A rigorous definition of mass in special relativity

The axiomatic definition of mass in classical mechanics, outlined by Mach in the second half of 19th century and improved by several authors, is simplified and extended to the theory of special relativity. According to the extended definition presented here, the mass of a relativistic particle is independent of its velocity and coincides with the rest mass, i.e., with the mass defined in classical mechanics. Then, force is defined as the product of mass and acceleration, both in the classical and in the relativistic framework.Comment: to be published in Il Nuovo Cimento

Recent Progress in the Definition of Thermodynamic Entropy

The principal methods for the definition of thermodynamic entropy are discussed with special reference to those developed by Carath\'eodory, the Keenan School, Lieb and Yngvason, and the present authors. An improvement of the latter method is then presented. Seven basic axioms are employed: three Postulates, which are considered as having a quite general validity, and four Assumptions, which identify the domains of validity of the definitions of energy (Assumption 1) and entropy (Assumptions 2, 3, 4). The domain of validity of the present definition of entropy is not restricted to stable equilibrium states. For collections of simple systems, it coincides with that of the proof of existence and uniqueness of an entropy function which characterizes the relation of adiabatic accessibility proposed by Lieb and Yngvason. However, our treatment does not require the formation of scaled copies so that it applies not only to collections of simple systems, but also to systems contained in electric or magnetic fields and to small and few-particle systems.Comment: 23 pages, 5 figure

Rigorous and General Definition of Thermodynamic Entropy

The physical foundations of a variety of emerging technologies --- ranging from the applications of quantum entanglement in quantum information to the applications of nonequilibrium bulk and interface phenomena in microfluidics, biology, materials science, energy engineering, etc. --- require understanding thermodynamic entropy beyond the equilibrium realm of its traditional definition. This paper presents a rigorous logical scheme that provides a generalized definition of entropy free of the usual unnecessary assumptions which constrain the theory to the equilibrium domain. The scheme is based on carefully worded operative definitions for all the fundamental concepts employed, including those of system, property, state, isolated system, environment, process, separable system, system uncorrelated from its environment, and parameters of a system. The treatment considers also systems with movable internal walls and/or semipermeable walls, with chemical reactions and/or external force fields, and with small numbers of particles. The definition of reversible process is revised by introducing the new concept of scenario. The definition of entropy involves neither the concept of heat nor that of quasistatic process; it applies to both equilibrium and nonequilibrium states. The role of correlations on the domain of definition and on the additivity of energy and entropy is discussed: it is proved that energy is defined and additive for all separable systems, while entropy is defined and additive only for separable systems uncorrelated from their environment; decorrelation entropy is defined. The definitions of energy and entropy are extended rigorously to open systems. Finally, to complete the discussion, the existence of the fundamental relation for stable equilibrium states is proved, in our context, for both closed and open systems.Comment: 19 pages, RevTex