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Related Concept Videos

Phase Transitions02:31

Phase Transitions

Whether solid, liquid, or gas, a substance's state depends on the order and arrangement of its particles (atoms, molecules, or ions). Particles in the solid pack closely together, generally in a pattern. The particles vibrate about their fixed positions but do not move or squeeze past their neighbors. In liquids, although the particles are closely spaced, they are randomly arranged. The position of the particles are not fixed—that is, they are free to move past their neighbors to occupy...
Phase Transitions01:21

Phase Transitions

A phase transition is the process in which a substance changes from one state of matter to another, like from a solid to a liquid, liquid to gas, or vice versa, at a specific temperature and under given pressure conditions. This change is spontaneous and is affected by alterations in temperature and pressure. These parameters impact the strength of the forces between molecules (intermolecular forces) in the substance.During a phase transition, both the initial and final phases of the substance...
Basic Discrete Time Signals01:16

Basic Discrete Time Signals

The unit step sequence is defined as 1 for zero and positive values of the integer n. This sequence can be graphically displayed using a set of eight sample points, showing a step function starting from n=0 and remaining constant thereafter.
The unit impulse or sample sequence is mathematically expressed as zero for all n values except at n=0, where it is one. The unit impulse sequence, denoted by δ(n), is the first difference of the unit step sequence, while the unit step sequence u(n) is the...
Binomial Expansion Using Pascal's Triangle01:30

Binomial Expansion Using Pascal's Triangle

Expanding a binomial expression such as (a + b)n results in a predictable sequence of terms that can be systematically derived using Pascal’s Triangle. This triangular array of numbers plays a central role in understanding and computing the coefficients of binomial expansions.Pascal’s Triangle is constructed such that each row corresponds to the coefficients of a binomial raised to a power. The topmost row, known as the zeroth row, corresponds to (a + b)0, and each successive row gives the...
Transition State Theory01:25

Transition State Theory

Transition-state theory, also known as activated-complex theory, provides a molecular-level explanation of reaction rates in both gas-phase and solution-phase reactions. It extends earlier kinetic models by considering the formation of a short-lived, high-energy configuration during a reaction.The progress of a chemical reaction can be represented using a reaction profile, which plots potential energy against the reaction coordinate. As two reactant molecules approach one another, their...
Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.

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Related Experiment Video

Updated: Jul 13, 2026

A Tactile Automated Passive-Finger Stimulator (TAPS)
19:44

A Tactile Automated Passive-Finger Stimulator (TAPS)

Published on: June 3, 2009

Phase transition in a stochastic prime-number generator.

Bartolo Luque1, Lucas Lacasa, Octavio Miramontes

  • 1Departamento de Matemática Aplicada y Estadística, ETSI Aeronáuticos, Universidad Politécnica de Madrid, Madrid 28040, Spain.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 7, 2007
PubMed
Summary

This study introduces a novel stochastic algorithm for prime number generation. The algorithm exhibits a phase transition, distinguishing between effective prime reduction and a low-density frozen state, with critical slowing-down observed.

Related Experiment Videos

Last Updated: Jul 13, 2026

A Tactile Automated Passive-Finger Stimulator (TAPS)
19:44

A Tactile Automated Passive-Finger Stimulator (TAPS)

Published on: June 3, 2009

Area of Science:

  • Number theory
  • Computational mathematics
  • Statistical physics

Background:

  • Prime numbers are fundamental in number theory and cryptography.
  • Generating large primes efficiently is a computationally challenging problem.
  • Stochastic algorithms offer alternative approaches to deterministic methods.

Purpose of the Study:

  • To introduce a new stochastic algorithm for prime number generation.
  • To investigate the phase transition dynamics of the algorithm.
  • To analyze the algorithm's performance and behavior.

Main Methods:

  • Development of a novel stochastic algorithm.
  • Numerical simulations to observe algorithm dynamics.
  • Analytical approach using annealed approximation for data collapse.
  • Identification of critical slowing-down phenomena.

Main Results:

  • The algorithm successfully generates prime numbers.
  • A continuous phase transition was identified, separating two distinct phases.
  • One phase allows for efficient integer-to-prime reduction, while the other leads to a low prime density 'frozen state'.
  • Annealed approximation effectively collapses simulation data.
  • Evidence of critical slowing-down near the phase transition.

Conclusions:

  • The stochastic algorithm presents a new method for prime number generation.
  • The observed phase transition is a key characteristic of the algorithm's dynamics.
  • The findings provide insights into the behavior of prime-generating stochastic systems.
  • Further research can explore optimizations and applications of this algorithm.