Aspects of entropy in classical and in quantum physics

Abstract

Entropy has played an essential role in the history of physics. Its mathematical definition and applications have changed over time till today. In this paper, we first review the historical evolution of these various points of view, from the thermodynamic definition to information entropy from Shannon in classical physics, up to the modern concept of Neumann’s quantum entropy. As a specific example, we consider entanglement entropy and compare the phase space approach in classical physics to the Hilbert space approach in quantum physics in simple model systems. We derive a general expression for the entanglement entropy of fermions and bosons in arbitrary partitions of Hilbert space, valid beyond the thermodynamic limit. Next, we compare thermodynamic heat engines with quantum heat engines. Finally, we proceed to the more general concept of quantum (computational) complexity and argue, using the concept of entanglement entropy, that the Heisenberg time in classically chaotic systems coincides with the time when maximal complexity is reached in the quantum case for systems with all-all interactions.

A Quantum-Classical Model of Brain Dynamics

“…In truth 'microscopic statistical disorder' is a misnomer that stands for the great number of microscopic states that correspond to the same macroscopic state [142,143]. Von Neumann entropy (and its quantum-classical generalization defined in terms of the Mixed Weyl of the statistical operator) is a property of the macrostates given in terms of the probability of microstates [144,145]. The belief that the passage to macroscopic 'order' is associated with entropy decrease is mistaken.…”

“…Another example is given by the phenomenon of reentrant phase transitions [146][147][148][149], where the macroscopic 'ordered' phase has higher entropy than the microscopic 'disordered' one because of the unfreezing of certain DOF. Irreversible microscopic dynamics, such as diffusive motion, does not conserve the number of accessible microstates of the system conditioned by the macroscopic constraints and, thus, lead to an increase of entropy [144,145]. This is the essence of Schrödinger's "order from disorder" mechanism [90]: in our macrocosm we are surrounded by structures that we classify as ordered but that are based on microscopic disorder in agreement with the Second Law of Thermodynamics.…”

A Quantum-Classical Model of Brain Dynamics

“…In truth, 'microscopic statistical disorder' is a misnomer that stands for the great number of microscopic states that correspond to the same macroscopic state [145,146]. Von Neumann entropy (and its quantum-classical generalization defined in terms of the mixed Weyl of the statistical operator) is a property of the macrostates given in terms of the probability of microstates [147,148]. The belief that the passage to macroscopic 'order' is associated with an entropy decrease is mistaken.…”

“…Irreversible microscopic dynamics, such as diffusive motion, does not conserve the number of accessible microstates of the system conditioned by the macroscopic constraints and thus leads to an increase in entropy [147,148]. This is the essence of Schrödinger's 'order from disorder' mechanism [115]: in our macrocosm, we are surrounded by structures that we classify as ordered but that are based on microscopic disorder in agreement with the second law of thermodynamics.…”

A Quantum–Classical Model of Brain Dynamics

“…In truth, ‘microscopic statistical disorder’ is a misnomer that stands for the great number of microscopic states that correspond to the same macroscopic state [ 145 , 146 ]. Von Neumann entropy (and its quantum–classical generalization defined in terms of the mixed Weyl of the statistical operator) is a property of the macrostates given in terms of the probability of microstates [ 147 , 148 ]. The belief that the passage to macroscopic ‘order’ is associated with an entropy decrease is mistaken.…”