Unitary evolution for a two-level quantum system in fractional-time scenario.

The time-evolution operator obtained from the fractional-time Schrödinger equation (FTSE) is said to be nonunitary since it does not preserve the norm of the vector state in time. As done in the time-dependent non-Hermitian quantum formalism, for a traceless non-Hermitian two-level quantum system, we demonstrate that it is possible to map the nonunitary time-evolution operator in a unitary one. It is done by considering a dynamical Hilbert space with a time-dependent metric operator, constructed from a Hermitian time-dependent Dyson map, in respect to which the system evolves in a unitary way, and the standard quantum mechanics interpretation can be made properly. To elucidate our approach, we consider three examples of Hamiltonian operators and their corresponding unitary dynamics obtained from the solutions of FTSE, and the respective Dyson maps.

Efficiency of random search with space-dependent diffusivity.

We address the problem of random search for a target in an environment with a space-dependent diffusion coefficient D(x). Considering a general form of the diffusion differential operator that includes Itô, Stratonovich, and Hänggi-Klimontovich interpretations of the associated stochastic process, we obtain and analyze the first-passage-time distribution and use it to compute the search efficiency E=〈1/t〉. For the paradigmatic power-law diffusion coefficient D(x)=D_{0}|x|^{α}, where x is the distance from the target and α<2, we show the impact of the different interpretations. For the Stratonovich framework, we obtain a closed-form expression for E, valid for arbitrary diffusion coefficient D(x). This result depends only on the distribution of diffusivity values and not on its spatial organization. Furthermore, the analytical expression predicts that a heterogeneous diffusivity profile leads to a lower efficiency than the homogeneous one with the same average level within the space between the target and the searcher initial position, but this efficiency can be exceeded for other interpretations.

Critical patch size reduction by heterogeneous diffusion.

Population survival depends on a large set of factors and on how they are distributed in space. Due to landscape heterogeneity, species can occupy particular regions that provide the ideal scenario for development, working as a refuge from harmful environmental conditions. Survival occurs if population growth overcomes the losses caused by adventurous individuals that cross the patch edge. In this work, we consider a single species dynamics in a patch with a space-dependent diffusion coefficient. We show analytically, within the Stratonovich framework, that heterogeneous diffusion reduces the minimal patch size for population survival when contrasted with the homogeneous case with the same average diffusivity. Furthermore, this result is robust regardless of the particular choice of the diffusion coefficient profile. We also discuss how this picture changes beyond the Stratonovich framework. Particularly, the Itô case, which is nonanticipative, can promote the opposite effect, while Hänggi-Klimontovich interpretation reinforces the reduction effect.

Dynamical aspects of supercooled TIP3P-water in the grooves of DNA.

We investigate by molecular dynamics simulations the mobility of the water located at the DNA minor and major grooves. We employ the TIP3P water model, and our system is analyzed for a range of temperatures 190-300 K. For high temperatures, the water at the grooves shows an Arrhenius behavior similar to that observed in the bulk water. At lower temperatures, a departure from the bulk behavior is observed. This slowing down in the dynamics is compared with the dynamics of the hydrogen of the DNA at the grooves and with the autocorrelation functions of the water hydrogen bonds. Our results indicate that the hydrogen bonds of the water at the minor grooves are highly correlated, which suggests that this is the mechanism for the slow dynamics at this high confinement.

Entropic nonadditivity, H theorem, and nonlinear Klein-Kramers equations.

We use the H theorem to establish the entropy and the entropic additivity law for a system composed of subsystems, with the dynamics governed by the Klein-Kramers equations, by considering relations among the dynamics of these subsystems and their entropies. We start considering the subsystems governed by linear Klein-Kramers equations and verify that the Boltzmann-Gibbs entropy is appropriated to this dynamics, leading us to the standard entropic additivity, S_{BG}^{(1∪2)}=S_{BG}^{1}+S_{BG}^{2}, consistent with the fact that the distributions of the subsystem are independent. We then extend the dynamics of these subsystems to independent nonlinear Klein-Kramers equations. For this case, the results show that the H theorem is verified for a generalized entropy, which does not preserve the standard entropic additivity for independent distributions. In this scenario, consistent results are obtained when a suitable coupling among the nonlinear Klein-Kramers equations is considered, in which each subsystem modifies the other until an equilibrium state is reached. This dynamics, for the subsystems, results in the Tsallis entropy for the system and, consequently, verifies the relation S_{q}^{(1∪2)}=S_{q}^{1}+S_{q}^{2}+(1-q)S_{q}^{1}S_{q}^{2}/k, which is a nonadditive entropic relation.