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Related Concept Videos

Entropy02:39

Entropy

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...
Entropy01:18

Entropy

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...
Entropy and the Second Law of Thermodynamics01:26

Entropy and the Second Law of Thermodynamics

Consider an isolated system in which a hot object is placed in contact with a cold one. This is an irreversible process that eventually leads both objects to reach the same equilibrium temperature. It is crucial to note that the constituents of any substance exhibit increased disorder at higher temperatures. As a cold substance absorbs heat, its constituents become more disordered. The energy transfer from a hotter object to a cooler one increases the system's disorder or randomness. This...
Entropy and the Second Law of Thermodynamics01:20

Entropy and the Second Law of Thermodynamics

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...
Random Error01:04

Random Error

Random or indeterminate errors originate from various uncontrollable variables, such as variations in environmental conditions, instrument imperfections, or the inherent variability of the phenomena being measured. Usually, these errors cannot be predicted, estimated, or characterized because their direction and magnitude often vary in magnitude and direction even during consecutive measurements. As a result, they are difficult to eliminate. However, the aggregate effect of these errors can be...
The Second Law of Thermodynamics01:14

The Second Law of Thermodynamics

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 put...

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Sealable Femtoliter Chamber Arrays for Cell-free Biology
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Ergodic theory, randomness, and "chaos".

D S Ornstein

    Science (New York, N.Y.)
    |January 13, 1989
    PubMed
    Summary

    Ergodic theory reveals deterministic chaos in systems governed by Newton's laws, linking them to the baker's transformation. This provides a new framework for understanding random behavior and statistical properties.

    Area of Science:

    • Dynamical Systems
    • Ergodic Theory
    • Chaos Theory

    Background:

    • Ergodic theory studies the long-term statistical behavior of dynamical systems.
    • The baker's transformation is a key model in ergodic theory, illustrating deterministic chaos.
    • Previous understanding suggested an analogy between the baker's transformation and chaotic systems.

    Purpose of the Study:

    • To demonstrate a fundamental connection between systems governed by Newton's laws and the baker's transformation.
    • To organize and understand the diverse types of random behavior in dynamical systems.
    • To establish a statistical analog of structural stability in chaotic systems.

    Main Methods:

    • Abstract mathematical analysis of dynamical systems.
    • Formalizing the baker's transformation as a model for Newtonian systems.

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  • Investigating the statistical properties and behavior of these systems.

    • Main Results:

    • A deep connection, beyond analogy, is established between Newtonian systems and the baker's transformation at an abstract level.
    • A framework is developed to categorize and understand various forms of random behavior.
    • Concrete results include demonstrating that Newtonian and coin-tossing mechanisms can produce identical processes.

    Conclusions:

    • Newtonian dynamics can exhibit the same characteristics as the baker's transformation, unifying deterministic chaos and statistical behavior.
    • The abstract framework provides new insights into structural stability and random processes.
    • This work bridges the gap between deterministic laws and observed randomness in physical systems.