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Entropy

2007 Schools Wikipedia Selection. Related subjects: General Physics

   Ice melting - classic example of entropy increasing described in 1862
   by Rudolf Clausius as an increase in the disgregation of the molecules
   of the body of ice.
   Enlarge
   Ice melting - classic example of entropy increasing described in 1862
   by Rudolf Clausius as an increase in the disgregation of the molecules
   of the body of ice.

   The concept of entropy in thermodynamics is central to the second law
   of thermodynamics, which deals with physical processes and whether they
   occur spontaneously. Spontaneous changes occur with an increase in
   entropy. In contrast the first law of thermodynamics deals with the
   concept of energy, which is conserved. Entropy change has often been
   defined as a change to a more disordered state at a microscopic level.
   In recent years, entropy has been interpreted in terms of the "
   dispersal" of energy. Entropy is an extensive state function that
   accounts for the effects of irreversibility in thermodynamic systems.

   Quantitatively, entropy, symbolized by S, is defined by the
   differential quantity dS = δQ / T, where δQ is the amount of heat
   absorbed in a reversible process in which the system goes from one
   state to another, and T is the absolute temperature. Entropy is one of
   the factors that determines the free energy of the system.

   When a system's energy is defined as the sum of its "useful" energy,
   (e.g. that used to push a piston), and its "useless energy", i.e. that
   energy which cannot be used for external work, then entropy may be
   (most concretely) visualized as the "scrap" or "useless" energy whose
   energetic prevalence over the total energy of a system is directly
   proportional to the absolute temperature of the considered system, as
   is the case with the Gibbs free energy or Helmholtz free energy
   relations.

   In terms of statistical mechanics, the entropy describes the number of
   the possible microscopic configurations of the system. The statistical
   definition of entropy is generally thought to be the more fundamental
   definition, from which all other important properties of entropy
   follow. Although the concept of entropy was originally a thermodynamic
   construct, it has been adapted in other fields of study, including
   information theory, psychodynamics, thermoeconomics, and evolution.

History

   Rudolf Clausius - originator of the concept of "entropy" S
   Enlarge
   Rudolf Clausius - originator of the concept of "entropy" S

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

   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 recent 1789 views of Count Rumford who
   showed that heat could be created by friction as when cannons bored.
   Accordingly, Carnot reasoned that if the body of the working substance,
   such as a body of steam, is brought back 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 60s, German physicist Rudolf Clausius gravely objected
   to this latter 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. This was in contrast to earlier
   views, based on the theories of Isaac Newton, that heat was an
   indestructible particle that had mass. Later, scientists such as Ludwig
   Boltzmann, Willard Gibbs, and James Clerk Maxwell gave entropy a
   statistical basis. Carathéodory linked entropy with a mathematical
   definition of irreversibility, in terms of trajectories and
   integrability.

Definitions and descriptions

   In science, the term "entropy" is generally interpreted in three
   distinct, but semi-related, ways, i.e. from macroscopic viewpoint (
   classical thermodynamics), a microscopic viewpoint ( statistical
   thermodynamics), and an information viewpoint ( information theory).
   Entropy in information theory is a fundamentally different concept from
   thermodynamic entropy. However, at a philosophical level, some argue
   that thermodynamic entropy can be interpreted as an application of the
   information entropy concept to a very particular set of physical
   questions.

Macroscopic viewpoint (classical thermodynamics)

                                                  Conjugate variables
                                                        of thermodynamics
                                                     Pressure       Volume
                                                    ( Stress)    ( Strain)
                                                  Temperature      Entropy
                                              Chem. potential Particle no.

   In a thermodynamic system, a "universe" consisting of "surroundings"
   and "systems" and made up of quantities of matter, its pressure
   differences, density differences, and temperature differences all tend
   to equalize over time. 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 heat energy from the warm surroundings
   become spread out to the cooler system of ice and water.
   Thermodynamic System
   Enlarge
   Thermodynamic System

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

   A special case of entropy increase, the entropy of mixing, occurs when
   two or more different substances are mixed. If the substances are at
   the same temperature and pressure, there will be no net exchange of
   heat or work - the entropy increase will be entirely due to the mixing
   of the different substances.

   From a macroscopic perspective, in classical thermodynamics the entropy
   is interpreted simply as a state function of a thermodynamic system:
   that is, a property depending only on the current state of the system,
   independent of how that state came to be achieved. The state function
   has the important property that, when multiplied by a reference
   temperature, it can be understood as a measure of the amount of energy
   in a physical system that cannot be used to do thermodynamic work;
   i.e., work mediated by thermal energy. More precisely, in any process
   where the system gives up energy ΔE, and its entropy falls by ΔS, a
   quantity at least T[R] ΔS of that energy must be given up to the
   system's surroundings as unusable heat (T[R] is the temperature of the
   system's external surroundings). Otherwise the process will not go
   forward.

   In 1862, Clausius stated what he calls the “theorem respecting the
   equivalence-values of the transformations” or what is now known as the
   second law of thermodynamics, as such:

          The algebraic sum of all the transformations occurring in a
          cyclical process can only be positive, or, as an extreme case,
          equal to nothing.

   Quantitatively, Clausius states the mathematical expression for this
   theorem is as follows. Let δQ be an element of the heat given up by the
   body to any reservoir of heat during its own changes, heat which it may
   absorb from a reservoir being here reckoned as negative, and T the
   absolute temperature of the body at the moment of giving up this heat,
   then the equation:

          \int \frac{\delta Q}{T} = 0

   must be true for every reversible cyclical process, and the relation:

          \int \frac{\delta Q}{T} \ge 0

   must hold good for every cyclical process which is in any way possible.
   This is the essential formulation of the second law and one of the
   original forms of the concept of entropy. It can be seen that the
   dimensions of entropy are energy divided by temperature, which is the
   same as the dimensions of Boltzmann's constant (k_B) and heat capacity.
   The SI unit of entropy is " joule per kelvin" (J•K^−1). In this manner,
   the quantity "ΔS" is utilized as a type of internal ordering energy,
   which accounts for the effects of irreversibility, in the energy
   balance equation for any given system. In the Gibbs free energy
   equation, i.e. ΔG = ΔH - TΔS, for example, which is a formula commonly
   utilized to determine if chemical reactions will occur, the energy
   related to entropy changes TΔS is subtracted from the "total" system
   energy ΔH to give the "free" energy ΔG of the system, as during a
   chemical process or as when a system changes state.

Microscopic viewpoint (statistical mechanics)

   From a microscopic perspective, in statistical thermodynamics the
   entropy is a measure of the number of microscopic configurations that
   are capable of yielding the observed macroscopic description of the
   thermodynamic system:

          S = k_B \ln \Omega \!

   where Ω is the number of microscopic configurations, and k[B] is
   Boltzmann's constant. 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.

   In 1877, thermodynamicist Ludwig Boltzmann visualized a probabilistic
   way to measure the entropy of an ensemble of ideal gas particles, in
   which he defined entropy to be proportional to the logarithm of the
   number of microstates such a gas could occupy. Henceforth, the
   essential problem in statistical thermodynamics, i.e. according to
   Erwin Schrödinger, has been to determine the distribution of a given
   amount of energy E over N identical systems.

   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 measures the degree to which the probability of
   the system is spread out over different possible quantum states. The
   more states available to the system with higher probability, and thus
   the greater the entropy. In essence, the most general interpretation of
   entropy is as a measure of our ignorance 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 quantities; maximizing the entropy maximizes our ignorance
   about the details of the system.

   On the molecular scale, the two definitions match up because adding
   heat to a system, which increases its classical 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.

Entropy in chemical thermodynamics

   Thermodynamic entropy is central in chemical thermodynamics, enabling
   changes to be quantified and the outcome of reactions predicted. The
   second law of thermodynamics states that entropy in the combination of
   a system and its surroundings (or in an isolated system by itself)
   increases during all spontaneous chemical and physical processes.
   Spontaneity in chemistry means “by itself, or without any outside
   influence”, and has nothing to do with speed. The Clausius equation of
   δq[rev]/T = ΔS introduces the the measurement of entropy change, ΔS.
   Entropy change describes the direction and quantitates the magnitude of
   simple changes such as heat transfer between systems – always from
   hotter to cooler spontaneously. Thus, when a mole of substance at 0 K
   is warmed by its surroundings to 298 K, 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 298 K. 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 of complex
   chemical reactions, i.e. whether a process will go as written or
   proceed in the opposite direction. For such applications, ΔS must be
   incorporated in an expression that includes both the system and its
   surroundings, Δ S[universe] = ΔS[surroundings] + Δ S [system]. This
   expression becomes, via some steps, the Gibbs free energy 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].

The second law

   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 (i.e. the system and its
   surroundings), assumed as an isolated system, tends to increase. 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 can
   operate 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 " perpetual motion" 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.

   In general, according to the second law, the entropy of a system that
   is not isolated may decrease. An air conditioner, for example, cools
   the air in a room, thus reducing the entropy of the air. The heat,
   however, involved in operating the air conditioner always makes a
   bigger contribution to the entropy of the environment than the decrease
   of the entropy of the air. Thus the total entropy of the room and the
   environment increases, in agreement with the second law.

Entropy balance equation for open systems

   In chemical engineering, the principles of thermodynamics are commonly
   applied to " open systems", i.e. those in which heat, work, and mass
   flow across the system boundary. In a system in which there are flows
   of both heat ( \dot{S} ) and work, i.e. \dot{W}_S (shaft work) and
   P(dV/dt) (pressure-volume work), across the system boundaries, the heat
   flow, but not the work flow, causes a change in the entropy of the
   system. This rate of entropy change is \dot{S}/T , where T is the
   absolute thermodynamic temperature of the system at the point of the
   heat flow. If, in addition, there are mass flows across the system
   boundaries, the total entropy of the system will also change due to
   this convected flow.
   During steady-state continuous operation, an entropy balance applied to
   an open system accounts for system entropy changes related to heat flow
   and mass flow across the system boundary.
   Enlarge
   During steady-state continuous operation, an entropy balance applied to
   an open system accounts for system entropy changes related to heat flow
   and mass flow across the system boundary.

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

          \frac{dS}{dt} = \sum_{k=1}^K \dot{M}_k \hat{S}_k +
          \frac{\dot{Q}}{T} + \dot{S}_{gen}

   where

          \sum_{k=1}^K \dot{M}_k \hat{S}_k = the net rate of entropy flow
          due to the flows of mass into and out of the system (where
          \hat{S} = entropy per unit mass).

          \frac{\partial \dot{Q}}{\partial T} = the rate of entropy flow
          due to the flow of heat across the system boundary.

          \dot{S}_{gen} = the rate of internal generation of entropy
          within the system.

   Note, also, that if there are multiple heat flows, the term \dot{Q}/T
   is to be replaced by \sum \dot{Q}_j/T_j , where \dot{Q}_j is the heat
   flow and T[j] is the temperature at the jth heat flow port into the
   system.

Standard textbook definitions

     * Entropy – energy broken down in irretrievable heat.
     * – Boltzmann's constant times the logarithm of a multiplicity; where
       the multiplicity of a macrostate is the number of microstates that
       correspond to the macrostate.
     * – the number of ways of arranging things in a system (times the
       Boltzmann's constant).
     * – a non-conserved thermodynamic state function, measured in terms
       of the number of microstates a system can assume, which corresponds
       to a degradation in usable energy.
     * – a direct measure of the randomness of a system.
     * – a measure of energy dispersal at a specific temperature.
     * – a measure of the partial loss of the ability of a system to
       perform work due to the effects of irreversibility.
     * – an index of the tendency of a system towards spontaneous change.
     * – a measure of the unavailability of a system’s energy to do work;
       also a measure of disorder; the higher the entropy the greater the
       disorder.
     * – a parameter representing the state of disorder of a system at the
       atomic, ionic, or molecular level.
     * – a measure of disorder in the universe or of the availability of
       the energy in a system to do work.

Approaches to understanding entropy

Order and disorder

   Entropy, historically, has often been associated with the amount of
   order, disorder, and or chaos in a thermodynamic system. The
   traditional definition 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, a number of authors,
   in recent years, 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, which is based on a
   combination of thermodynamics and information theory arguments.
   Landsberg 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 the following
   expression:

          Disorder=C_D/C_I\,

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

          Order=1-C_O/C_I\,

   In which C[D] is the "disorder" capacity of the system, which is the
   entropy of the parts contained in the permitted ensemble, C[I] is the
   "information" capacity of the system, an expression similar to
   Shannon's channel capacity, and C[O] is the "order" capacity of the
   system.

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 theory, entropy
   changes have been described in terms of the mixing or "spreading" of
   the total energy of each constituent of a system over its particular
   quantized energy levels.

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

Entropy and Information theory

   In information theory, entropy is the measure of the amount of
   information that is missing before reception and is sometimes referred
   to as Shannon entropy.. Shannon entropy is a very general concept which
   finds applications in information theory as well as thermodynamics. It
   was originally devised by Claude Shannon in 1948 to study the amount of
   information in a transmitted message. The definition of the information
   entropy is, however, very general, and is expressed in terms of a
   discrete set of probabilities p[i]. In the case of transmitted
   messages, these probabilities were the probabilities that a particular
   message was actually transmitted, and the entropy of the message system
   was a measure of how much information was in the 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 hotly debated topic. Many authors argue that there is a
   link between the two, while others will argue that they have absolutely
   nothing to do with each other.

   The expressions for the two entropies are very similar. The information
   entropy H for equal probabilities p[i] is:

          H=K\ln(1/p_i)\,

   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:

          S=k\ln(1/p)\,

   where p  is the probability of a system being in a particular
   microstate, given that it is in a particular macrostate, and k  is
   Boltzmann's constant. It can be seen that one may think of the
   thermodynamic entropy as Boltzmann's constant, divided by ln(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.

   The problem with linking thermodynamic entropy to information entropy
   is that the entire body of thermodynamics which deals with the physical
   nature of entropy is missing. The second law of thermodynamics which
   governs the behaviour of thermodynamic systems in equilibrium, and the
   first law which expresses heat energy as the product of temperature and
   entropy are physical concepts rather than informational concepts. If
   thermodynamic entropy is seen as including all of the physical dynamics
   of entropy as well as the equilibrium statistical aspects, then
   information entropy gives only part of the description of thermodynamic
   entropy. Some authors, like Tom Schneider, argue for dropping the word
   entropy for the H function of information theory and using Shannon's
   other term "uncertainty" instead.

Ice melting example

   The illustration for this article is a classic example in which entropy
   increases in a small 'universe', a thermodynamic system consisting of
   the 'surroundings' (the warm room) and 'system' (glass, ice, cold
   water). In this universe, some heat energy δQ from the warmer room
   surroundings (at 298 K or 25 C) will spread out to the cooler system of
   ice and water at its constant temperature T of 273 K (0 C), the melting
   temperature of ice. The entropy of the system will change by the amount
   dS = δQ/T, in this example δQ/273 K. (The heat δQ for this process is
   the energy required to change water from the solid state to the liquid
   state, and is called the enthalpy of fusion, i.e. the ΔH for ice
   fusion.) The entropy of the surroundings will change by an amount dS =
   -δQ/298 K. So in this example, the entropy of the system increases,
   whereas the entropy of the surroundings decreases.

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

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

Topics in entropy

Entropy and life

   For over a century and a half, beginning with Clausius' 1863 memoir "On
   the Concentration of Rays of Heat and Light, and on the Limits of its
   Action", much writing and research has been devoted to the relationship
   between thermodynamic entropy and the evolution of life. The argument
   that that life feeds on negative entropy or negentropy as put forth in
   the 1944 book What is Life? by physicist Erwin Schrödinger served as a
   further stimulus to this research. Recent writings have utilized the
   concept of Gibbs free energy to elaborate on this issue. In other
   cases, some creationists have argued that entropy rules out evolution.

   In the popular textbook 1982 textbook Principles of Biochemistry by
   noted American biochemist Albert Lehninger, for example, it is 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. In short, according to Lehninger,
   "living organisms preserve their internal order by taking from their
   surroundings free energy, in the form of nutrients or sunlight, and
   returning to their surroundings an equal amount of energy as heat and
   entropy."

   Evolution related definitions:
     * Negentropy - a shorthand colloquial phrase for negative entropy.
     * Ectropy - a measure of the tendency of a dynamical system to do
       useful work and grow more organized.
     * Syntropy - a tendency towards order and symmetrical combinations
       and designs of ever more advantageous and orderly patterns.
     * Extropy – a metaphorical term defining the extent of a living or
       organizational system's intelligence, functional order, vitality,
       energy, life, experience, and capacity and drive for improvement
       and growth.
     * Ecological entropy - a measure of biodiversity in the study of
       biological ecology.

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. Hence, from one perspective, entropy measurement is
   thought of as a kind of clock.

Entropy and cosmology

   We have previously mentioned that a finite 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. 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 generally increasing entropy,
   then - as Roger Penrose has pointed out - gravity plays an important
   role in the increase because gravity causes dispersed matter to
   accumulate into stars, which collapse eventually into black holes.
   Jacob Bekenstein and Stephen Hawking have shown that black holes have
   the maximum possible entropy of any object of equal size. This makes
   them likely end points of all entropy-increasing processes, if they are
   totally effective matter and energy traps. Hawking has, however,
   recently changed his stance on this aspect.

   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 the model of 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. Other complicating factors, such
   as the energy density of the vacuum and macroscopic quantum effects,
   are difficult to reconcile with thermodynamical models, making any
   predictions of large-scale thermodynamics extremely difficult.

Other relations

Generalized entropy

   Many generalizations of entropy have been studied, two of which,
   Tsallis and Rényi entropies, are widely used and the focus of active
   research.

   The Rényi entropy is an information measure for fractal systems.

          H_\alpha(X) = \frac{1}{1-\alpha}\log\Bigg(\sum_{i=1}^n
          p_i^\alpha\Bigg) .

   where α > 0 is the 'order' of the entropy, p[i] are the probabilities
   of {x[1], x[2] ... x[n]}. For α = 1 we recover the standard entropy
   form.

   The Tsallis entropy is employed in Tsallis statistics to study
   nonextensive thermodynamics.

          S_q(p) = {1 \over q - 1} \left( 1 - \sum_x p^q(x) \right).

   where p denotes the probability distribution of interest, and q is a
   real parameter that measures the non- extensitivity of the system of
   interest. In the limit as q → 1, we again recover the standard entropy.

Other mathematical definitions

     * Kolmogorov-Sinai entropy - a mathematical type of entropy in
       dynamical systems related to measures of partitions.
     * Topological entropy - a way of defining entropy in an iterated
       function map in ergodic theory.
     * Relative entropy - is a natural distance measure from a "true"
       probability distribution P to an arbitrary probability distribution
       Q.

Sociological definitions

   The concept of entropy has also entered the domain of sociology,
   generally as a metaphor for chaos, disorder or dissipation of energy,
   rather than as a direct measure of thermodynamic or information
   entropy:
     * Entropology – the study or discussion of entropy or the name
       sometimes given to thermodynamics without differential equations.
     * Psychological entropy - the distribution of energy in the psyche,
       which tends to seek equilibrium or balance among all the structures
       of the psyche.
     * Economic entropy – a quantitative measure of the irrevocable
       dissipation and degradation of natural materials and available
       energy with respect to economic activity.
     * Social entropy – a measure of social system structure, having both
       theoretical and statistical interpretations, i.e. society
       (macrosocietal variables) measured in terms of how the individual
       functions in society (microsocietal variables); also related to
       social equilibrium.
     * Corporate entropy - energy waste as red tape and business team
       inefficiency, i.e. energy lost to waste.

Quotes & humor

     * As stated by John von Neumann in conversation with Claude Shannon
       in 1949:


   Entropy

     Nobody knows what entropy really is, so in a debate you will always
                             have the advantage.


   Entropy

     * As stated by Frederic Keffer:


Entropy

       The future belongs to those who can manipulate entropy; those who
                understand but energy will be only accountants.


                                                                        Entropy

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