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Entropy unit8/30/2023 ![]() ![]() When this condition does not occur, we can increase efficiency by extending the source, using a binary Shannon-Fano code. If, for each symbol, it is verified that, then we can prove that and. This ensures the uniqueness of the decoding. If codeword lengths satisfy the Kraft-McMillan inequality, then we can construct a prefix code with these codeword lengths. In this context, an important result is Kraft-McMillan’s inequality: For example, if we use variable length coding, we do not have a prefix code, because the codeword of is the prefix code of the codeword for (substring 11). An example of this type of encoding is the prefix code.Ī prefix code is a code where no codeword is the prefix of any other codeword. The value of is the limit to aim for when developing a compression algorithm.Īn intuitive result is that if any encoded string has only one possible source string producing it, then we have unique decodability. Note the similarity between the maximum thermodynamic entropy and the equality of physical properties in all points of the system, and the maximum information entropy that derives from equality in probabilities. This conjecture is known as heat death or entropic death of the universe. When the maximum entropy is reached, there will no longer be any gradient of energy that will allow any spontaneous process. The universe is an adiabatic and isolated system. The maximum entropy corresponds to the thermodynamical equilibrium. In isolated systems, the processes leading to an increase in entropy are spontaneous. The result is the zeroing of the gradient of some physical observable. The second principle often manifests by the establishment of physical processes that try to equal some property in the systems. If the two bodies have different masses, they will have different amounts of energy at the end of the process, but the energy per unit of volume will be the same. In this case, there will be a heat flow between the two bodies to equal their temperatures, which we can consider a measure of the concentration or density of energy. If we inject gas into a container full of air and wait for sufficient time, we can observe that the gas will spontaneously diffuse into the air until it reaches the same concentration at all points.Īnother example is the contact between two bodies at different temperatures. This fact has important macroscopic consequences. The result is also valid for irreversible processes in adiabatic systems, in which there is no heat transfer with the outside. We know that, in an isolated system, the disorder or entropy increases with each physical irreversible process until it reaches a maximum. If a Markov process leads to a statistic that is independent of the sample when the number of events is large, then we have an ergodic process.Įntropy is, therefore, a measure of uncertainty, surprise, or information related to a choice between a certain number of possibilities when we consider ergodic processes. ![]() When the choice of symbols of a stochastic process depends on the symbols or events previously chosen, we have a Markov process. When we have processes in which we choose symbols by a set of probabilities, we deal with stochastic processes. The probability that after a there is a vowel is much higher than the probability that there is an, for example. Think of a message written in English, in which we compose the words with the symbols of the usual alphabet. In real processes, the probability of choosing a symbol is not independent of previous choices. We compose a message choosing among possible symbols in an alphabet. The symbol sequences of a message have, in general, different probabilities. ![]()
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