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Fórmula de Entropia de Boltzmann

1877
  • Ludwig Boltzmann
19th-century scientific office with Boltzmann's Entropy Formula and thermodynamic equations.

(Imagem gerada apenas para fins ilustrativos)

Essa fórmula fundamental conecta o macroscópico termodinâmica quantity of entropy (S) with the number of possible microscopic arrangements, or microstates (W), corresponding to the system’s macroscopic state. The equation, [latex]S = k_B \ln W[/latex], reveals that entropy is a measure of statistical disorder or randomness. The constant [latex]k_B[/latex] is the Boltzmann constant, linking energy at the particle level with temperature. To write this equation in a report, a LaTeX cheat sheet lists the commands for subscripts, logarithms and the Boltzmann constant symbol, and a LaTeX math cheat sheet extends it to fractions and integrals.

Boltzmann’s entropy formula provides a statistical definition for the termodinâmica concept of entropy, which was previously defined by Rudolf Clausius in terms of heat transfer ([latex]dS = \frac{\delta Q}{T}[/latex]). Boltzmann’s breakthrough was to link this macroscopic quantity to the statistical properties of the system’s constituent particles. A ‘macrostate’ is defined by macroscopic variables like pressure, volume, and temperature. A ‘microstate’ is a specific configuration of the positions and momenta of all individual particles. The key insight is that a single macrostate can be realized by an enormous number of different microstates. The quantity W, sometimes called the statistical weight or thermodynamic probability, is this number.

A fórmula implica que o estado de equilíbrio de um sistema isolado, que é o estado de máxima entropia de acordo com a Segunda Lei da Termodinâmica, is simply the most probable macrostate—the one with the largest number of corresponding microstates (largest W). The logarithmic relationship is crucial because it ensures that entropy is an extensive property. If you combine two independent systems, their total entropy is the sum of their individual entropies ([latex]S_{tot} = S_1 + S_2[/latex]), while the total number of microstates is the product ([latex]W_{tot} = W_1 W_2[/latex]). The logarithm turns this product into a sum: [latex]k_B \ln(W_1 W_2) = k_B \ln W_1 + k_B \ln W_2[/latex]. This formula is famously engraved on Boltzmann’s tombstone in Vienna.

UNESCO Nomenclature: 2211
Termodinâmica

Tipo

Sistema abstrato

Interrupção

Revolucionário

Uso

Uso generalizado

Precursores

  • A formulação de Rudolf Clausius sobre segunda lei of thermodynamics and the classical definition of entropy
  • O trabalho de James Clerk Maxwell sobre a distribuição estatística das velocidades moleculares em um gás
  • Desenvolvimento da teoria da probabilidade por matemáticos como Pierre-Simon Laplace
  • A teoria cinética dos gases

Aplicações

  • teoria da informação (entropia de Shannon)
  • black hole termodinâmica (entropia de Bekenstein-Hawking)
  • ciência dos materiais para prever a estabilidade de fase
  • química computacional para o cálculo de entropias de reação
  • física da transição vítrea

Patentes:

NA

Ideias de Inovação Potencial

  • design of high-entropy alloys with tailored disorder for extreme environments
  • development of more efficient data compression algorithms based on entropy minimization
  • creating self-assembling nanomaterials by programming their most probable (highest w) structures
  • optimizing drug-receptor binding by calculating the entropic cost of conformational changes
  • modeling urban growth and traffic flow as systems tending towards maximum statistical complexity
  • advanced cryptographic systems using principles of statistical randomness
  • ai algorithms for protein design that maximize the conformational entropy of flexible regions
  • predictive models for ecosystem stability based on species diversity as a form of entropy
Relacionado a: entropia, Boltzmann, microestados, macroestados, termodinâmica, probabilidade, mecânica estatística, constante de Boltzmann.

Contexto histórico

Fórmula de Entropia de Boltzmann

1868
1870
1873
1877
1880
1882-01-01
1883
1865
1869
1871
1876
1877
1880
1882-01-01
1884

(Caso a data seja desconhecida ou irrelevante, por exemplo, "mecânica dos fluidos", é fornecida uma estimativa aproximada de seu surgimento notável)

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