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Related Concept Videos

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.
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A probability histogram is a visual representation of a probability distribution. Similar a typical histogram, the probability histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents. The vertical axis is labeled with probability. Each rectangular bar in the histogram is 1 unit wide, which suggests that the area under each bar equals the probability, P(x), where x is 1, 2, 3, and so on.
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A Poisson probability distribution is a discrete probability distribution. It gives the probability of a number of events occurring in a fixed interval of time or space if these events happen at a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on average, there are five words spelled incorrectly in 100 pages. The interval is 100 pages.
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To understand intra-specific interactions in populations, scientists measure the spatial arrangement of species individuals. This geographic arrangement is known as the species distribution or dispersion. Highly territorial species exhibit a uniform distribution pattern, in which individuals are spaced at relatively equal distances from one another. Species that are highly tied to particular resources, such as food or shelter, tend to concentrate around those resources, and thus exhibit a...
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In concrete, the pore size distribution significantly influences the material's properties. Capillary pores, markedly larger than gel pores, form a vast network within partially hydrated cement paste, reducing the concrete's strength and increasing its permeability. This heightened permeability leads to a greater risk of damage from environmental factors like freeze-thaw cycles and chemical attacks, with the extent of vulnerability also being tied to the water-to-cement ratio.
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The distribution law or Nernst's distribution law is the law that governs the distribution of a solute between two immiscible solvents. This law, also known as the partition law, states that if a solute is added to the mixture of two immiscible solvents at a constant temperature, the solute is distributed between the two solvents in such a way that the ratio of solute concentrations in the solvents remains constant at equilibrium.
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Related Experiment Video

Updated: Jan 13, 2026

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
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Distribution of the Number of Paths in Two-Dimensional Directed Percolation.

Leon Seeger1, Alexander K Hartmann1

  • 1Institut for Physics, University of Oldenburg, 26111 Oldenburg, Germany.

Entropy (Basel, Switzerland)
|October 28, 2025
PubMed
Summary

This study numerically investigates directed percolating paths in 2D diluted systems. We analyzed path distribution and entropy, revealing insights into system behavior across different phases.

Keywords:
Monte Carlo algorithmsbiaslarge deviationslatticepathspercolationsimulations

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Percolation Theory

Background:

  • Percolating systems exhibit numerous spanning paths.
  • Understanding path distribution is crucial for characterizing system properties.

Purpose of the Study:

  • Numerically investigate the number of directed percolating paths in 2D diluted systems.
  • Analyze the average entropy and distribution of these paths.
  • Explore system behavior in percolating, non-percolating, and critical phases.

Main Methods:

  • Numerical simulations on L×L diluted lattices.
  • Calculation of average entropy (⟨S⟩ = ⟨logN⟩) as a function of occupation density (p).
  • Large-deviation approaches to determine the path distribution P(S) for low probabilities.

Main Results:

  • Comparison of calculated average entropy with existing mathematical results.
  • Detailed analysis of the path distribution P(S) across various system phases.
  • Characterization of path structures for specific entropy and density values.

Conclusions:

  • The study provides numerical insights into directed percolating path statistics.
  • Large-deviation techniques enable exploration of rare path configurations.
  • Findings contribute to understanding complex system behavior near critical points.