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Related Concept Videos

Constraints and Statical Determinacy01:26

Constraints and Statical Determinacy

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In structural engineering, the equilibrium of a system is not only determined by its equations of equilibrium but also with the help of constraints. Constraints refer to restrictions on the motion of a system. The proper combinations of constraints can minimize the total number of constraints needed to maintain a system in mechanical equilibrium. When this happens, the system is said to be statically determinate. For such systems, the unknown reaction supports can be estimated using equilibrium...
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A pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K) may be described by a single microstate, as its purity, perfect crystallinity,and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.
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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...
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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...
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In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Processes that involve an increase in entropy of the system (ΔS > 0) are very often spontaneous; however, examples to the contrary are plentiful. By expanding consideration of entropy changes to include the surroundings, a significant conclusion regarding the relation between this property and spontaneity may be reached. In thermodynamic...
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Related Experiment Video

Updated: Sep 30, 2025

Bulk and Thin Film Synthesis of Compositionally Variant Entropy-stabilized Oxides
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Maximum entropy and constraints in composite systems.

John D Ramshaw1

  • 1Department of Physics, Portland State University, Portland, Oregon 97207, USA.

Physical Review. E
|March 16, 2022
PubMed
Summary

The principle of maximum entropy (PME) is refined by unconstrained maximization of total entropy. This reveals correct noncanonical distributions for systems in equilibrium with finite heat baths, offering a simpler alternative to Jaynes PME.

Area of Science:

  • Statistical Mechanics
  • Thermodynamics
  • Information Theory

Background:

  • The principle of maximum entropy (PME) by Jaynes maximizes Boltzmann-Gibbs-Shannon (BGS) entropy under linear constraints.
  • This typically yields canonical (exponential) probability distributions.
  • However, the basis for linear constraints is unclear, and noncanonical distributions occur.

Purpose of the Study:

  • To derive the correct noncanonical probability distribution for systems in equilibrium with finite heat baths.
  • To provide a more robust foundation for the principle of maximum entropy.
  • To generalize the method for arbitrary composite systems.

Main Methods:

  • Unconstrained maximization of the total BGS entropy of a system and its finite heat bath.

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  • Demonstrating equivalence to maximizing system entropy subject to a nonlinear constraint.
  • Analyzing the derived constraint for infinite and finite heat bath scenarios.

    • Main Results:

    • The unconstrained maximization of total entropy correctly yields noncanonical distributions for finite heat baths.
    • This procedure is equivalent to a nonlinear constraint on system entropy, simplifying to linear or logarithmic constraints.
    • A power-law distribution is implied for systems coupled to large, finite heat baths.

    Conclusions:

    • The unconstrained maximization of total entropy provides a clear derivation of probability distributions, resolving ambiguities in constraint selection.
    • This approach offers a simpler and more generalizable alternative to the traditional Jaynes PME.
    • It clarifies the emergence of noncanonical and power-law distributions in statistical mechanics.