**decrease in entropy**- This is the spontaneous DECREASE in entropy. (tripod.com)
- I have heard people say that entropy can not decrease, but just look around and there are many examples of a spontaneous decrease in entropy. (tripod.com)
- The gravitational collapse to form planets and the sun represents a decrease in entropy. (tripod.com)
- A friend of mine always argued that 'we' are evidence of significant decrease in entropy. (physicsforums.com)
- If heat is transferred out the sign would be reversed giving a decrease in entropy of the system. (wikipedia.org)

**displaystyle**- In the mid-19th century, the change in entropy (ΔS) of a system undergoing a thermodynamically reversible process was defined by Rudolf Clausius as: Δ S = ∫ δ Q rev T , {\displaystyle \Delta S=\int {\frac {\delta Q_{\text{rev}}}{T}},} where T is the absolute temperature of the system, dividing an incremental reversible transfer of heat into that system (δQrev). (wikipedia.org)
- The rate of entropy production, denoted by S ˙ i {\displaystyle {\dot {S}}_{i}} , is a key element of the second law of thermodynamics for open inhomogeneous systems which reads d S d t = Σ k Q ˙ k T k + Σ k S ˙ k + Σ k S ˙ i k with S ˙ i k ≥ 0. (wikipedia.org)
- S ˙ i k {\displaystyle {\dot {S}}_{ik}} represents the entropy production rates due to internal processes. (wikipedia.org)
- The index i in S ˙ i k {\displaystyle {\dot {S}}_{ik}} refers to the fact that the entropy is produced due to irreversible processes. (wikipedia.org)
- Complete graphs on n {\displaystyle n} vertices have entropy lg n {\displaystyle \lg n} , where lg {\displaystyle \lg } is the binary logarithm. (wikipedia.org)
- Complete balanced k-partite graphs have entropy lg k {\displaystyle \lg k} where l g {\displaystyle lg} is the binary logarithm. (wikipedia.org)
- Complete bipartite graphs with n {\displaystyle n} vertices in one partition and m {\displaystyle m} in the other have entropy H ( n m + n ) {\displaystyle H\left({\frac {n}{m+n}}\right)} , where H {\displaystyle H} is the binary entropy function. (wikipedia.org)
- Here, we use properties of graph entropy to provide a simple proof that a complete graph G {\displaystyle G} on n {\displaystyle n} vertices cannot be expressed as the union of fewer than lg n {\displaystyle \lg n} bipartite graphs. (wikipedia.org)
- Therefore, the union of fewer than lg n {\displaystyle \lg n} bipartite graphs cannot have the same entropy as G {\displaystyle G} , so G {\displaystyle G} cannot be expressed as such a union. (wikipedia.org)
- Henceforth, we shall drop the subscript ρ {\displaystyle \rho } from the min entropy when it is obvious from the context on what state it is evaluated. (wikipedia.org)
- Because of this, the crystal is locked into a state with 2 N {\displaystyle 2^{N}} different corresponding microstates, giving a residual entropy of S = N k ln ( 2 ) {\displaystyle S=Nk\ln(2)} , rather than zero. (wikipedia.org)
- The Rényi entropy for any α ≥ 0 {\displaystyle \alpha \geq 0} is Schur concave. (wikipedia.org)
- Collision entropy, sometimes just called "Rényi entropy", refers to the case α = 2, H 2 ( X ) = − log ∑ i = 1 n p i 2 = − log P ( X = Y ) {\displaystyle H_{2}(X)=-\log \sum _{i=1}^{n}p_{i}^{2}=-\log P(X=Y)} where X and Y are independent and identically distributed. (wikipedia.org)

**approximate entropy**- Sample entropy (SampEn) is a modification of approximate entropy (ApEn), used for assessing the complexity of physiological time-series signals, diagnosing diseased states. (wikipedia.org)
- Like approximate entropy (ApEn), Sample entropy (SampEn) is a measure of complexity. (wikipedia.org)
- In statistics, an approximate entropy (ApEn) is a technique used to quantify the amount of regularity and the unpredictability of fluctuations over time-series data. (wikipedia.org)

**system's entropy**- No matter how large the number of particles is, at equilibrium there is an equal chance that the system's entropy will increase and decrease/ system will heat or cool. (physicsforums.com)
- The second law of thermodynamics states that an isolated system's entropy never decreases. (wikipedia.org)

**calculates the entropy**- The Java plugin I am going to present calculates the entropy of files according to the method presented on Ero Carrera's blog . (the-interweb.com)

**ApEn**- ApEn was developed by Steve M. Pincus to handle these limitations by modifying an exact regularity statistic, Kolmogorov-Sinai entropy. (wikipedia.org)

**Generalization**- Constantino Tsallis has proposed a nonextensive entropy (Tsallis entropy), which is a generalization of the traditional Boltzmann-Gibbs entropy. (wikipedia.org)
- As with the classical Shannon entropy and its quantum generalization, the von Neumann entropy, one can define a conditional versions of min entropy. (wikipedia.org)

**lower entropy**- It enters into econophysics both in an ontological way through the Second Law of Thermodynamics as this drives the world economy from its ecological foundations as solar energy passes through food chains in dissipative process of entropy rising and production fundamentally involving the replacement of lower entropy energy states with higher entropy ones. (springer.com)
- The second law of thermodynamics allows higher order (lower entropy) to appear locally, but it still disallows organized complexity. (talkorigins.org)

**relative entropy**- Related concepts such as joint, conditional differential entropy, and relative entropy are defined in a similar fashion. (wikipedia.org)

**logarithm**- The entropy H of a random variable X with possible outcomes of { X1 … X n } is the product of the probability of outcome i times the base 2 logarithm of one over the probability of outcome i . (cisco.com)
- The base 2 logarithm (also known as the binary logarithm ) is most often used to compute entropy because of its close connection with the binary numbering system, and this has become the standard unit of measure when discussing information entropy. (cisco.com)
- Specifically, assuming for simplicity that each of the microscopic configurations is equally probable, the entropy of the system is the natural logarithm of that number of configurations, multiplied by the Boltzmann constant kB. (wikipedia.org)
- The min entropy, in information theory, is the smallest of the Rényi family of entropies, corresponding to the most conservative way of measuring the unpredictability of a set of outcomes, as the negative logarithm of the probability of the most likely outcome. (wikipedia.org)
- The min entropy is never greater than the ordinary or Shannon entropy (which measures the average unpredictability of the outcomes) and that in turn is never greater than the Hartley or max entropy, defined as the logarithm of the number of outcomes. (wikipedia.org)
- In the limit for α → 0, the Rényi entropy is just the logarithm of the size of the support of X. The limit for α → 1 is the Shannon entropy. (wikipedia.org)
- As with its discrete analog, the units of differential entropy depend on the base of the logarithm, which is usually 2 (i.e., the units are bits). (wikipedia.org)

**quantify**- Entropy is a key component of information theory , a branch of mathematics designed to quantify information. (cisco.com)
- Entropy gives us a way to quantify the average number of bits needed to convey a message when considering all possible outcomes. (cisco.com)
- Entropies quantify the diversity, uncertainty, or randomness of a system. (wikipedia.org)

**decreases**- The entropy of the system remains the same or the entropy decreases. (tripod.com)
- The 2nd law is statistical in nature and so there can be small, local fluctuations where entropy momentarily decreases. (physicsforums.com)
- Another notion (not by myself) along these lines can be stated that if the entropy in one part of the universe decreases (say the Earth) the entropy in another part must increase. (physicsforums.com)

**Neumann**- Von Neumann told him he should call it the "entropy" because nobody really knows what entropy is (indeed it does appear to be a loaded term ) and thus Shannon would have an advantage when debating his burgeoning field with pundits. (cisco.com)
- These quantities can be seen as generalizations of the von Neumann entropy. (wikipedia.org)
- The smoothed entropies share many interesting properties with the von Neumann entropy. (wikipedia.org)

**Shannon**- In information theory, the Rényi entropy generalizes the Hartley entropy, the Shannon entropy, the collision entropy and the min entropy. (wikipedia.org)
- In particular, the min-entropy is never larger than the Shannon entropy. (wikipedia.org)
- merely having a large Shannon entropy does not suffice for this task. (wikipedia.org)
- Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Shannon to extend the idea of (Shannon) entropy, a measure of average surprisal of a random variable, to continuous probability distributions. (wikipedia.org)
- Unfortunately, Shannon did not derive this formula, and rather just assumed it was the correct continuous analogue of discrete entropy, but it is not. (wikipedia.org)

**theoretical compute**- In theoretical computer science, the min-entropy is used in the context of randomness extractors. (wikipedia.org)
- In mathematics and theoretical computer science, entropy compression is an information theoretic method for proving that a random process terminates, originally used by Robin Moser to prove an algorithmic version of the Lovász local lemma. (wikipedia.org)

**define**- Before we can adequately define entropy, we need to first define information . (cisco.com)

**probability**- A. Rényi, On measures of entropy and information, in Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics, and Probability 1960, Volume 1: Contributions to the Theory of Statistics , edited by J. Neyman (University of California Press, Berkeley, 1961). (springer.com)
- So that's what sets the probability of exciting the atom, the fact that the reservoir has a higher probability of being in a state of higher entropy. (physicsforums.com)
- As α approaches infinity, the Rényi entropy is increasingly determined by the events of highest probability. (wikipedia.org)
- One must take care in trying to apply properties of discrete entropy to differential entropy, since probability density functions can be greater than 1. (wikipedia.org)

**uncertainty**- Entropy is a measurement of uncertainty. (cisco.com)
- Alice has access to system A and Bob to system B. The conditional entropy measures the average uncertainty Bob has about Alice's state upon sampling from his own system. (wikipedia.org)

**thermodynamics**- Hitherto this has led to much confusion since the term entropy also plays a central role in many areas of science, including thermodynamics, heat, and energy. (cisco.com)
- In the meantime, Callisto reflects on the concept of entropy, on the possibility that the universe will ultimately suffer heat death and cease to act at all (the first sign of which is the constant thermometer reading outside his window), and on the possible implications of the laws of thermodynamics for social existence (this with some help from those investigators who had appropriated the term "entropy" from physics to information theory). (enotes.com)
- Entropy was discovered when it was noticed to be a quantity that behaves as a function of state, as a consequence of the second law of thermodynamics. (wikipedia.org)
- The absolute entropy (S rather than ΔS) was defined later, using either statistical mechanics or the third law of thermodynamics, an otherwise arbitrary additive constant is fixed such that the entropy of a pure substance at absolute zero is zero. (wikipedia.org)
- Social entropy is a sociological theory that evaluates social behaviours using a method based on the second law of thermodynamics. (wikipedia.org)
- describes software entropy as follows: The second law of thermodynamics, in principle, states that a closed system's disorder cannot be reduced, it can only remain unchanged or increase. (wikipedia.org)
- This term is used in condensed matter physics to describe the entropy at zero kelvin of a glass or plastic crystal referred to the crystal state, whose entropy is zero according to the third law of thermodynamics. (wikipedia.org)

**inequality**- Katok's entropy inequality was recently exploited to obtain a tight asymptotic bound for the systolic ratio of surfaces of large genus, see systoles of surfaces. (wikipedia.org)

**consequence**- The story ends with the death of the bird and with Aubade's breaking the glass that separates her and Callisto from the cold outside, as the two of them await the equilibrating of the temperatures between outside and inside, the ultimate consequence, for them at least, of the principle of entropy. (enotes.com)

**irreversible processes**- However, irreversible processes increase the combined entropy of the system and its environment. (wikipedia.org)
- Entropy is produced in irreversible processes. (wikipedia.org)
- The importance of avoiding irreversible processes (hence reducing the entropy production) was recognized as early as 1824 by Carnot. (wikipedia.org)

**randomness**- It is often said that entropy is an expression of the disorder, or randomness of a system, or of our lack of information about it. (wikipedia.org)
- The existence of these multiple configurations (choices for each H of orientation along O--O axis) that meet the rules of absolute zero (2-in 2-out for each O) amounts to randomness, or in other words, entropy. (wikipedia.org)

**calculation**- Finally, we will cover some practical applications of entropy calculation and analysis. (cisco.com)
- However, accurate entropy calculation requires vast amounts of data, and the results will be greatly influenced by system noise, therefore it is not practical to apply these methods to experimental data. (wikipedia.org)

**macroscopic**- In statistical mechanics, entropy (usual symbol S) is related to the number of microscopic configurations Ω that a thermodynamic system can have when in a state as specified by some macroscopic variables. (wikipedia.org)
- Entropy may be understood as a measure of disorder within a macroscopic system. (wikipedia.org)
- The above definition is sometimes called the macroscopic definition of entropy because it can be used without regard to any microscopic description of the contents of a system. (wikipedia.org)
- These have residual entropy, because the atom-by-atom microscopic structure can be arranged in a huge number of different ways across a macroscopic system. (wikipedia.org)

**conditional**- The conditional quantum min entropy is a one-shot, or conservative, analog of conditional quantum entropy. (wikipedia.org)

**asymptotic**- 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. (wikipedia.org)
- The volume entropy (or asymptotic volume growth) is defined as the limit where B(R) is the ball of radius R in centered at and vol is the Riemannian volume in the universal cover with the natural Riemannian metric. (wikipedia.org)
- Sabourau, S.: Entropy of systolically extremal surfaces and asymptotic bounds. (wikipedia.org)

**Physics**- Entropy is a central concept of statistical mechanics, which is the main branch of physics that underlies econophysics, the application of physics concepts to understand economic phenomena. (springer.com)

**invariant**- Differential entropy is translation invariant, i.e., h(X + c) = h(X) for a constant c. (wikipedia.org)
- Differential entropy is in general not invariant under arbitrary invertible maps. (wikipedia.org)
- However, differential entropy does not have other desirable properties: It is not invariant under change of variables, and is therefore most useful with dimensionless variables. (wikipedia.org)

**microscopic**- In the modern microscopic interpretation of entropy in statistical mechanics, entropy is the amount of additional information needed to specify the exact physical state of a system, given its thermodynamic specification. (wikipedia.org)

**theory**- In this blog post you will learn about entropy within the domain of information theory. (cisco.com)
- In contrast the mathematics of entropy as appearing in information theory becomes the basis for modeling financial market dynamics as well as income and wealth distribution dynamics. (springer.com)
- This work will be of value to physicists, chemists, biologists and engineers interested in the theory and applications of entropy and its generation. (springer.com)
- Entropy is also the term used for lost information in information theory. (conservapedia.com)
- The rationale behind the theory is that Gibbs-Boltzmann entropy leads to systems that have a strong dependence on initial conditions. (wikipedia.org)
- Social Entropy Theory. (wikipedia.org)
- In information theory, the graph entropy is a measure of the information rate achievable by communicating symbols over a channel in which certain pairs of values may be confused. (wikipedia.org)
- This concept is closely related with other notions of entropy found in dynamical systems and plays an important role in differential geometry and geometric group theory. (wikipedia.org)

**context**- In this context the term "entropy" was first proposed in 1865, by Rudolf Clausius . (conservapedia.com)
- In the context of fractal dimension estimation, the Rényi entropy forms the basis of the concept of generalized dimensions. (wikipedia.org)

**statistical**- Maximum entropy is a powerful method for constructing statistical models of classification tasks, such as part of speech tagging in Natural Language Processing. (sourceforge.net)
- The second law is now often seen as an expression of the fundamental postulate of statistical mechanics through the modern definition of entropy. (wikipedia.org)
- Nonextensive entropy leads to nonextensive statistical mechanics, whose typical functions are power laws, instead of the traditional exponentials. (wikipedia.org)

**increases**- This argument is even more controversial as the implication is that the state of entropy in the universe is somewhat constant, and when entropy increases in one part of the universe, it must decrease in another part. (physicsforums.com)
- Non-isolated systems may lose entropy, provided their environment's entropy increases by at least that amount. (wikipedia.org)

**statistic**- The method getStatsDescription() returns the description of the file statistic as displayed in the tab header of the File Statistics dialog ('Entropy' in this case). (the-interweb.com)

**system**- Once you have given the box a good shake, you have pretty much maximized the entropy of the system. (tripod.com)
- Note that the collapse of the proto-solar system did not cause an increase in the entropy of some larger system as we would find in the classic refrigerator example. (tripod.com)
- The entropy inside the icebox is lower than that of the surrounding kitchen, but in running the motors of the fridge we have heated up the kitchen and the entropy of a larger system was increased in order to lower the entropy around our cold cuts and beer. (tripod.com)
- Entropy CAN in fact be reduced in a closed system, and this happens spontaneously according to the fluctuation theorem. (physicsforums.com)
- If your system includes the particle + the rest of the liquid, then in this closed system the total entropy would not decrease. (physicsforums.com)
- In either case, entropy of a closed system does not decrease. (physicsforums.com)
- Since entropy is a function of the state of the system, a change in entropy of a system is determined by its initial and final states. (wikipedia.org)
- Understanding the role of thermodynamic entropy in various processes requires an understanding of how and why that information changes as the system evolves from its initial to its final state. (wikipedia.org)
- The equivalent of entropy in a social system is considered to be wealth or residence location. (wikipedia.org)

**Thus**- Thus systems that can take multiple configurations at or near absolute zero are said to have residual entropy. (wikipedia.org)
- A great deal of research has thus been undertaken into finding other systems that exhibit residual entropy. (wikipedia.org)
- This material is thus analogous to water ice, with the exception that the spins on the corners of the tetrahedra can point into or out of the tetrahedra, thereby producing the same 2-in, 2-out rule as in water ice, and therefore the same residual entropy. (wikipedia.org)
- Thus, differential entropy does not share all properties of discrete entropy. (wikipedia.org)

**smallest**- The name min-entropy stems from the fact that it is the smallest entropy measure in the family of Rényi entropies. (wikipedia.org)

**measure**- A measure of this disorder is entropy. (wikipedia.org)
- The various Rényi entropies are all equal for a uniform distribution, but measure the unpredictability of a nonuniform distribution in different ways. (wikipedia.org)
- The Rényi entropy is also important in quantum information, where it can be used as a measure of entanglement. (wikipedia.org)

**concept**- Erwin Schrodinger applied the concept of entropy to biology in 1944, concluding that hereditary carriers had to be small in size and permanent in form. (conservapedia.com)
- The concept of entropy has been found to be generally useful and has several other formulations. (wikipedia.org)
- In 1865 Rudolf Clausius expanded his previous work from 1854 on the concept of "unkompensierte Verwandlungen" (uncompensated transformations), which, in our modern nomenclature, would be called the entropy production. (wikipedia.org)

**state**- The all heads condition is just one possible state for the entropy machine of 2^100 possible states. (tripod.com)
- In the state of maximum entropy we would expect to find about 50 cents showing heads and fifty cents showing tails. (tripod.com)
- Once you have achieved this state, entropy can not increase any further so it must stay the same or decrease. (tripod.com)
- Such systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. (wikipedia.org)
- The min entropy can be interpreted as the distance of a state from a maximally entangled state. (wikipedia.org)
- Residual entropy is the difference in entropy between a non-equilibrium state and crystal state of a substance close to absolute zero. (wikipedia.org)
- Entropy monitors produce two numbers (RE - Response Entropy, SE - State Entropy) that are related to frequency bandpass used. (wikipedia.org)
- The reason for using higher frequency bandpass in response entropy is to allow faster response from the monitor in relation to clinical state. (wikipedia.org)

**discrete**- The actual continuous version of discrete entropy is the limiting density of discrete points (LDDP). (wikipedia.org)
- Differential entropy (described here) is commonly encountered in the literature, but it is a limiting case of the LDDP, and one that loses its fundamental association with discrete entropy. (wikipedia.org)
- For the direct analogue of discrete entropy extended to the continuous space, see limiting density of discrete points. (wikipedia.org)

**second**- Entropy" was the second professional story published by Pynchon, and this comic but grim tale established one of the dominant themes of his entire body of work. (enotes.com)

**locally**- It is of considerable interest in differential geometry to find the Riemannian metric on a given smooth manifold which minimizes the volume entropy, with locally symmetric spaces forming a basic class of examples. (wikipedia.org)
- Locally symmetric spaces minimize entropy when the volume is prescribed. (wikipedia.org)
- French) [Entropy and rigidity of locally symmetric spaces with strictly negative curvature] Geom. (wikipedia.org)

**outcome**- Entropy, which is typically measured in bits, is actually a measurement of how much information is NOT available when we don't know the outcome of an information source. (cisco.com)

**example**- For example, the entropy of a coin flip with 2 equally likely outcomes is less than that of the roll of die with 6 equally likely outcomes. (cisco.com)

**higher**- The entropy was higher when it was a cloud of gas. (tripod.com)
- Response Entropy incorporates higher frequency components that include that of electromyogram activity. (wikipedia.org)

**thermal**- The entropy does decrease because high entropy energy (thermal energy of liquid) becomes transformed into low entropy energy (gravitational potential of particle). (physicsforums.com)
- Entropy production determines the performance of thermal machines such as power plants, heat engines, refrigerators, heat pumps, and air conditioners. (wikipedia.org)

**fundamental**- Today much progress is being made in our understanding of entropy and entropy generation in both fundamental aspects and application to concrete problems. (springer.com)

**systems**- You have probably heard that the main statement of entropy is that systems always tend to become more disordered. (tripod.com)
- S. Thurner, R. Hanel, The entropy of non-ergodic complex systems- a derivation from first principles, Int. J. Mod. (springer.com)
- Geometrically frustrated systems in particular often exhibit residual entropy. (wikipedia.org)

**case**- This is because as anaesthesia "deepens", there are predictable changes in the EEG including slowing, synchronicity, and burst suppression, that, in the case of BIS or Entropy, are converted to a number roughly correlated to the likelihood of anaesthesia awareness. (wikipedia.org)

**various**- It also provides the basis for an alternative view of stochastic price equilibria in economics, as well providing a crucial link between econophysics and sociophysics, keeping in mind the essential unity of the various concepts of entropy. (springer.com)
- Regularity was originally measured by exact regularity statistics, which has mainly centered on various entropy measures. (wikipedia.org)

**energy**- Boltzmann's constant, and therefore entropy, have dimensions of energy divided by temperature. (wikipedia.org)

**spin**- One of the interesting properties of geometrically frustrated magnetic materials such as spin ice is that the level of residual entropy can be controlled by the application of an external magnetic field. (wikipedia.org)
- In the Heisenberg XY spin chain model, the Rényi entropy as a function of α can be calculated explicitly by virtue of the fact that it is an automorphic function with respect to a particular subgroup of the modular group. (wikipedia.org)

**applications**- Mathematical Foundations and Applications of Graph Entropy. (wikipedia.org)

**below**- The screenshot below shows the entropy distribution of Notepad.exe. (the-interweb.com)
- Shannon's classic formula for computing entropy is shown below. (cisco.com)
- This is consistent with 19th-century formulas for entropy in terms of heat and temperature, as discussed below. (wikipedia.org)

**important**- It is important to make the distinction that entropy is not information. (cisco.com)
- The purpose of this volume is to present some of these recent and important results in a manner that not only appeals to the entropy specialist but also makes them accessible to the nonspecialist looking for an overview of the field. (springer.com)
- The Rényi entropy is important in ecology and statistics as index of diversity. (wikipedia.org)

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