For other senses of the term, see entropy (disambiguation).

The Thermodynamic entropy S, often simply called the entropy in the context of thermodynamics, is a measure of the amount of energy in a physical system that cannot be used to do work. In simpler terms, it is also a measure of the disorder and randomness present in a system. In thermodynamics and statistical mechanics, it is a key physical variable in describing a thermodynamic system. Entropy is "a quantitative measure of the amount of thermal energy not available to do work" within a closed thermodynamic system.[1]

The SI unit of entropy is a joule per kelvin (J·K⁻¹), which is the same as the unit of heat capacity, and entropy is said to be thermodynamically conjugate to temperature. The entropy depends only on the current state of the system, not its detailed previous history, and so it is a state function of the parameters like pressure, temperature, etc., which describe the observable macroscopic properties of the system.

There is an important connection between entropy and the amount of internal energy in the system which is not available to perform work. In any process where the system gives up an energy ΔE, and its entropy falls by ΔS, a quantity at least T_R ΔS of that energy must be given up to the system's surroundings as unusable heat. Otherwise the process will not go forward. (T_R is the temperature of the system's external surroundings, which may not be the same as the system's current temperature T ).

Introduction

Many quantities of matter tend to equalize their thermodynamic parameters - reducing differentials towards zero. Pressure differences, density differences, and temperature differences, all tend towards equalizing. Entropy is a measure of how far along this process of equalization has come. Entropy increases as this equalization process advances. For example, the combined entropy of "a cup of hot water in a cool room" is less than the entropy of "the room and the water after it has cooled (and warmed the room slightly)," because the heat is more evenly distributed. The entropy of the room and the empty cup after the water has evaporated is even higher.

An important law of physics, the second law of thermodynamics, states that the total entropy of any isolated thermodynamic system tends to increase over time, approaching a maximum value; and so, by implication, the entropy of the universe as a whole (i.e. the system and its surroundings) tends to increase. We will consider the meaning of the "second law" further in a subsequent section. Two important consequences are that heat cannot of itself pass from a colder to a hotter body: i.e., it is impossible to transfer heat from a cold to a hot reservoir without at the same time converting a certain amount of work to heat. It is also impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work; it can only get useful work out of the heat if heat is at the same time transferred from a hot to a cold reservoir. This means that there is no possibility of a 'perpetuum mobile' which is isolated. Also, from this it follows, that a reduction in the increase of entropy in a specified process, such as a chemical reaction, means that it is energetically more efficient.

Statistical mechanics explains entropy as the amount of uncertainty (or "mixedupness" in the phrase of Gibbs) which remains about a system, after its observable macroscopic properties have been taken into account. For a given set of macroscopic quantities, like temperature and volume, the entropy is a function of the probability that the system is in various quantum states. The more states available to the system with higher probability, the greater the "disorder" and thus, the greater the entropy.

(Note that it is important to distinguish the definition of disorder in the context of entropy and the definition of disorder in the context of everyday usage. In physics, the term "disorder" in this sense refers to a specific, well-defined quantity, while disorder in everyday usage is more akin to disorganization. A more thorough exploration of this concept can be found below).

The two definitions match up because adding heat to a system, which increases its classic thermodynamic entropy, also increases the system's thermal fluctuations, so giving an increased lack of information about the exact microscopic state of the system, i.e. an increased statistical mechanical entropy. This will be considered in more detail below.

The entropy in statistical mechanics can be considered as a specific application of Shannon entropy, according to a viewpoint known as MaxEnt thermodynamics. Roughly speaking, Shannon entropy is proportional to the minimum number of yes/no questions you have to ask to get the answer to some question. The statistical mechanical entropy is then proportional to the minimum number of yes/no questions you have to ask in order to determine the microstate, given that you know the macrostate.

History

Thermodynamic entropy was first introduced in the context of classical thermodynamics by Rudolf Clausius in 1850, in his analysis of Sadi Carnot's 1824 work on thermodynamic efficiency; although it was not until 1865 that Clausius singled the quantity out and gave it the name entropy.

In 1877, Ludwig Boltzmann formulated the alternative definition of entropy as a measure of statistical "mixedupness" or disorder, soon refined by J. Willard Gibbs, which is now regarded as one of the cornerstones of the theory of statistical mechanics.

Claude Shannon introduced the very general concept of information entropy, used in information theory, in 1948. The view of seeing entropy in statistical mechanics as merely a particular application of Shannon's information entropy is largely due to the campaigning of E. T. Jaynes, starting in 1957.

When necessary, to disambiguate between the statistical thermodynamic concept of entropy, and entropy-like formulae put forward by different researchers, the statistical thermodynamic entropy is most properly referred to as the Gibbs entropy. The terms Boltzmann-Gibbs entropy or BG entropy, and Boltzmann-Gibbs-Shannon entropy or BGS entropy are also seen in the literature.

Thermodynamic definition of entropy

In this section, we discuss the original definition of entropy, as introduced by Clausius in the context of classical thermodynamics. Clausius defined the change in entropy of a thermodynamic system, during a reversible process in which an amount of heat $δ Q$ is introduced at constant absolute temperature T, as

$dS = \frac{\delta Q}{T} \,\!$

This expression comes from the Clausius theorem. This definition makes sense when absolute temperature has been defined. Note that the small amount $δ Q$ of energy transferred by heating is denoted by $δ Q$ rather than $d Q$ , because Q is not a state function while the entropy is.

Clausius gave the quantity S the name "entropy", from the Greek word τρoπή, "transformation". Since this definition involves only differences in entropy, the entropy itself is only defined up to an arbitrary additive constant.

When a process is irreversible, the above definition must be replaced by the statement that the entropy change is equal to the amount of energy required to return the system to its original state by a reversible transformation at a constant temperature, divided by that temperature. This is explained in more detail below.

Heat engines

Clausius' identification of S as a significant quantity was motivated by the study of reversible and irreversible thermodynamic transformations. A thermodynamic transformation is a change in a system's thermodynamic properties, such as temperature and volume. A transformation is reversible if it is quasistatic which means that it is infinitesimally close to thermodynamic equilibrium at all times. Otherwise, the transformation is irreversible. To illustrate this, consider a gas enclosed in a piston chamber, whose volume may be changed by moving the piston. If we move the piston slowly enough, the density of the gas is always homogeneous, so the transformation is reversible. If we move the piston quickly, pressure waves are created, so the gas is not in equilibrium, and the transformation is irreversible.

A heat engine is a thermodynamic system that can undergo a sequence of transformations which ultimately return it to its original state. Such a sequence is called a cyclic process, or simply a cycle. During some transformations, the engine may exchange energy with the environment. The net result of a cycle is (i) mechanical work done by the system (which can be positive or negative, the latter meaning that work is done on the engine), and (ii) heat energy transferred from one part of the environment to another. By the conservation of energy, the net energy lost by the environment is equal to the work done by the engine.

If every transformation in the cycle is reversible, the cycle is reversible, and it can be run in reverse, so that the energy transfers occur in the opposite direction and the amount of work done switches sign.

Definition of temperature

In thermodynamics, absolute temperature is defined in the following way. Suppose we have two heat reservoirs, which are systems sufficiently large that their temperatures do not change when energy flows into or out of them. A reversible cycle exchanges heat with the two heat reservoirs. If the cycle absorbs an amount of heat Q from the first reservoir and delivers an amount of heat Q′ to the second, then the respective reservoir temperatures T and T′ obey

$\frac{Q}{T} = \frac{Q'}{T'} \,\!$

Proof: Introduce an additional heat reservoir at an arbitrary temperature T₀, as well as N cycles with the following property: the j-th such cycle operates between the T₀ reservoir and the T_j reservoir, transferring energy $δ Q j$ to the latter. From the above definition of temperature, the energy extracted from the T₀ reservoir by the j-th cycle is

$\delta Q_{0,j} = T_0 \frac{\delta Q_j}{T_j} \,\!$

Now consider one cycle of the heat engine, accompanied by one cycle of each of the smaller cycles. At the end of this process, each of the N reservoirs have zero net energy loss (since the energy extracted by the engine is replaced by the smaller cycles), and the heat engine has done an amount of work equal to the energy extracted from the T₀ reservoir,

$W = \sum_{j=1}^N \delta Q_{0,j} = T_0 \sum_{j=1}^N \frac{\delta Q_j}{T_j} \,\!$

If this quantity is positive, this process would be a perpetual motion machine of the second kind, which is impossible. Thus,

$\sum_{i=1}^N \frac{\delta Q_i}{T_i} \le 0 \,\!$

Now repeat the above argument for the reverse cycle. The result is

$\sum_{i=1}^N \frac{\delta Q_i}{T_i} = 0 \,\!$ (reversible cycles)

Now consider a reversible cycle in which the engine exchanges heats $\delta Q_1,\delta Q_2,\cdots,\delta Q_N$ with a sequence of N heat reservoirs with temperatures T₁, ..., T_N. A negative $δ Q$ means that energy flows from the reservoir to the engine, and a positive $δ Q$ means that energy flows from the engine to the reservoir. We can show (see the box on the right) that

$\sum_{i=1}^N \frac{\delta Q_i}{T_i} = 0 \,\!$ .

Since the cycle is reversible, the engine is always infinitesimally close to equilibrium, so its temperature is equal to any reservoir with which it is contact. In the limiting case of a reversible cycle consisting of a continuous sequence of transformations,

$\oint \frac{\delta Q}{T} = 0 \,\!$ (reversible cycles)

where the integral is taken over the entire cycle, and T is the temperature of the system at each point in the cycle. This is a particular case of the Clausius theorem.

Entropy as a state function

We can now deduce an important fact about the entropy change during any thermodynamic transformation, not just a cycle. First, consider a reversible transformation that brings a system from an equilibrium state A to another equilibrium state B. If we follow this with any reversible transformation which returns that system to state A, our above result says that the net entropy change is zero. This implies that the entropy change in the first transformation depends only on the initial and final states.

This allows us to define the entropy of any equilibrium state of a system. Choose a reference state R and call its entropy S_R. The entropy of any equilibrium state X is

$S_X = S_R + \int_R^X \frac{\delta Q}{T} \,\!$

Since the integral is independent of the particular transformation taken, this equation is well-defined.

Entropy change in irreversible transformations

We now consider irreversible transformations. It can be shown that the entropy change during any transformation between two equilibrium states is

$\Delta S \ge \int \frac{\delta Q}{T} \,\!$

where the equality holds if the transformation is reversible.

Notice that if $δ Q = 0$ , then ΔS ≥ 0. This is the Second Law of Thermodynamics, which we have discussed earlier.

Suppose a system is thermally and mechanically isolated from the environment. For example, consider an insulating rigid box divided by a movable partition into two volumes, each filled with gas. If the pressure of one gas is higher, it will expand by moving the partition, thus performing work on the other gas. Also, if the gases are at different temperatures, heat can flow from one gas to the other provided the partition is an imperfect insulator. Our above result indicates that the entropy of the system as a whole will increase during these process (it could in principle remain constant, but this is unlikely.) Typically, there exists a maximum amount of entropy the system may possess under the circumstances. This entropy corresponds to a state of stable equilibrium, since a transformation to any other equilibrium state would cause the entropy to decrease, which is forbidden. Once the system reaches this maximum-entropy state, no part of the system can perform work on any other part. It is in this sense that entropy is a measure of the energy in a system that "cannot be used to do work".

Measuring entropy

In real experiments, it is quite difficult to measure the entropy of a system. The techniques for doing so are based on the thermodynamic definition of the entropy, and require extremely careful calorimetry.

For simplicity, we will examine a mechanical system, whose thermodynamic state may be specified by its volume V and pressure P. In order to measure the entropy of a specific state, we must first measure the heat capacity at constant volume and at constant pressure (denoted C_V and C_P respectively), for a successive set of states intermediate between a reference state and the desired state. The heat capacities are related to the entropy S and the temperature T by

$C_X = T \left(\frac{\partial S}{\partial T}\right)_X \,\!$

where the X subscript refers to either constant volume or constant pressure. This may be integrated numerically to obtain a change in entropy:

$\Delta S = \int \frac{C_X}{T} dT \,\!$

We can thus obtain the entropy of any state (P,V) with respect to a reference state (P₀,V₀). The exact formula depends on our choice of intermediate states. For example, if the reference state has the same pressure as the final state,

$S(P,V) = S(P, V_0) + \int^{T(P,V)}_{T(P,V_0)} \frac{C_P(P,V(T,P))}{T} dT \,\!$

In addition, if the path between the reference and final states lies across any first order phase transition, the latent heat associated with the transition must be taken into account.

The entropy of the reference state must be determined independently. Ideally, one chooses a reference state at an extremely high temperature, at which the system exists as a gas. The entropy in such a state would be that of a classical ideal gas plus contributions from molecular rotations and vibrations, which may be determined spectroscopically. Choosing a low temperature reference state is sometimes problematic since the entropy at low temperatures may behave in unexpected ways. For instance, a calculation of the entropy of ice by the latter method, assuming no entropy at zero temperature, falls short of the value obtained with a high-temperature reference state by 3.41 J/(mol·K). This is due to the "zero-point" entropy of ice mentioned earlier.

Statistical entropy

Entropy and disorder

As stated above, when discussing entropy, the term disorder does not necessarily mean disorganization. Many textbooks utilize a bedroom as an example of a hypothetical system in which disorganization is spontaneously increasing, and those textbooks say this is an example of entropy. A statement like this must be carefully made or else it can be misleading. For example, if all of the socks in the room were in perfect rows in a drawer, would that configuration have less entropy than if the socks were strewn about the room? The answer is to be found in the Shannon definition of entropy. One must ask "how many distinct ways can socks be folded in the drawers", and "how many different ways can they be strewn about the room"? If these questions can be answered objectively, then the configuration with the most possibilities (probably the socks strewn about the room) would have the highest entropy, although this entropy would not be true thermodynamic entropy, but it would qualify as a type of information entropy. It is only when the information entropy concept is applied to the individual particles of a thermodynamic system that the information entropy and the thermodynamic entropy become the same.

Mathematical description

The macroscopic state of the system is defined by a distribution on the microstates that are accessible to a system in the course of its thermal fluctuations. So the entropy is defined over two different levels of description of the given system. For a quantum system with a discrete set of microstates, if $E i$ is the energy of microstate i, and $p i$ is its probability that it occurs during the system's fluctuations, then the entropy of the system is

$S = -k_B\,\sum_i p_i \ln \,p_i$

Entropy changes for systems in a canonical state

A system with a well-defined temperature, ie one in thermal equilibrium with a thermal reservoir, has a probability of being in a microstate i given by Boltzmann's distribution.

Changes in the entropy caused by changes in the external constraints are then given by:

$dS = -k_B\,\sum_i dp_i \ln p_i$

$\,\,\, = -k_B\,\sum_i dp_i (-E_i/k_BT -\ln Z)$

$\,\,\, = \sum_i E_i dp_i / T$

$\,\,\, = \sum_i [d (E_i p_i) - (dE_i) p_i] / T$

where we have twice used the conservation of probability, ∑ dp_i=0 .

Now, ∑_i d (E_i p_i) is the expectation value of the change in the total energy of the system.

If the changes are sufficiently slow, so that the system remains in the same microscopic state, but the state slowly (and reversibly) changes, then ∑_i (dE_i) p_i is the expectation value of the work done on the system through this reversible process, dw_rev.

But from the first law of thermodynamics, δE = δw +δq. Therefore,

$dS = \frac{\delta\langle q_{rev} \rangle}{T}$

In the thermodynamic limit, the fluctuation of the macroscopic quantities from their average values becomes negligible; so this reproduces the definition of entropy from classical thermodynamics, given above.

The quantity $k B$ is a physical constant known as Boltzmann's constant, which, like the entropy, has units of heat capacity. The logarithm is dimensionless.

This definition is valid including far away from equilibrium. Other definitions assume that the system is in thermal equilibrium, either as an isolated system, or as a system in exchange with its surroundings. The set of microstates on which the sum is to be done is called a statistical ensemble. Each statistical ensemble (micro-canonical, canonical, grand-canonical, etc.) describes a different configuration of the system's exchanges with the outside, from an isolated system to a system that can exchange one more quantity with a reservoir, like energy, volume or molecules. In every ensemble, the equilibrium configuration of the system is dictated by the maximization of the entropy of the union of the system and its reservoir, according to the second law of thermodynamics (see the statistical mechanics article).

Note the above expression of the statistical entropy is a discretized version of Shannon entropy

Boltzmann's principle

In Boltzmann's definition, entropy is a measure of the number of possible microscopic states (or microstates) of a system in thermodynamic equilibrium, consistent with its macroscopic thermodynamic properties (or macrostate). To understand what microstates and macrostates are, consider the example of a gas in a container. At a microscopic level, the gas consists of a vast number of freely moving atoms, which occasionally collide with one another and with the walls of the container. The microstate of the system is a description of the positions and momenta of all the atoms. In principle, all the physical properties of the system are determined by its microstate. However, because the number of atoms is so large, the motion of individual atoms is mostly irrelevant to the behavior of the system as a whole. Provided the system is in thermodynamic equilibrium, the system can be adequately described by a handful of macroscopic quantities, called "thermodynamic variables": the total energy E, volume V, pressure P, temperature T, and so forth. The macrostate of the system is a description of its thermodynamic variables.

There are three important points to note. Firstly, to specify any one microstate, we need to write down an impractically long list of numbers, whereas specifying a macrostate requires only a few numbers (E, V, etc.). However, and that is the second point, the usual thermodynamic equations only describe the macrostate of a system adequately when this system is in equilibrium; non-equilibrium situations can generally not be described by a small number of variables. For example, if a gas is sloshing around in its container, even a macroscopic description would have to include, e.g., the velocity of the fluid at each different point. Actually, the macroscopic state of the system will be described by a small number of variables only if the system is at global thermodynamic equilibrium. Thirdly, more than one microstate can correspond to a single macrostate. In fact, for any given macrostate, there will be a huge number of microstates that are consistent with the given values of E, V, etc.

We are now ready to provide a definition of entropy. Let Ω be the number of microstates consistent with the given macrostate. The entropy S is defined as

$S = k_B \ln \Omega \,\!$

where k_B is Boltzmann's constant.

The statistical entropy reduces to Boltzman's entropy when all the accessible microstates of the system are equally likely. It is also the configuration corresponding to the maximum of a system's entropy for a given set of accessible microstates, in other words the macroscopic configuration in which the disorder (or lack of information) is maximal. As such, according to the second law of thermodynamics, it is the equilibrium configuration of an isolated system. Boltzman's entropy is the expression of entropy at thermodynamic equilibrium in the micro-canonical ensemble.

This postulate, which is known as Boltzmann's principle, may be regarded as the foundation of statistical mechanics, which describes thermodynamic systems using the statistical behaviour of its constituents. It turns out that S is itself a thermodynamic property, just like E or V. Therefore, it acts as a link between the microscopic world and the macroscopic. One important property of S follows readily from the definition: since Ω is a natural number (1,2,3,...), S is either zero or positive (this is a property of the logarithm.)

Disorder and the second law of thermodynamics

We can view Ω as a measure of the disorder in a system. This is reasonable because what we think of as "ordered" systems tend to have very few configurational possibilities, and "disordered" systems have very many. As an illustration of this idea, consider a set of 100 coins, each of which is either heads up or tails up. The macrostates are specified by the total number of heads and tails, whereas the microstates are specified by the facings of each individual coin. For the macrostates of 100 heads or 100 tails, there is exactly one possible configuration, corresponding to the most "ordered" state in which all the coins are facing the same way. The most "disordered" macrostate consists of 50 heads and 50 tails in any order, for which there are 100891344545564193334812497256 (100 choose 50) possible microstates.

Even when a system is entirely isolated from external influences, its microstate is constantly changing. For instance, the particles in a gas are constantly moving, and thus occupy a different position at each moment of time; their momenta are also constantly changing as they collide with each other or with the container walls. Suppose we prepare the system in an artificially highly-ordered equilibrium state. For instance, imagine dividing a container with a partition and placing a gas on one side of the partition, with a vacuum on the other side. If we remove the partition and watch the subsequent behavior of the gas, we will find that its microstate evolves according to some chaotic and unpredictable pattern, and that on average these microstates will correspond to a more disordered macrostate than before. It is possible, but extremely unlikely, for the gas molecules to bounce off one another in such a way that they remain in one half of the container. It is overwhelmingly probable for the gas to spread out to fill the container evenly, which is the new equilibrium macrostate of the system.

This is an illustration of a principle that we will prove rigorously in a subsequent section, known as the Second Law of Thermodynamics. This states that

the total entropy of any isolated thermodynamic system tends to increase over time, approaching a maximum value.

Since its discovery, the idea that disorder tends to increase has been the focus of a great deal of thought, some of it confused. A chief point of confusion is the fact that the Second Law applies only to isolated systems. For example, the Earth is not an isolated system because it is constantly receiving energy in the form of sunlight. Nevertheless, it has been pointed out that the universe may be considered an isolated system, so that its total disorder should be constantly increasing. We will discuss the implications of this idea in the section on Entropy and cosmology.

Once again, it is important to distinguish the meaning of "disorder" in the context of entropy and the colloquial definition, which is a vague term associated with "chaos". The "disorder" to which we refer in this article is a specific, well-defined quantity.

Counting of microstates

In classical statistical mechanics, the number of microstates is actually uncountably infinite, since the properties of classical systems are continuous. For example, a microstate of a classical ideal gas is specified by the positions and momenta of all the atoms, which range continuously over the real numbers. If we want to define Ω, we have to come up with a method of grouping the microstates together to obtain a countable set. This procedure is known as coarse graining. In the case of the ideal gas, we count two states of an atom as the "same" state if their positions and momenta are within δx and δp of each other. Since the values of δx and δp can be chosen arbitrarily, the entropy is not uniquely defined. It is defined only up to an additive constant. (As we will see, the thermodynamic definition of entropy is also defined only up to a constant.)

This ambiguity can be resolved with quantum mechanics. The quantum state of a system can be expressed as a superposition of "basis" states, which can be chosen to be energy eigenstates (i.e. eigenstates of the quantum Hamiltonian.) Usually, the quantum states are discrete, even though there may be an infinite number of them. In quantum statistical mechanics, we can take Ω to be the number of energy eigenstates consistent with the thermodynamic properties of the system.

An important result, known as Nernst's theorem or the third law of thermodynamics, states that the entropy of a system at zero absolute temperature is a well-defined constant. This is due to the fact that a system at zero temperature exists in its lowest-energy state, or ground state, so that its entropy is determined by the degeneracy of the ground state. Many systems, such as crystal lattices, have a unique ground state, and (since ln(1) = 0) this means that they have zero entropy at absolute zero. Other systems have more than one state with the same, lowest energy, and have a non-vanishing "zero-point entropy". For instance, ordinary ice has a zero-point entropy of 3.41 J/(mol·K), due to the fact that its underlying crystal structure possesses multiple configurations with the same energy (a phenomenon known as geometrical frustration).

The third law of thermodynamics states that the entropy of a perfect crystal at absolute zero or 0 Kelvin is zero. This means that in a perfect crystal, at 0 Kelvin, nearly all molecular motion should cease in order to achieve ΔS=0. A perfect crystal is one in which the internal lattice structure is the same at all times; in other words, it is fixed and non-moving, and does not have rotational or vibrational energy. This means that there is only one way in which this order can be attained: when every particle of the structure is in its proper place.

However, the equation for predicting quantized vibrational levels shows that even when the vibrational quantum number is 0, the molecule still has vibrational energy. This means that no matter how cold the temperature gets, the molecule will always have vibration. This is in keeping with the Heisenberg uncertainty principle, which states that both the position and the momentum of a particle cannot be known precisely, at a given time:

Ev = h(v0)[v+(1/2)],

where h=Planck's constant, v0=characteristic frequency of the vibration, and v=the vibrational quantum number. Note that even when v=0 (the zero-point energy), Ev does not equal 0.

Since all molecules will have some vibrational energy at all times, the entropy of such a molecule will not be 0. However, the third law of thermodynamics requires the entropy of a perfect crystal to be 0, at absolute zero. Therefore, it can be inferred that absolute zero is not attainable, since a perfect crystal configuration cannot be achieved.

The arrow of time

Entropy is the only quantity in the physical sciences that "picks" a particular direction for time, sometimes called an arrow of time. As we go "forward" in time, the Second Law of Thermodynamics tells us that the entropy of an isolated system can only increase or remain the same; it cannot decrease. In contrast, all physical processes occurring at the microscopic level, such as mechanics, do not pick out an arrow of time. Going forward in time, we might see an atom moving to the left, whereas going backward in time, we would see the same atom moving to the right; the behavior of the atom is not qualitatively different in either case. In contrast, we would be shocked if a gas that originally filled a container evenly, spontaneously shrinks to occupy only half the container.

The reader may have noticed that the Second Law allows for the entropy remaining the same. If the entropy is constant in either direction of time, there would be no preferred direction. However, the entropy can only be a constant if the system is in the highest possible state of disorder, such as a gas that always was, and always will be, uniformly spread out in its container. The existence of a thermodynamic arrow of time implies that the system is highly ordered in one time direction, which would by definition be the "past".

Unlike most other laws of physics, the Second Law of Thermodynamics is statistical in nature, and its reliability arises from the huge number of particles present in macroscopic systems. It is not impossible, in principle, for all 10²³ atoms in a gas to spontaneously migrate to one half of container; it is only fantastically unlikely -- so unlikely that no macroscopic violation of the Second Law has ever been observed.

In 1867, James Clerk Maxwell introduced a now-famous thought experiment that highlighted the contrast between the statistical nature of entropy and the deterministic nature of the underlying physical processes. This experiment, known as Maxwell's demon, consists of a hypothetical "demon" that guards a trapdoor between two containers filled with gases at equal temperatures. By allowing fast molecules through the trapdoor in only one direction and only slow molecules in the other direction, the demon raises the temperature of one gas and lowers the temperature of the other, apparently violating the Second Law. Maxwell's thought experiment was only resolved in the 20th century by Leó Szilárd, Charles H. Bennett, and others. The key idea is that the demon itself necessarily possesses a non-negligible amount of entropy that increases even as the gases lose entropy, so that the entropy of the system as a whole increases. This is because the demon has to contain many internal "parts" if it is to perform its job reliably, and therefore has to be considered a macroscopic system with non-vanishing entropy. An equivalent way of saying this is that the information possessed by the demon on which atoms are considered "fast" or "slow", can be considered a form of entropy known as information entropy.

Unsolved problems in physics: Arrow of time: Why did the universe have such low entropy in the past, resulting in the distinction between past and future and the second law of thermodynamics?

Many physicists believe that all phenomena that behave differently in one time direction can ultimately be linked to the Second Law of Thermodynamics. This includes the fact that ice cubes melt in hot coffee rather than assembling themselves out of the coffee, that a block sliding on a rough surface slows down rather than speeding up, and that we can remember the past rather than the future. (This last phenomenon, called the "psychological arrow of time", has deep connections with Maxwell's demon and the physics of information.) If the thermodynamic arrow of time is indeed the only arrow of time, then the ultimate reason for a preferred time direction is that the universe as a whole was in a highly ordered state at the Big Bang. The question of why this highly ordered state existed, and how to describe it, remains an area of research.

Entropy and cosmology

We have previously mentioned that the universe may be considered an isolated system. As such, it may be subject to the Second Law of Thermodynamics, so that its total entropy is constantly increasing. If the entropy of the universe keeps on increasing then this violates some of our physical laws because at the time of the big crunch the matter ends comes together and they come in a highly orderly form so the entropy decreases so the second law contradicts the law of gravitation

It has been speculated that the universe is fated to a heat death in which all the energy ends up as a homogeneous distribution of thermal energy, so that no more work can be extracted from any source.

If the universe can be considered to have increasing entropy, then, as Roger Penrose has pointed out, an important role in the disordering process is played by gravity, which causes dispersed matter to accumulate into stars, which collapse eventually into black holes. Jacob Bekenstein and Stephen Hawking have shown that black holes have the maximum possible entropy of any object of equal size. This makes them likely end points of all entropy-increasing processes.

The role of entropy in cosmology remains a controversial subject. Recent work has cast extensive doubt on the heat death hypothesis and the applicability of any simple thermodynamic model to the universe in general. Although entropy does increase in an expanding universe, the maximum possible entropy rises much more rapidly and leads to an "entropy gap," thus pushing the system further away from equilibrium with each time increment. Complicating factors, such as the energy density of the vacuum and macroscopic quantum effects, are difficult to reconcile with thermodynamical models, making any predictions of large-scale thermodynamics extremely difficult.

Entropy in literature

Isaac Asimov's The Last Question, a short science fiction story about entropy
Thomas Pynchon, an American author who deals with entropy in many of his novels
Diane Duane's Young Wizards series, in which the protagonists' ultimate goal is to slow down entropy and delay heat death.
Gravity Dreams by L.L. Modesitt Jr.
The Arrow of Time, an essay by journalist K.C. Cole, takes the physical law of entropy and metaphorically applies it to real life.
Jeremy Rifkin's Entropy: A New World View, a notorious misinterpretation of entropy [2]
The Planescape setting for Dungeons & Dragons includes the Doomguard faction, who worship entropy.
Arcadia by Tom Stoppard, explores entropy, the arrow of time, and heat death.

External links

Entropy Explained - Correction of common misconceptions, and explaining the true meaning of entropy.
Entropy and chaos
Entropy concepts at HyperPhysics
Entropy is Simple...If You Avoid the Briar Patches
Dictionary of the History of Ideas:Entropy
Java "entropy pool" for cryptographically-secure unguessable random numbers
Entropy in Relation to Free Energy
Entropy Is Simple, Qualitatively article by Frank Lambert on http://EntropySite.com/
Entropy and the second law of thermodynamics Molecular approach to entropy.
Entropy for students in general chemistry Simple approach to the second law, in Q and A form.
All About Entropy The laws of thermodynamics, and order from disorder.

References

Fermi, Enrico (1937). Thermodynamics, Prentice Hall. ISBN 048660361X.
Kroemer, Herbert; Charles Kittel (1980). Thermal Physics, 2nd Ed., W. H. Freeman Company. ISBN 0716710889.
Penrose, Roger (2005). The Road to Reality : A Complete Guide to the Laws of the Universe. ISBN 0679454438.
Reif, F. (1965). Fundamentals of statistical and thermal physics, McGraw-Hill. ISBN 0070518009.

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Entropy

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