Generalized time reversal symmetry in systems coupled with a magnetic field: analysis and outcomes on statistical properties

The thesis work starts from some recent papers about time reversal symmetry for systems in a static constant magnetic field published by Prof. L. Rondoni and collaborators. Going into details, their dissertation focuses on how Onsager relations may hold in a system of classical or quantum spinless particles coupled with a constant homogeneous magnetic field. The thesis originates from these works and it is organized in three Chapters. The first one concerns T-symmetry. In the very first part I retrace their results in the classical case; for a system of free particles, they proved the existence of involutional maps that yield time reversal invariance on phase space even when the canonical map (i.e. the inversion of all the momenta) does not. After the introduction, I proceed in order to generalize the results present in literature; specifically, I commence deducing the most general form of a time reversal transformation from the Hamilton equations, clarifying the relation between the magnetic vector potential and these transformations. In the case in study, we found a formula expressing the number of these operations, also including the novel case involving permutations of coordinates. In the subsequent part, I extend the formula to quantum spinless non-relativistic systems using the similarities between Quantum and Classical Mechanics; on the other hand, exploiting the differences between these two frameworks, I found other generalized time reversal operators whose use is forbidden in a classical context. Then, I analysed the case of particles with spin 1/2: retracing the motivation at the basis of the canonical procedure used by Sachs for example, I found other generalized time reversal operator on spin space other than σy. The second Chapter is about Response Theory and it uniquely concerns the theory necessary to show the importance of the generalized time reversal operations. In fact, the main consequence already showed by Rondoni et al. is that under the Kubo correlation integral one can find for any trajectory forward in time an equal weighted backward trajectory, even if it is not the original one travelled in the opposite direction. These considerations disprove the universally accepted conjecture that Onsager relations must be in general replaced by the Onsager-Casimir ones if the system is coupled with a magnetic field; see apropos the original paper by Casimir (1945), and the classical textbooks by Landau, Kubo, De Groot and Mazur. Other than this. the generalization becomes fundamental whenever it appears a hypothesis of TRI, as for instance in the context of nonequilibrium statistical mechanics (e.g. regarding Fluctuation Relations). In the final Chapter I concentrate on the functional form of the magnetic field, showing in which cases each generalized T-symmetry operation can persist and expressing these necessary conditions for a magnetic field with any orientation and functional expression. Thereafter, some notable examples of forces, both interparticle and external, are analysed using a lemma that I proved that concerns the compatibility condition between a generalized T-symmetries and a force potential. I treat central potentials, the PIM and RIM (that model intermolecular forces), the Coulomb ring-shaped and a momentum dependent potential.