The notion of entropy has been at the core of thermodynamics and statistical physics since the 19th century, originally mentioned in the statement of the second law of thermodynamics as ``entropy is non-decreasing in an isolated system.'' In the context of both classical and quantum mechanics, there are cases where the second law is violated. In other words, entropy of the system may fluctuate downward. The question of how low or high the entropy of a quantum system can get is of interest in the context of small systems that can maintain quantum coherence for short times, and in the context of our universe, assuming it is an isolated quantum system. We explore the rare and extreme fluctuations of two well-developed notions of entropy that are relevant and interesting in isolated thermodynamic quantum systems, namely the observational and entanglement entropy. We find several fundamental differences in the behavior of the two when fluctuating to their extremes. We also provide an analytical proof of a tight upper bound on entanglement entropy for systems constrained to conservation laws.
In Chapter 3 the introduction of entropy as a state function was facilitated by the realization that there exist possible and impossible processes, and by an examination of the heat and work effects occurring during these processes. From the formal statement of the Second Law of Thermodynamics, as developed from Classical Thermodynamics arguments, it is difficult to assign a physical significance or a physical quality to entropy. In this respect entropy differs from internal energy in spite of the fact that, within the scope of Classical Thermodynamics, both properties are simply mathematical functions of the state of a system. The ready acceptance of the First Law of Thermodynamics, after its enunciation, was due to the easily understood physical significance of internal energy, whereas the lack of corresponding understanding of entropy caused the acceptance of the Second Law of Thermodynamics to be slow. From the classical viewpoint the Second Law is only valid as a direct result of the fact that, to date, human ingenuity has failed to invent a perpetual motion machine. Thus, within the scope of Classical Thermodynamics, the Second Law is a “law” only because it has not yet been disproved. The physical interpretation of entropy had to await the development of quantum theory and statistical mechanics.
In this paper, heat and Carnot's working substance are defined respectively by the equations
E=3pV and
E=3/2pV, relating the energy
E, the pressure
p and the volume
V of these two corpuscular systems. The thermodynamic-kinetic theoretical analysis of their equilibrium reveals that the constraints imposed by the second law on the working substance are the consequence of the relativistic and the quantum taut constraints, respectively,
∣v∣≤c and
∣ΔpΔq∣≥h, imposed on the motion of its molecules. It is further shown that the action of these restrictions is described by the ladder operators. By revealing the laws that rule the entropy formation mechanism, thereby explaining the time asymmetry, the present approach supersedes statistical mechanics. Furthermore, it establishes the epistemological chain connecting the laws of thermodynamics to the axioms of quantum mechanics, leading to yet another unorthodox interpretation of that theory.