Thermodynamics, Statistical Mechanics and Entropy

Swendsen, Robert H.

doi:10.3390/e19110603

Cited by 31 publications

(48 citation statements)

References 49 publications

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“…States of apparent negative thermodynamic temperatures were first produced in 1951 [1] by forced inversion of quantum states in select systems, and to the present day still garner widespread attention, e.g., Refs. [2][3][4][5][6].…”

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confidence: 99%

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Work Storage in States of Apparent Negative Thermodynamic Temperature

Struchtrup

2018

Phys. Rev. Lett.

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Inverted quantum states provide a challenge to classical thermodynamics, since they appear to contradict the classical formulation of the second law of thermodynamics. Ramsey interpreted these states as stable equilibrium states of negative thermodynamic temperature, and added a provision to allow these states to the Kelvin-Planck statement of the second law [N. F. Ramsey, Phys. Rev. 103, 20 (1956)PHRVAO0031-899X10.1103/PhysRev.103.20]. Since then, Ramsey's interpretation has prevailed in the literature. Here, we present an alternative option to accommodate inverted states within thermodynamics, which strictly enforces the original Kelvin-Planck statement of the second law, and reconciles inverted states and the second law by interpreting the former as unstable states, for which no temperature-positive or negative-can be defined. Specifically, we recognize inverted quantum states as temperature-unstable states, for which all processes are in agreement with the original Kelvin-Planck statement of the second law, and positive thermodynamic temperatures in stable equilibrium states. These temperature-unstable states can only be created by work done to the system, which is stored as energy in the unstable states, and can be released as work again, just as in a battery or a spring.

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confidence: 99%

“…Recent attempts to repudiate negative temperatures led to a vivid discussion on the definition of equilibrium entropy in statistical mechanics (Boltzmann versus Gibbs entropy) [3][4][5][6]. The following discussion relies on the Boltzmann definition of entropy, which leads to apparent negative temperatures.…”

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confidence: 99%

Work Storage in States of Apparent Negative Thermodynamic Temperature

Struchtrup

2018

Phys. Rev. Lett.

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“…The results of [37], insofar as they barely discriminate between two different single-particle spectra that share the same asymptotic N (λ), are very powerful results. This is clear in the steps from Equations (38)- (42). Thus, the inclusion or exclusion of the zero from the energy eigenvalues does not change the emerging physics, since it just results in a shift of one place in the sequence Λ.…”

Section: The Physics Of Partitions and Compositions With Squares Of Imentioning

confidence: 98%

“…Here, we assume Dirichlet bounday conditions, i.e., n ∈ Z + . We want to use Equation (42). We point out that the ground state has an energy E 0 such that,…”

Section: The Physics Of Partitions and Compositions With Squares Of Imentioning

confidence: 99%

Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose–Einstein Gas in a Box

Gregorio

Rondoni

2018

Entropy

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From basic principles, we review some fundamentals of entropy calculations, some of which are implicit in the literature. We mainly deal with microcanonical ensembles to effectively compare the counting of states in continuous and discrete settings. When dealing with non-interacting elements, this effectively reduces the calculation of the microcanonical entropy to counting the number of certain partitions, or compositions of a number. This is true in the literal sense, when quantization is assumed, even in the classical limit. Thus, we build on a moderately dated, ingenuous mathematical work of Haselgrove and Temperley on counting the partitions of an arbitrarily large positive integer into a fixed (but still large) number of summands, and show that it allows us to exactly calculate the low energy/temperature entropy of a one-dimensional Bose-Einstein gas in a box. Next, aided by the asymptotic analysis of the number of compositions of an integer as the sum of three squares, we estimate the entropy of the three-dimensional problem. For each selection of the total energy, there is a very sharp optimal number of particles to realize that energy. Therefore, the entropy is 'large' and almost independent of the particles, when the particles exceed that number. This number scales as the energy to the power of (2/3)-rds in one dimension, and (3/5)-ths in three dimensions. In the one-dimensional case, the threshold approaches zero temperature in the thermodynamic limit, but it is finite for mesoscopic systems. Below that value, we studied the intermediate stage, before the number of particles becomes a strong limiting factor for entropy optimization. We apply the results of moments of partitions of Coons and Kirsten to calculate the relative fluctuations of the ground state and excited states occupation numbers. At much lower temperatures than threshold, they vanish in all dimensions. We briefly review some of the same results in the grand canonical ensemble to show to what extents they differ.

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“…Instead of an expression "heat flow-in to a body, we use more convenient "the internal energy of a body increase due to its heating". Additionally, having a proper definition of "internal energy" we are sure that the Callen structure of thermostatics can be easily adopted [21] and the consistent interrelations with statistical mechanics can be established [22].…”

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confidence: 99%

Neoclassical Navier–Stokes Equations Considering the Gyftopoulos–Beretta Exposition of Thermodynamics

2020

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The seminal Navier–Stokes equations were stated even before the creation of the foundations of thermodynamics and its first and second laws. There is a widespread opinion in the literature on thermodynamic cycles that the Navier–Stokes equations cannot be taken as a thermodynamically correct model of a local “working fluid”, which would be able to describe the conversion of “heating” into “working” (Carnot’s type cycles) and vice versa (Afanasjeva’s type cycles). Also, it is overall doubtful that “cycle work is converted into cycle heat” or vice versa. The underlying reason for this situation is that the Navier–Stokes equations come from a time when thermodynamic concepts such as “internal energy” were still poorly understood. Therefore, this paper presents a new exposition of thermodynamically consistent Navier–Stokes equations. Following that line of reasoning—and following Gyftopoulos and Beretta’s exposition of thermodynamics—we introduce the basic concepts of thermodynamics such as “heating” and “working” fluxes. We also develop the Gyftopoulos and Beretta approach from 0D into 3D continuum thermodynamics. The central role within our approach is played by “internal energy” and “energy conversion by fluxes.” Therefore, the main problem of exposition relates to the internal energy treated here as a form of “energy storage.” Within that context, different forms of energy are discussed. In the end, the balance of energy is presented as a sum of internal, kinetic, potential, chemical, electrical, magnetic, and radiation energies in the system. These are compensated by total energy flux composed of working, heating, chemical, electrical, magnetic, and radiation fluxes at the system boundaries. Therefore, the law of energy conservation can be considered to be the most important and superior to any other law of nature. This article develops and presents in detail the neoclassical set of Navier–Stokes equations forming a thermodynamically consistent model. This is followed by a comparison with the definition of entropy (for equilibrium and non-equilibrium states) within the context of available energy as proposed in the Gyftopoulos and Beretta monograph. The article also discusses new possibilities emerging from this “continual” Gyftopoulos–Beretta exposition with special emphasis on those relating to extended irreversible thermodynamics or Van’s “universal second law”.

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Thermodynamics, Statistical Mechanics and Entropy

Cited by 31 publications

References 49 publications

Work Storage in States of Apparent Negative Thermodynamic Temperature

Work Storage in States of Apparent Negative Thermodynamic Temperature

Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose–Einstein Gas in a Box

Neoclassical Navier–Stokes Equations Considering the Gyftopoulos–Beretta Exposition of Thermodynamics

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