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Related Concept Videos

Entropy02:39

Entropy

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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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Entropy01:18

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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...
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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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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 models, the...
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Second Law of Thermodynamics00:53

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The Second Law of Thermodynamics states that entropy, or the amount of disorder in a system, increases each time energy is transferred or transformed. Each energy transfer results in a certain amount of energy that is lost—usually in the form of heat—that increases the disorder of the surroundings. This can also be demonstrated in a classic food web. Herbivores harvest chemical energy from plants and release heat and carbon dioxide into the environment. Carnivores harvest the...
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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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Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
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Stochastic Entropy Solutions for Stochastic Nonlinear Transport Equations.

Rongrong Tian1, Yanbin Tang1

  • 1School of Mathematics and Statistics, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary

This study establishes the existence and uniqueness of stochastic entropy solutions for nonlinear transport equations with stochastic perturbations. The findings confirm solution stability concerning coefficient and function variations.

Keywords:
existencenonlinear transport equationstochastic (strong) entropy solutionuniqueness

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Area of Science:

  • Stochastic Analysis
  • Partial Differential Equations
  • Nonlinear Dynamics

Background:

  • Nonlinear transport equations are fundamental in modeling phenomena across physics and engineering.
  • Stochastic perturbations introduce complexity, necessitating advanced mathematical frameworks.
  • Entropy solutions provide a robust concept for handling discontinuities and weak solutions.

Purpose of the Study:

  • To investigate the existence and uniqueness of stochastic entropy solutions for nonlinear transport equations with stochastic perturbations.
  • To establish the continuous dependence of these solutions on key equation parameters.
  • To contribute to the theoretical understanding of stochastic partial differential equations.

Main Methods:

  • Uniqueness is proven using the doubling variable method.
  • Existence is established through a novel parabolic approximation scheme.
  • The scheme is inspired by the vanishing viscosity method from Feng and Nualart (2008).

Main Results:

  • The paper confirms the existence and uniqueness of stochastic entropy solutions.
  • Continuous dependence of strong entropy solutions on the coefficient 'b' and nonlinear function 'f' is demonstrated.
  • The parabolic approximation scheme is shown to be effective for this class of equations.

Conclusions:

  • The study provides a rigorous mathematical framework for stochastic nonlinear transport equations.
  • The results enhance the understanding of solution behavior under uncertainty.
  • This work has implications for the numerical analysis and application of stochastic PDEs.