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Thermodynamic Systems01:06

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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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Statements of the Second Law of Thermodynamics01:15

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The second law of thermodynamics can be stated in several different ways, and all of them can be shown to imply the others. The Clausius’ statement of the second law of thermodynamics is based on the irreversibility of spontaneous heat flow. It states that heat will not flow from the colder body to the hotter body unless some other process is involved. Additionally, as per the Kelvin’s statement, it is impossible to convert the heat from a single source into work without any other...
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A change in the internal energy of a system depends on the the net heat transfer into the system and the net work done by the system. The first law of thermodynamics, which is a generalized form of energy conservation, relates these three quantities mathematically. It states that the change in the internal energy equals the difference between the heat transfer and work done by the system.
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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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Entropy01:18

Entropy

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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.
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First Law Of Thermodynamics: Problem-Solving01:21

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The first law of thermodynamics states that the change in internal energy of the system is equal to the net heat transfer into the system minus the net work done by the system. This equation is a generalized form of energy conservation and can be applied to any thermodynamic process.
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Thermodynamic Formalism for General Iterated Function Systems with Measures.

Jader E Brasil1, Elismar R Oliveira1, Rafael Rigão Souza1

  • 1Instituto de Matemática e Estatística - UFRGS, Av. Bento Gonçalves, 9500, Porto Alegre, RS 91509-900 Brazil.

Qualitative Theory of Dynamical Systems
|December 26, 2022
PubMed
Summary

This study introduces Thermodynamic Formalism for Iterated Function Systems with Measures (IFSm). It proves the existence and uniqueness of equilibrium states for continuous potentials, advancing dynamical systems theory.

Keywords:
EntropyEquilibrium statesErgodic theoryIterated function systemPressureThermodynamic formalismTransfer operator

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Area of Science:

  • Dynamical Systems and Ergodic Theory
  • Mathematical Physics

Background:

  • Iterated Function Systems (IFSs) are fundamental in fractal geometry.
  • Thermodynamic Formalism provides tools to study complex dynamical systems.
  • Extending these concepts to systems with measures (IFSm) is a key challenge.

Purpose of the Study:

  • Introduce a novel theory of Thermodynamic Formalism for Iterated Function Systems with Measures (IFSm).
  • Investigate spectral properties of associated operators.
  • Develop variational principles for topological entropy and pressure.
  • Establish existence and uniqueness of equilibrium states.

Main Methods:

  • Spectral analysis of Transfer and Markov operators.
  • Variational formulations for topological entropy and pressure.
  • Definition and analysis of equilibrium states.

Main Results:

  • Established a theory of Thermodynamic Formalism for IFSm.
  • Characterized spectral properties of relevant operators.
  • Proved existence of equilibrium states for continuous potentials.
  • Demonstrated uniqueness of equilibrium states under Gâteaux differentiability conditions.

Conclusions:

  • The developed theory provides a robust framework for analyzing IFSm.
  • The results contribute to a deeper understanding of complex dynamical systems and their statistical properties.
  • This work opens avenues for further research in fractal dynamics and statistical physics.