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Entropy02:39

Entropy

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Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
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Entropy01:18

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The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...
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Entropy and the Second Law of Thermodynamics01:20

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The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
The relation  between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...
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A pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K) may be described by a single microstate, as its purity, perfect crystallinity,and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.
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Probability Distributions01:32

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 The probability of a random variable x  is the likelihood of its occurrence. A probability distribution represents the probabilities of a random variable using a formula, graph, or table. There are two types of probability distribution– discrete probability distribution and continuous probability distribution.
A discrete probability distribution is a probability distribution of discrete random variables. It can be categorized into binomial probability distribution and Poisson...
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The Second Law of Thermodynamics01:14

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In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Scientists refer to the measure of randomness or disorder within a system as entropy. High entropy means high disorder and low energy. To better understand entropy, think of a student’s bedroom. If no energy or work were put into it, the room would quickly become messy. It would exist in a very disordered state, one of high entropy. Energy must be...
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Minimum and Maximum Entropy Distributions for Binary Systems with Known Means and Pairwise Correlations.

Badr F Albanna1,2,3, Christopher Hillar3,4, Jascha Sohl-Dickstein3,5,6,7

  • 1Department of Natural Sciences, Fordham University, New York, NY 10023, USA.

Entropy (Basel, Switzerland)
|February 4, 2021
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Summary

Maximum entropy models explore neural population activity. This study reveals minimum entropy distributions, showing logarithmic entropy scaling with system size and implications for information transmission.

Keywords:
Ising modelcompressed sensinginformation theorymaximum entropyminimum entropyneural networkspairwise correlationsstatistical mechanics

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Area of Science:

  • Computational neuroscience
  • Statistical physics
  • Information theory

Background:

  • Maximum entropy models are widely used for neural population activity analysis.
  • Existing models often overlook the full spectrum of probability distributions.
  • Understanding entropy bounds is crucial for characterizing neural coding.

Purpose of the Study:

  • To explore the space of probability distributions for neural populations.
  • To derive entropy bounds for minimum entropy distributions.
  • To investigate the relationship between system size and entropy.

Main Methods:

  • Derivation of upper and lower bounds on entropy.
  • Construction of specific low-entropy distributions.
  • Analysis of entropy scaling with system size.

Main Results:

  • Minimum entropy distributions exhibit logarithmic scaling with system size.
  • Certain low-order statistics are only realizable in small systems.
  • Small amounts of randomness can mimic high-entropy distribution properties.

Conclusions:

  • Provides a theoretical framework for minimum entropy distributions in neural systems.
  • Highlights the surprising efficiency of low-entropy states in mimicking complex statistics.
  • Suggests implications for biological and engineered information transmission systems.