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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...
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A spontaneous process is one that occurs naturally under certain conditions. A nonspontaneous process, on the other hand, will not take place unless it is “driven” by the continual input of energy from an external source. Processes have a natural tendency to occur in one direction under a given set of conditions. Water will naturally flow downhill (spontaneous process), but uphill flow (nonspontaneous process) requires outside intervention such as the use of a pump. Iron exposed to...
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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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In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Processes that involve an increase in entropy of the system (ΔS > 0) are very often spontaneous; however, examples to the contrary are plentiful. By expanding consideration of entropy changes to include the surroundings, a significant conclusion regarding the relation between this property and spontaneity may be reached. In thermodynamic...
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Eigenstate Thermalization Hypothesis and Free Probability.

Silvia Pappalardi1, Laura Foini2, Jorge Kurchan1

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This summary is machine-generated.

We uncovered a link between quantum thermalization and free probability theory. This connection simplifies understanding complex correlations in quantum systems using free cumulants.

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Area of Science:

  • Quantum physics
  • Statistical mechanics

Background:

  • Quantum thermalization is explained by the eigenstate thermalization hypothesis (ETH).
  • ETH describes correlations of matrix elements using typicality arguments and local rotations.

Purpose of the Study:

  • To explore the relationship between ETH and free probability theory.
  • To provide a mathematical framework for understanding quantum thermalization correlations.

Main Methods:

  • Applied free probability theory to thermal ensembles and energy shells.
  • Utilized invariance under local rotations of nearby energy levels.
  • Derived an explicit formula for free cumulants.

Main Results:

  • Established a direct connection between ETH and free probability theory.
  • Demonstrated that higher-order correlation functions decompose into free cumulants.
  • Showed that local functions of ETH operators also satisfy ETH.

Conclusions:

  • Free probability theory offers a powerful tool for analyzing quantum thermalization.
  • The study expands the scope of free probability and its relevance to quantum mechanics.