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Related Concept Videos

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.
Propagation of Uncertainty from Random Error00:59

Propagation of Uncertainty from Random Error

An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...
Entropy Changes Accompanying Specific Processes01:21

Entropy Changes Accompanying Specific Processes

Entropy, a measure of disorder in a system, changes during phase transitions like freezing or boiling. At the transition temperature Ttrs, where two phases are in equilibrium, the phase transition is a reversible process. The entropy change can be calculated from a substance's enthalpy of transition using the equation ΔStrs = ΔtrsH /Ttrs.When a perfect gas expands isothermally from one volume to another, entropy increases logarithmically with volume. Conversely, isothermal compression results...
Distribution of Molecular Speeds01:27

Distribution of Molecular Speeds

The motion of molecules in a gas is random in magnitude and direction for individual molecules, but a gas of many molecules has a predictable distribution of molecular speeds. This predictable distribution of molecular speeds is known as the Maxwell-Boltzmann distribution. The distribution of molecular speeds in liquids is comparable to that of gases but not identical and can help to understand the phenomenon of the boiling and vapor pressure of a liquid. Consider that a molecule requires a...
Propagation of Uncertainty from Systematic Error01:10

Propagation of Uncertainty from Systematic Error

The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this particular...
Mutation, Gene Flow, and Genetic Drift01:09

Mutation, Gene Flow, and Genetic Drift

In a population that is not at Hardy-Weinberg equilibrium, the frequency of alleles changes over time. Therefore, any deviations from the five conditions of Hardy-Weinberg equilibrium can alter the genetic variation of a given population. Conditions that change the genetic variability of a population include mutations, natural selection, non-random mating, gene flow, and genetic drift (small population size).

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Related Experiment Video

Updated: May 26, 2026

Sealable Femtoliter Chamber Arrays for Cell-free Biology
13:44

Sealable Femtoliter Chamber Arrays for Cell-free Biology

Published on: March 11, 2015

Noise can speed convergence in Markov chains.

Brandon Franzke1, Bart Kosko

  • 1Center for Quantum Information Science and Technology, Signal and Image Processing Institute, Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 21, 2011
PubMed
Summary

Noise can accelerate convergence to equilibrium in discrete finite-state Markov chains by enhancing state exploration. This finding is demonstrated through a new theorem and simulations across various models, showing significant speedups.

Related Experiment Videos

Last Updated: May 26, 2026

Sealable Femtoliter Chamber Arrays for Cell-free Biology
13:44

Sealable Femtoliter Chamber Arrays for Cell-free Biology

Published on: March 11, 2015

Area of Science:

  • Mathematical Physics
  • Computational Science
  • Stochastic Processes

Background:

  • Markov chains are fundamental models in various scientific disciplines.
  • Understanding and accelerating their convergence to equilibrium is crucial for accurate predictions.
  • The role of noise in such systems has been a topic of ongoing research.

Purpose of the Study:

  • To introduce a new theorem demonstrating that noise can accelerate convergence to equilibrium in discrete finite-state Markov chains.
  • To explore the conditions under which noise provides a benefit, specifically relating to vector-norm inequalities.
  • To develop and evaluate algorithms for harnessing noise-induced benefits.

Main Methods:

  • Development of a new theorem establishing conditions for noise-induced convergence benefits in Markov chains.
  • Formulation of a noise-benefit algorithm requiring steady-state knowledge and a blind alternative using historical data.
  • Empirical validation through simulations on three established Markov models: Ehrenfest diffusion, Wright-Fisher, and zeolite crystallization.

Main Results:

  • A theorem proves that noise, applied to state density, can speed up convergence by aiding exploration of improbable states.
  • Noise benefits are guaranteed for states satisfying specific vector-norm inequalities, leading to reduced norm components.
  • Simulations showed convergence rate increases of up to 64% for states meeting the theorem's criteria and 53% for those meeting the corollary's criteria.

Conclusions:

  • Noise can be a powerful tool for accelerating convergence in Markov chains, particularly those with slow convergence or weak absorbing states.
  • The developed algorithms, both knowledge-based and blind, effectively leverage noise for improved performance.
  • The findings have broad implications for fields utilizing Markov chain modeling, from physics to population genetics and chemical reactions.