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The Product Rule

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In calculus, the Product Rule provides a method for differentiating expressions that are the product of two functions. It states that the derivative of the product of two differentiable functions equals the first function times the rate of change of the second, plus the second function times the rate of change of the first.This rule ensures that the rate of change of the product accounts for the simultaneous variation of both functions.A compelling way to understand the Product Rule is through...

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Concentration and Rate Law

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The rate of a reaction is affected by the concentrations of reactants. Rate laws (differential rate laws) or rate equations are mathematical expressions describing the relationship between the rate of a chemical reaction and the concentration of its reactants.
For example, in a generic reaction aA + bB ⟶ products, where a and b are stoichiometric coefficients, the rate law can be written as:

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Related Experiment Video

Updated: May 29, 2026

Research and Development of High-performance Explosives

Research and Development of High-performance Explosives

Published on: February 20, 2016

Explosive site percolation with a product rule.

Woosik Choi¹, Soon-Hyung Yook, Yup Kim

¹Department of Physics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, Korea.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|September 21, 2011

Summary

This study investigates site percolation using the Achlioptas process and a product rule on a 2D lattice. Results reveal a discontinuous phase transition, distinct from sum rule transitions.

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Area of Science:

Statistical Physics
Complex Systems
Network Science

Background:

Percolation theory models connectivity in random networks.
Achlioptas processes introduce dynamic rules for network growth.
Understanding phase transitions is crucial for complex system analysis.

Purpose of the Study:

To analyze site percolation using the Achlioptas process with a product rule.
To determine the nature of the phase transition in a 2D square lattice.
To compare the product rule transition with sum rule transitions.

Main Methods:

Site percolation simulation on a 2D square lattice.
Analysis of cluster size distribution P(s).
Examination of order parameter hysteresis loops.

Main Results:

A robust power-law regime followed by a stable hump in P(s) was observed.
The cluster size distribution indicates a discontinuous phase transition.
Hysteresis in the order parameter confirms the discontinuous nature of the transition.
The product rule transition differs from the sum rule transition in 2D.

Conclusions:

The Achlioptas process with a product rule induces a discontinuous phase transition in 2D site percolation.
This transition behavior is distinct from that observed with sum rules.
The findings contribute to understanding phase transitions in complex network models.