy change is
where
T is the absolute
thermodynamic temperatureThermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...
of the system at the point of the heat flow. If, in addition, there are mass flows across the system boundaries, the total entropy of the system will also change due to this convected flow.
To derive a generalized entropy balanced equation, we start with the general balance equation for the change in any extensive quantity Θ in a
thermodynamic systemA thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....
, a quantity that may be either conserved, such as energy, or nonconserved, such as entropy. The basic generic balance expression states that dΘ/dt, i.e. the rate of change of Θ in the system, equals the rate at which Θ enters the system at the boundaries, minus the rate at which Θ leaves the system across the system boundaries, plus the rate at which Θ is generated within the system. Using this generic balance equation, with respect to the rate of change with time of the extensive quantity entropy
S, the
entropy balance equation for an open thermodynamic system is:
where
= the net rate of entropy flow due to the flows of mass into and out of the system (where
= entropy per unit mass).
= the rate of entropy flow due to the flow of heat across the system boundary.
= the rate of internal generation of entropy within the system.
Note, also, that if there are multiple heat flows, the term
is to be replaced by
where
is the heat flow and
is the temperature at the
jth heat flow port into the system.
Entropy in quantum mechanics (von Neumann entropy)
In
quantum statistical mechanicsQuantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. A statistical ensemble is described by a density operator S, which is a nonnegative, selfadjoint, traceclass operator of trace 1 on the Hilbert space H describing the quantum system. This can be...
, the concept of entropy was developed by
John von NeumannJohn von Neumann was a HungarianAmerican mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
and is generally referred to as "
von Neumann entropyIn quantum statistical mechanics, von Neumann entropy, named after John von Neumann, is the extension of classical entropy concepts to the field of quantum mechanics....
", namely
where
is the
density matrixIn quantum mechanics, a density matrix is a selfadjoint positivesemidefinite matrix of trace one, that describes the statistical state of a quantum system...
and Tr is the
traceIn linear algebra, the trace of an nbyn square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
operator.
This upholds the
correspondence principleIn physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....
, because in the
classical limitThe classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict nonclassical behavior...
, i.e. whenever the classical notion of probability applies, this expression is equivalent to the familiar classical definition of entropy,
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 matrixIn quantum mechanics, a density matrix is a selfadjoint positivesemidefinite matrix of trace one, that describes the statistical state of a quantum system...
he extended the classical concept of entropy into the quantum domain.
Order and disorder
Entropy has often been loosely associated with the amount of order,
disorderRandomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....
, and/or
chaosChaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
in a
thermodynamic systemA thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....
. The traditional qualitative description of entropy is that it refers to changes in the status quo of the system and is a measure of "molecular disorder" and the amount of wasted energy in a dynamical energy transformation from one state or form to another. In this direction, several recent authors have derived exact entropy formulas to account for and measure disorder and order in atomic and molecular assemblies. One of the simpler entropy order/disorder formulas is that derived in 1984 by thermodynamic physicist Peter Landsberg, based on a combination of
thermodynamicsThermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
and
information theoryInformation theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
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:
Similarly, the total amount of "order" in the system is given by:
In which
C_{D} is the "disorder" capacity of the system, which is the entropy of the parts contained in the permitted ensemble,
C_{I} is the "information" capacity of the system, an expression similar to Shannon's
channel capacityIn electrical engineering, computer science and information theory, channel capacity is the tightest upper bound on the amount of information that can be reliably transmitted over a communications channel...
, and
C_{O} is the "order" capacity of the system.
Energy dispersal
The concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature. Similar terms have been in use from early in the history of classical thermodynamics, and with the development of statistical thermodynamics and
quantum theoryQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, entropy changes have been described in terms of the mixing or "spreading" of the total energy of each constituent of a system over its particular quantized energy levels.
Ambiguities in the terms
disorder and
chaos, which usually have meanings directly opposed to equilibrium, contribute to widespread confusion and hamper comprehension of entropy for most students. As the
second law of thermodynamicsThe second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
shows, in an
isolated systemIn the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy  mass stays constant...
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 thermodynamicsThe first law of thermodynamics is an expression of the principle of conservation of work.The law states that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed...
(compare discussion in next section). Physical chemist
Peter AtkinsPeter William Atkins is a British chemist and former Professor of Chemistry at the University of Oxford and a Fellow of Lincoln College. He is a prolific writer of popular chemistry textbooks, including Physical Chemistry, Inorganic Chemistry, and Molecular Quantum Mechanics...
, 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".
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. 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 UniverseThe heat death of the universe is a suggested ultimate fate of the universe, in which the universe has diminished to a state of no thermodynamic free energy and therefore can no longer sustain motion or life. Heat death does not imply any particular absolute temperature; it only requires that...
".
Ice melting example
The illustration for this article is a classic example in which entropy increases in a small "universe", a
thermodynamic systemA thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....
consisting of the "surroundings" (the warm room) and "system" (glass, ice, cold water). In this universe, some
thermal energyThermal energy is the part of the total internal energy of a thermodynamic system or sample of matter that results in the system's temperature....
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 fusionThe enthalpy of fusion is the change in enthalpy resulting from heating one mole of a substance to change its state from a solid to a liquid. The temperature at which this occurs is the melting point....
, i.e. the Δ
H 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 systemA thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....
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
workIn thermodynamics, work performed by a system is the energy transferred to another system that is measured by the external generalized mechanical constraints on the system. As such, thermodynamic work is a generalization of the concept of mechanical work in mechanics. Thermodynamic work encompasses...
, 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,
isothermicAn isothermal process is a change of a system, in which the temperature remains constant: ΔT = 0. This typically occurs when a system is in contact with an outside thermal reservoir , and the change occurs slowly enough to allow the system to continually adjust to the temperature of the reservoir...
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 UniverseThe heat death of the universe is a suggested ultimate fate of the universe, in which the universe has diminished to a state of no thermodynamic free energy and therefore can no longer sustain motion or life. Heat death does not imply any particular absolute temperature; it only requires that...
".
Entropy and adiabatic accessibility
A definition of entropy based entirely on the relation of
adiabatic accessibilityAdiabatic accessibility denotes a certain relation between two equilibrium states of a thermodynamic system . The concept was coined by Constantin Carathéodory in 1909 and taken up 90 years later by Elliott Lieb and J. Yngvason in their axiomatic approach to the foundations of thermodynamics . It...
between equilibrium states was given by
E.H.LiebElliott H. Lieb is an eminent American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis....
and
J. YngvasonJakob Yngvason is an Icelandic/Austrian physicist. He is professor of mathematical physics at the University of Vienna and Scientific Director of the Erwin Schrödinger International Institute for Mathematical Physics in Vienna...
in 1999. This approach has several predecessors, including the pioneering work of
Constantin CarathéodoryConstantin Carathéodory was a Greek mathematician. He made significant contributions to the theory of functions of a real variable, the calculus of variations, and measure theory...
from 1909 and the monograph by R. Giles from 1964. In the setting of Lieb and Yngvason one starts by picking, for a unit amount of the substance under consideration, two reference states
and
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
is defined as the largest number
such that
is adiabatically accessible from a composite state consisting of an amount
in the state
and a complementary amount,
, in the state
. 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
The following is a list of additional definitions of entropy from a collection of textbooks:
 a measure of energy dispersal at a specific temperature.
 a measure of disorder in the universe or of the availability of the energy in a system to do work.
Interdisciplinary applications of entropy
Although the concept of entropy was originally a thermodynamic construct, it has been adapted in other fields of study, including
information theoryInformation theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
,
psychodynamicsPsychodynamics is the theory and systematic study of the psychological forces that underlie human behavior, especially the dynamic relations between conscious motivation and unconscious motivation...
,
thermoeconomicsThermoeconomics, also referred to as biophysical economics, is a school of heterodox economics that applies the laws of thermodynamics to economic theory. The term "thermoeconomics" was coined in 1962 by American engineer Myron Tribus, and developed by the statistician and economist Nicholas...
, and
evolutionEvolution is any change across successive generations in the heritable characteristics of biological populations. Evolutionary processes give rise to diversity at every level of biological organisation, including species, individual organisms and molecules such as DNA and proteins.Life on Earth...
.
Thermodynamic and statistical mechanics concepts
 Entropy unit – a nonS.I. unit of thermodynamic entropy, usually denoted "e.u." and equal to one calorie
The calorie is a preSI metric unit of energy. It was first defined by Nicolas Clément in 1824 as a unit of heat, entering French and English dictionaries between 1841 and 1867. In most fields its use is archaic, having been replaced by the SI unit of energy, the joule...
per Kelvin per mole, or 4.184 Joules per Kelvin per mole.
 Gibbs entropy – the usual statistical mechanical entropy of a thermodynamic system.
 Boltzmann entropy
In thermodynamics, specifically in statistical mechanics, the Boltzmann entropy is an approximation to the normal Gibbs entropy.The Boltzmann entropy is obtained if one assumes one can treat all the component particles of a thermodynamic system as statistically independent...
– a type of Gibbs entropy, which neglects internal statistical correlations in the overall particle distribution.
 Tsallis entropy
In physics, the Tsallis entropy is a generalization of the standard BoltzmannGibbs entropy. In the scientific literature, the physical relevance of the Tsallis entropy is highly debated...
– a generalization of the standard BoltzmannGibbs entropy.
 Standard molar entropy
In chemistry, the standard molar entropy is the entropy content of one mole of substance, under standard conditions ....
– is the entropy content of one mole of substance, under conditions of standard temperature and pressure.
 Residual entropy
Residual entropy is small amount of entropy which is present even after a substance is cooled arbitrarily close to absolute zero. It occurs if a material can exist in many different microscopic states when cooled to absolute zero...
– the entropy present after a substance is cooled arbitrarily close to absolute zeroAbsolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....
.
 Entropy of mixing
In thermodynamics the entropy of mixing is the increase in the total entropy of a compound system, when different and chemically nonreacting chemical substances or material components are mixed by removing partition between the system's initially separate volumes...
– the change in the entropy when two different chemical substanceIn chemistry, a chemical substance is a form of matter that has constant chemical composition and characteristic properties. It cannot be separated into components by physical separation methods, i.e. without breaking chemical bonds. They can be solids, liquids or gases.Chemical substances are...
s or componentsIn thermodynamics, a component is a chemically distinct constituent ofa system. Calculating the number of components in a system is necessary, for example, when applying Gibbs phase rule in determination of the number of degrees of freedom of a system....
are mixed.
 Loop entropy – is the entropy lost upon bringing together two residues of a polymer within a prescribed distance.
 Conformational entropy
Conformational entropy is the entropy associated with the physical arrangement of a polymer chain that assumes a compact or globular state in solution. The concept is most commonly applied to biological macromolecules such as proteins and RNA, but can also be used for polysaccharides and other...
– is the entropy associated with the physical arrangement of a polymerA polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
chain that assumes a compact or globularGlobular proteins, or spheroproteins are one of the two main protein classes, comprising "globe"like proteins that are more or less soluble in aqueous solutions...
state in solution.
 Entropic force
In physics, an entropic force acting in a system is a phenomenological force resulting from the entire system's statistical tendency to increase its entropy, rather than from a particular underlying microscopic force.Polymers:...
– a microscopic force or reaction tendency related to system organization changes, molecular frictional considerations, and statistical variations.
 Free entropy
A thermodynamic free entropy is an entropic thermodynamic potential analogous to the free energy. Also known as a Massieu, Planck, or MassieuPlanck potentials , or free information. In statistical mechanics, free entropies frequently appear as the logarithm of a partition function...
– an entropic thermodynamic potential analogous to the free energy.
 Entropic explosion
An entropic explosion is an explosion in which the reactants undergo a large change in volume without releasing a large amount of heat. The chemical decomposition of triacetone triperoxide or TATP is an example of an entropic explosion. It is not a thermochemically highly favored event...
– 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
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
transferred dQ_{rev} divided by the absolute temperature T of the systemA thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....
in this interval.
 SackurTetrode entropy – the entropy of a monatomic classical ideal gas determined via quantum considerations.
Entropy and life
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
evolutionEvolution is any change across successive generations in the heritable characteristics of biological populations. Evolutionary processes give rise to diversity at every level of biological organisation, including species, individual organisms and molecules such as DNA and proteins.Life on Earth...
of
lifeLife is a characteristic that distinguishes objects that have signaling and selfsustaining processes from those that do not, either because such functions have ceased , or else because they lack such functions and are classified as inanimate...
. The argument that life feeds on negative entropy or
negentropyThe negentropy, also negative entropy or syntropy, of a living system is the entropy that it exports to keep its own entropy low; it lies at the intersection of entropy and life...
as asserted in the 1944 book
What is Life?What Is Life? is a 1944 nonfiction science book written for the lay reader by physicist Erwin Schrödinger. The book was based on a course of public lectures delivered by Schrödinger in February 1943, under the auspices of the Dublin Institute for Advanced Studies at Trinity College, Dublin...
by
physicistA physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from subatomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
Erwin SchrödingerErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...
served as a further stimulus to this research. Recent writings have used the concept of
Gibbs free energyIn thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or processinitiating work obtainable from a thermodynamic system at a constant temperature and pressure...
to elaborate on this issue.
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 energyThe thermodynamic free energy is the amount of work that a thermodynamic system can perform. The concept is useful in the thermodynamics of chemical or thermal processes in engineering and science. The free energy is the internal energy of a system less the amount of energy that cannot be used to...
, in the form of nutrients or sunlight, and returning to their surroundings an equal amount of energy as
heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
and entropy."
Evolutionrelated concepts:
 Negentropy
The negentropy, also negative entropy or syntropy, of a living system is the entropy that it exports to keep its own entropy low; it lies at the intersection of entropy and life...
– a shorthand colloquial phrase for negative entropy.
 Ectropy
In thermodynamics, ectropy is a measure of the tendency of a dynamical system to do useful work and grow more organized. Ectropy, in a loose sense, can be thought of as the opposite of entropy. Ectropy is minus entropy...
– a measure of the tendency of a dynamical system to do useful work and grow more organized.
 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
Biodiversity is the degree of variation of life forms within a given ecosystem, biome, or an entire planet. Biodiversity is a measure of the health of ecosystems. Biodiversity is in part a function of climate. In terrestrial habitats, tropical regions are typically rich whereas polar regions...
in the study of biological ecologyEcology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...
.
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 HelsinkiThe University of Helsinki is a university located in Helsinki, Finland since 1829, but was founded in the city of Turku in 1640 as The Royal Academy of Turku, at that time part of the Swedish Empire. It is the oldest and largest university in Finland with the widest range of disciplines available...
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.
Cosmology
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
energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
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 holeA black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
s. The entropy of a black hole is proportional to the surface area of the black hole's event horizon.
Jacob BekensteinJacob David Bekenstein is an Israeli theoretical physicist who has contributed to the foundation of black hole thermodynamics and to other aspects of the connections between information and gravitation.Biography:...
and
Stephen HawkingStephen William Hawking, CH, CBE, FRS, FRSA is an English theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity...
have shown that black holes have the maximum possible entropy of any object of equal size. This makes them likely end points of all entropyincreasing processes, if they are totally effective matter and energy traps. Hawking has, however, recently changed his stance on this aspect.
The role of entropy in cosmology remains a controversial subject. Recent work has cast some doubt on the heat death hypothesis and the applicability of any simple thermodynamic model to the universe in general. Although entropy does increase in the model of an expanding universe, the maximum possible entropy rises much more rapidly, moving the universe further from the heat death with time, not closer. This results in an "entropy gap" pushing the system further away from the posited heat death equilibrium. Other complicating factors, such as the energy density of the vacuum and macroscopic
quantumQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
effects, are difficult to reconcile with thermodynamical models, making any predictions of largescale thermodynamics extremely difficult.
The entropy gap is widely believed to have been originally opened up by the early rapid exponential expansion of the universe.
Information theory
In
information theoryInformation theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
,
entropy is the measure of the amount of information that is missing before reception and is sometimes referred to as
Shannon entropy. Shannon entropy is a broad and general concept which finds applications in information theory as well as
thermodynamicsIn physics, maximum entropy thermodynamics views equilibrium thermodynamics and statistical mechanics as inference processes. More specifically, MaxEnt applies inference techniques rooted in Shannon information theory, Bayesian probability, and the principle of maximum entropy...
. 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
:
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.
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, a few argue that they have nothing to do with each other.
The expressions for the two entropies are similar. The information entropy
H for equal probabilities
is
where
k is a constant which determines the units of entropy. For example, if the units are bits, then k = 1/ln(2). The thermodynamic entropy
S, from a statistical mechanical point of view, was first expressed by Boltzmann:
where
p is the probability of a system's being in a particular microstate, given that it is in a particular macrostate, and
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.
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.
Mathematics
 KolmogorovSinai entropy – a mathematical type of entropy in dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s related to measures of partitions.
 Relative entropy – is a natural distance measure from a "true" probability distribution P to an arbitrary probability distribution Q.
 Rényi entropy
In information theory, the Rényi entropy, a generalisation of Shannon entropy, is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system...
– a generalized entropy measure for fractal systems.
 Topological entropy
In mathematics, the topological entropy of a topological dynamical system is a nonnegative real number that measures the complexity of the system. Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew. Their definition was modelled after the definition of the...
– a way of defining entropy in an iterated function map in ergodic theoryErgodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
.
 Volume entropy
The volume entropy is an asymptotic invariant of a compact Riemannian manifold that measures the exponential growth rate of the volume of metric balls in its universal cover. This concept is closely related with other notions of entropy found in dynamical systems and plays an important role in...
– a Riemannian invariant measuring the exponential rate of volume growth.
Sociology
The concept of entropy has also entered the domain of
sociologySociology is the study of society. It is a social science—a term with which it is sometimes synonymous—which uses various methods of empirical investigation and critical analysis to develop a body of knowledge about human social activity...
, generally as a
metaphorA metaphor is a literary figure of speech that uses an image, story or tangible thing to represent a less tangible thing or some intangible quality or idea; e.g., "Her eyes were glistening jewels." Metaphor may also be used for any rhetorical figures of speech that achieve their effects via...
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
Red tape is excessive regulation or rigid conformity to formal rules that is considered redundant or bureaucratic and hinders or prevents action or decisionmaking...
and business team inefficiency, i.e. energy lost to waste. (This definition is comparable to von Clausewitz's concept of frictionFriction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
in war.)
 Economic entropy – a semiquantitative measure of the irrevocable dissipation and degradation of natural materials and available energy with respect to economic activity.
 Entropology – the study or discussion of entropy or the name sometimes given to thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
without differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s.
 Psychological entropy – the distribution of energy in the psyche, which tends to seek equilibrium or balance among all the structures of the psyche.
 Social entropy
Social entropy is a macrosociological systems theory. It is a measure of the natural decay within a social system. It can refer to the decomposition of social structure or of the disappearance of social distinctions...
– 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.
See also
 Autocatalytic reactions and order creation
 Baryogenesis
In physical cosmology, baryogenesis is the generic term for hypothetical physical processes that produced an asymmetry between baryons and antibaryons in the very early universe, resulting in the substantial amounts of residual matter that make up the universe today.Baryogenesis theories employ...
 Brownian ratchet
In the philosophy of thermal and statistical physics, the Brownian ratchet, or FeynmanSmoluchowski ratchet is a thought experiment about an apparent perpetual motion machine first analysed in 1912 by Polish physicist Marian Smoluchowski and popularised by American Nobel laureate physicist Richard...
 Clausius–Duhem inequality
 Configuration entropy
In statistical mechanics, configuration entropy is the portion of a system's entropy that is related to the position of its constituent particles rather than to their velocity or momentum. It is physically related to the number of ways of arranging all the particles of the system while maintaining...
 Departure function
In thermodynamics, a departure function is defined for any thermodynamic property as the difference between the property as computed for an ideal gas and the property of the species as it exists in the real world, for a specified temperature T and pressure P...
 Enthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
 Entropy rate
In the mathematical theory of probability, the entropy rate or source information rate of a stochastic process is, informally, the time density of the average information in a stochastic process...
 Geometrical frustration
In condensed matter physics, the term geometrical frustration means a phenomenon in which the geometrical properties of the crystal lattice or the presence of conflicting atomic forces forbid simultaneous minimization of the interaction energies acting at a given site.This may lead to highly...
 Laws of thermodynamics
The four laws of thermodynamics summarize its most important facts. They define fundamental physical quantities, such as temperature, energy, and entropy, in order to describe thermodynamic systems. They also describe the transfer of energy as heat and work in thermodynamic processes...
 Multiplicity function
The multiplicity function for a two state paramagnet, W, is the number of spin states such that n of the N spins point in the zdirection. This function is given by the combinatoric function C...
 Nonequilibrium thermodynamics
Nonequilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium; for they are changing or can be triggered to change over time, and are continuously and discontinuously...
 Orders of magnitude (entropy)
The following list shows different orders of magnitude of entropy.References:...
 Randomness
Randomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....
 Stirling's formula
 Thermodynamic databases for pure substances
Thermodynamic databases contain information about thermodynamic properties for substances, the most important being enthalpy, entropy, and Gibbs free energy. Numerical values of these thermodynamic properties are collected as tables or are calculated from thermodynamic datafiles...
 Thermodynamic potential
Further reading
External links