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Entropy02:39

Entropy

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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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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.
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One of the challenges of using the second law of thermodynamics to determine if a process is spontaneous is that it requires measurements of the entropy change for the system and the entropy change for the surroundings. An alternative approach involving a new thermodynamic property defined in terms of system properties only was introduced in the late nineteenth century by American mathematician Josiah Willard Gibbs. This new property is called the Gibbs free energy (G) (or simply the free...
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The free energy change for a reaction that occurs under the standard conditions of 1 bar pressure and at 298 K is called the standard free energy change. Since free energy is a state function, its value depends only on the conditions of the initial and final states of the system. A convenient and common approach to the calculation of free energy changes for physical and chemical reactions is by use of widely available compilations of standard state thermodynamic data. One method involves the...
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The free energy change for a process taking place with reactants and products present under nonstandard conditions (pressures other than 1 bar; concentrations other than 1 M) is related to the standard free energy change according to this equation:
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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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Bethe free-energy approximations for disordered quantum systems.

I Biazzo1, A Ramezanpour2

  • 1DISAT and Center for Computational Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|July 15, 2014
PubMed
Summary
This summary is machine-generated.

This study develops methods for creating globally consistent density matrices from local ones using the cavity method. These techniques are validated against the random transverse Ising model, offering improved approximations for entropy calculations.

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Area of Science:

  • Statistical Physics
  • Quantum Information Theory
  • Computational Physics

Background:

  • Locally consistent reduced density matrices are crucial for quantum many-body systems.
  • Global consistency is challenging to achieve from local information alone.
  • The cavity method offers a framework for approximating complex systems.

Purpose of the Study:

  • To construct globally consistent approximate density matrices from locally consistent ones.
  • To develop and compare methods for entropy approximation using the cavity method.
  • To validate these methods on the random transverse Ising model.

Main Methods:

  • Utilizing the cavity method of statistical physics.
  • Employing an annealing algorithm by decreasing temperature.
  • Minimizing an approximate Bethe free energy.
  • Approximating cavity messages for Bethe entropy calculations.

Main Results:

  • Construction of globally consistent density matrices for tree-structured trial density matrices.
  • Derivation of the classical Bethe entropy expression via a naive (mean-field) approximation.
  • Obtained an improved Bethe entropy expression dependent on diagonal elements of reduced density matrices.
  • Numerical simulations on the random transverse Ising model validated the methods.

Conclusions:

  • The cavity method provides a viable approach for constructing globally consistent density matrices.
  • The developed approximations for Bethe entropy show promise, especially at high temperatures.
  • The study successfully compares different approximation strategies against established models.