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A thermodynamic system is a set of objects whose thermodynamic properties are of interest. The system is considered to be embedded in its surroundings or the environment. The system and its environment can exchange heat and do work on each other through a boundary that separates them. However, the immediate surroundings of the system interact with it directly and therefore have a much stronger influence on its behavior and properties.
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Consider the two thermodynamic processes involving an ideal gas that are represented by paths AC and ABC in Figure 1:
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Multimachine stability analysis is crucial for understanding the dynamics and stability of power systems with multiple synchronous machines. The objective is to solve the swing equations for a network of M machines connected to an N-bus power system.
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Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
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Thermodynamic Potentials01:26

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Thermodynamic potentials are state functions that are extremely useful in analyzing a thermodynamic system. They have dimensions of energy. The four important thermodynamic potentials are internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy. These thermodynamic potentials can be expressed using two of the following variables: pressure, volume, temperature, and entropy. These two variables are expressed as the rate of change of the thermodynamic potential with respect to other...
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Maxwell's Thermodynamic Relations01:23

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Maxwell's thermodynamic relations are very useful in solving problems in thermodynamics. Each of Maxwell's relations relates a partial differential between quantities that can be hard to measure experimentally to a partial differential between quantities that can be easily measured. These relations are a set of equations derivable from the symmetry of the second derivatives and the thermodynamic potentials.
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Related Experiment Video

Updated: Jan 17, 2026

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
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Metriplectic four-bracket algorithm for constructing thermodynamically consistent dynamical systems.

Azeddine Zaidni1, Philip J Morrison2

  • 1Mohammed VI Polytechnic University, College of Computing, Lot 660, Hay Moulay Rachid, Ben Guerir 43150, Morocco.

Physical Review. E
|September 16, 2025
PubMed
Summary
This summary is machine-generated.

A novel unified thermodynamic algorithm constructs consistent dynamical systems, conserving energy while generating entropy. This approach, based on the metriplectic 4-bracket, generalizes key fluid dynamics and thermodynamics models.

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Area of Science:

  • Thermodynamics
  • Fluid Dynamics
  • Mathematical Physics

Background:

  • Dynamical systems require thermodynamic consistency, balancing energy conservation with entropy production.
  • Existing models often lack a unified framework for Hamiltonian and dissipative dynamics.

Purpose of the Study:

  • To present a unified thermodynamic algorithm for constructing thermodynamically consistent dynamical systems.
  • To generalize existing fluid dynamics and thermodynamics models.

Main Methods:

  • The algorithm is based on the metriplectic 4-bracket formulation.
  • It incorporates a force-flux relation: J^{α}=-L^{αβ}∇(δH/δξ^{β}).
  • The method is applied to Navier-Stokes-Fourier, Cahn-Hilliard-Navier-Stokes, and Brenner-Navier-Stokes-Fourier systems.

Main Results:

  • The unified algorithm successfully constructs thermodynamically consistent dynamical systems.
  • Significant generalizations of the Navier-Stokes-Fourier, Cahn-Hilliard-Navier-Stokes, and Brenner-Navier-Stokes-Fourier systems were achieved.
  • The force-flux relation provides a clear link between phenomenological coefficients, Hamiltonian, and dynamical variables.

Conclusions:

  • The presented algorithm offers a unified approach to modeling complex thermodynamic systems.
  • This framework enhances the consistency and applicability of fluid dynamics and thermodynamics models.
  • The generalization of established systems opens new avenues for research in nonequilibrium thermodynamics.