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Related Concept Videos

Entropy02:39

Entropy

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...
Entropy01:18

Entropy

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...
The Entropy as a State Function01:14

The Entropy as a State Function

Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...
Entropy and the Second Law of Thermodynamics01:20

Entropy and the Second Law of Thermodynamics

The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
The relation  between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...
Entropy and the Second Law of Thermodynamics01:26

Entropy and the Second Law of Thermodynamics

Consider an isolated system in which a hot object is placed in contact with a cold one. This is an irreversible process that eventually leads both objects to reach the same equilibrium temperature. It is crucial to note that the constituents of any substance exhibit increased disorder at higher temperatures. As a cold substance absorbs heat, its constituents become more disordered. The energy transfer from a hotter object to a cooler one increases the system's disorder or randomness. This...
Absolute Entropies and the Third Law of Thermodynamics01:23

Absolute Entropies and the Third Law of Thermodynamics

Ludwig Edward Boltzmann developed a definition for entropy, which stated that absolute entropy is proportional to the natural logarithm of the number of possible combinations of particles. Entropy stands alone among state functions as the only one whose absolute values can be determined.Consider a gas sample confined to a container. As the container expands, the energy levels of gas molecules become more closely spaced. This increases the number of available energy states, thereby increasing...

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Fluctuation-dissipation theorem for the microcanonical ensemble.

Marcus V S Bonança1

  • 1Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany. marcus.bonanca@physik.uni-regensburg.de

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 15, 2008
PubMed
Summary

This study derives the fluctuation-dissipation theorem for the microcanonical ensemble, extending its applicability beyond the canonical ensemble. The findings reveal a new relationship between equilibrium fluctuations and response functions in systems not in the thermodynamic limit.

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Area of Science:

  • Statistical Mechanics
  • Theoretical Physics

Background:

  • The fluctuation-dissipation theorem typically relates equilibrium fluctuations to system response in the canonical ensemble.
  • Its extension to other ensembles, like the microcanonical ensemble, is crucial for broader theoretical understanding.

Purpose of the Study:

  • To derive the fluctuation-dissipation theorem for the microcanonical ensemble.
  • To explore the relationship between equilibrium fluctuations and response functions beyond the thermodynamic limit.

Main Methods:

  • Utilizing linear response theory.
  • Analyzing frequency spectra of symmetric correlation and response functions.

Main Results:

  • A novel derivation of the fluctuation-dissipation theorem for the microcanonical ensemble.
  • Demonstration that dynamical fluctuations can dictate response in systems outside the thermodynamic limit.
  • Obtained dispersion relations and sum rules dependent on energy, not frequency.

Conclusions:

  • The fluctuation-dissipation theorem is not exclusive to the canonical ensemble.
  • The derived relations offer a new perspective on the interplay between fluctuations and response in statistical mechanics.