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...ks for composite systems, have a distinguished coordinate, the energy U , and N − 1 work coordinates, denoted collectively by V . Often, V is just the volume. A central concept in our approach (as in =-=[C]-=-, [L], [B], [FJ], [G]) is the relation of adiabatic accessibility. Its operational definition (inspired by Planck’s formulation of the Second Law, see [P]. p. 89) is as follows: A state Y is adiabatic...

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...f entropy this is equivalent to X ∼A ((1− λ)X0, λX1). (8) Any other entropy function S ′ also leads to (8) with λ replaced by some λ′. From the assumptions A1-A6 and X0 ≺≺ X1 it is easy to prove (see =-=[LY2]-=-, Lemma 2.2) that (8) can hold for at most one λ, i.e., λ = λ′. Hence the entropy is uniquely determined, up to the choice of the entropy of X0 and X1, i.e., up to an affine change of scale. We now co...

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...ight may have risen or fallen.1 It is important to note that the process taking X to Y need not be “quasi-static”, in fact, it can be arbitrarily violent. 1Such processes are called work processes in =-=[GB]-=-. 3 The following definitions and notations will be applied: If X ≺ Y or Y ≺ X we say that X and Y are adiabatically comparable (or comparable for short). If X ≺ Y but Y 6≺ X we write X ≺≺ Y , and if ...

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...ume. A central concept in our approach (as in [C], [L], [B], [FJ], [G]) is the relation of adiabatic accessibility. Its operational definition (inspired by Planck’s formulation of the Second Law, see =-=[P]-=-. p. 89) is as follows: A state Y is adiabatically accessible from a state X, in symbols X ≺ Y (read: “X precedes Y ”), if it is possible to change the state from X to Y in such a way that the final e...

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...thout further assumptions, however. These matters are discussed in [LY2], Sect. 6. 3 Non-equilibrium states There exist many variants of non-equilibrium thermodynamics. A concise overview is given in =-=[LJC]-=- where the following approaches are discussed, among others: Classical Irreversible Thermodynamics, Extended Irreversible Thermodynamics, Finite Time Thermodynamics, Theories with Internal Variables, ...

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...ent of models and its consequences are far reaching. Hence it deserves a simple and solid logical foundation! An approach to the foundational issues was developed in several papers by us in 1998-2003 =-=[LY1]-=--[LY4]. We emphasize that, contrary to possible first impression, our approach is not abstract but is based, in principle, on experimentally determined facts. We also emphasize that our approach is in...

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...e notion of increase cannot be properly quantified. Several definitions of entropy for non-equilibrium states have been proposed in the literature. (See, e.g., [LHC] for a review of these matters and =-=[ST]-=- for a discussion of steady-state thermodynamics.) These definitions do not necessarily fulfill the 1 main requirement of entropy, however, which, according to our view, is that entropy is a state fun...

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...ems, have a distinguished coordinate, the energy U , and N − 1 work coordinates, denoted collectively by V . Often, V is just the volume. A central concept in our approach (as in [C], [L], [B], [FJ], =-=[G]-=-) is the relation of adiabatic accessibility. Its operational definition (inspired by Planck’s formulation of the Second Law, see [P]. p. 89) is as follows: A state Y is adiabatically accessible from ...

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... U + ΦX0(X) and hence ΦX0(X) ≤ (U − U0)− T0(S− − S0). 3.2.1 Definition of entropy through maximum work In their textbook on engineering thermodynamics [GB], E. Gyftopolous and G. P. Beretta (see also =-=[BZ]-=-) take the concept of maximum work as a basis for their 14 definition of entropy. Paraphrasing their definition in our notation, they assume the maximum work ΦX0(X) to be a measurable quantity for arb...

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...r composite systems, have a distinguished coordinate, the energy U , and N − 1 work coordinates, denoted collectively by V . Often, V is just the volume. A central concept in our approach (as in [C], =-=[L]-=-, [B], [FJ], [G]) is the relation of adiabatic accessibility. Its operational definition (inspired by Planck’s formulation of the Second Law, see [P]. p. 89) is as follows: A state Y is adiabatically ...

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...posite systems, have a distinguished coordinate, the energy U , and N − 1 work coordinates, denoted collectively by V . Often, V is just the volume. A central concept in our approach (as in [C], [L], =-=[B]-=-, [FJ], [G]) is the relation of adiabatic accessibility. Its operational definition (inspired by Planck’s formulation of the Second Law, see [P]. p. 89) is as follows: A state Y is adiabatically acces...

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...ding explicitly on the heat flux can be considered and even computed in some simple cases. It has the property of increasing monotonously in time when heat conduction is described by Cattaneo’s model =-=[Cat]-=- with a hyperbolic heat transport equation rather than the parabolic Fourier’s law.8 The classical entropy, in contrast, may oscillate (see Fig. 7.2 in [LJC]), and does therefore not comply with the s...

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...e systems, have a distinguished coordinate, the energy U , and N − 1 work coordinates, denoted collectively by V . Often, V is just the volume. A central concept in our approach (as in [C], [L], [B], =-=[FJ]-=-, [G]) is the relation of adiabatic accessibility. Its operational definition (inspired by Planck’s formulation of the Second Law, see [P]. p. 89) is as follows: A state Y is adiabatically accessible ...

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... models and its consequences are far reaching. Hence it deserves a simple and solid logical foundation! An approach to the foundational issues was developed in several papers by us in 1998-2003 [LY1]-=-=[LY4]-=-. We emphasize that, contrary to possible first impression, our approach is not abstract but is based, in principle, on experimentally determined facts. We also emphasize that our approach is independ...