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• W1995763641 abstract "Abstract The quantum thermodynamics of the ideal gas is treated by means of the particle entropies σ = ka . The non-dimensional entropy numbers a are calculated by the extended, temperature-dependent Schrödinger equation applying periodic boundary conditions. The Boltzmann constant k represents the atomic entropy unit. The quantized particle entropies a are used to calculate the Boltzmann distribution f =exp( α − a ), the density of entropy levels D ( a ), the dimensionless chemical potential α = μ / kT , etc . – The thermodynamics of the ideal gas is treated on the basis of entropy quanta by means of the density matrix formalism. In this way we obtain the total particle entropy A , the thermodynamic entropy S (equation of Sackur-Tetrode), the internal and the Helmholtz free energy E and F , respectively, etc . Moreover, it can also be shown on the basis of entropy quanta that the entropy of a closed system in thermal equilibrium has a maximum value, S = S max . Against this, the entropy S ′ of an arbitrary non-equilibrium state is always smaller, S ′ < S max (second law of thermodynamics). – Application of ordinary density matrix theory to a closed system of N particles leads to the result that the entropy S does not change in time, d S /d t =0, regardless whether we consider an equilibrium state, S = S max , or a non-equilibrium state, S = S ′. Consequently, S cannot irreversibly change from S ′ to S max . However, any irreversible process is accompanied by a positive entropy production P =d S /d t >0. In order to overcome this obvious contradiction, we discuss the entropy evolution in time by means of the commutator equation G =i[ P , t ], which is deduced from very general assumptions. Here P and t are the operators of the entropy production P and the time t . Accordingly, P and t do not commute in general, and hence P and t are not sharply defined simultaneously. Instead we have uncertainties Δ P and Δ t , which are expressed by the uncertainty relation of the N -particle system, Δ P Δ t ≥(1/2)|〈 G 〉|=( γ /2) k . This P − t uncertainty relation easily allows a discussion of the evolution of the entropy S in time t from S ′ to S max . Now the irreversible steps are correctly described by the entropy production P =d S /d t > 0, and the thermal equilibrium by P =0, ∆ P =0, and thus the lifetime ∆ t =∞." @default.
• W1995763641 created "2016-06-24" @default.
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• W1995763641 date "2003-01-01" @default.
• W1995763641 modified "2023-09-26" @default.
• W1995763641 title "Particle Entropies and Entropy Quanta: IV. The Ideal Gas, the Second Law of Thermodynamics, and the <i>P</i>–<i>t</i> Uncertainty Relation" @default.
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• W1995763641 doi "https://doi.org/10.1524/zpch.217.1.55.18963" @default.
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