The 20th century has revealed two important limitations of scientific knowledge. On the one hand, the combination of Poincares nonlinear dynamics and Heisenbergs uncertainty principle leads to a world picture where physical reality is, in many respec
ts, intrinsically undetermined. On the other hand, Godels incompleteness theorems reveal us the existence of mathematical truths that cannot be demonstrated. More recently, Chaitin has proved that, from the incompleteness theorems, it follows that the random character of a given mathematical sequence cannot be proved in general (it is undecidable). I reflect here on the consequences derived from the indeterminacy of the future and the undecidability of randomness, concluding that the question of the presence or absence of finality in nature is fundamentally outside the scope of the scientific method.
We study the bosonic analog of Andreev reflection at a normal-superfluid interface where the superfluid is a boson condensate. We model the normal region as a zone where nonlinear effects can be neglected. Against the background of a decaying condens
ate, we identify a novel contribution to the current of reflected atoms. The group velocity of this Andreev reflected component differs from that of the normally reflected one. For a three-dimensional planar or two-dimensional linear interface Andreev reflection is neither specular nor conjugate.
Some features of nonadiabatic electron heat pumps are studied and connected to general questions of quantum cooling. Inelastic reflection is shown to contribute to heating if the external driving signal is time-symmetric. The quantum of cooling power
, $pi^2 k_B^2 T^2/6h$, is shown to be an upper limit to the cooling rate per transport channel in the presence of an arbitrary driving signal. The quantum limit to bulk atom cooling is also discussed. Within the electron tunneling limit, it is shown that electron cooling still occurs if the coherent ac source is replaced by a sufficiently hot thermal bath. A comparison with related refrigeration setups is presented.
We investigate the evolution of a central spin coupled to a spin bath without internal dynamics. We compare the cases where the bath couples to one or two components of the spin. It is found that the central spin dynamics is enhanced in the latter ca
se, which may be interpreted as a frustration of dissipation. However, the quantum purity of the spin decays fast in both scenarios. We conclude that symmetric coupling of the bath to two orthogonal components of the spin inhibits dissipation but not decoherence.
We propose that recent transport experiments revealing the existence of an energy gap in graphene nanoribbons may be understood in terms of Coulomb blockade. Electron interactions play a decisive role at the quantum dots which form due to the presenc
e of necks arising from the roughness of the graphene edge. With the average transmission as the only fitting parameter, our theory shows good agreement with the experimental data.
