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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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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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Experimentally, if object A is in equilibrium with object B, and object B is in equilibrium with object C, then object A is in equilibrium with object C. That statement of transitivity is called the "zeroth law of thermodynamics." For example, a cold metal block and a hot metal block are both placed on a metal plate at room temperature. Eventually, the cold block and the plate will be in thermal equilibrium. In addition, the hot block and the plate will be in thermal 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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Why Noether's theorem applies to statistical mechanics.

Sophie Hermann1, Matthias Schmidt1

  • 1Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, D-95447 Bayreuth, Germany.

Journal of Physics. Condensed Matter : an Institute of Physics Journal
|March 7, 2022
PubMed
Summary
This summary is machine-generated.

Noether's theorem, typically used for mechanics and field theory, is extended to thermal systems. This framework reveals connections between symmetries and conservation laws in statistical mechanics, even for active particles.

Keywords:
Noether’s theoremdensity functional theoryinvarianceliquid state theorypower functional theorystatistical mechanicssum rules

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Area of Science:

  • Statistical Mechanics
  • Theoretical Physics
  • Symmetry Principles

Background:

  • Noether's theorem classically links conservation laws to symmetries.
  • Existing applications focus on particle mechanics and field theory.
  • Thermal systems require a statistical mechanical description due to paramount fluctuations.

Purpose of the Study:

  • Extend Noether's theorem to thermal systems.
  • Provide a pedagogical introduction using the canonical ensemble.
  • Apply the framework to ideal sedimentation and active Brownian particles.

Main Methods:

  • Viewing thermodynamic quantities like free energy as functionals.
  • Employing systematic functional differentiation.
  • Analyzing macroscopic average forces and molecular correlations.

Main Results:

  • Demonstrated Noether's theorem's applicability to thermal systems.
  • Derived identities relating symmetries to conservation laws in statistical mechanics.
  • Extended the analysis to systems out-of-equilibrium and active matter.

Conclusions:

  • Noether's theorem provides a unified framework for conservation laws in diverse physical systems.
  • Functional methods offer a powerful approach for analyzing complex many-body systems.
  • The findings have implications for understanding both equilibrium and non-equilibrium statistical mechanics.