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Abstract:

Arguments inspired by algorithmic information theory predict an inverse relation between the probability and complexity of output patterns in a wide range of input-output maps. This phenomenon is known as simplicity bias. By viewing the parameters of dynamical systems as inputs, and the resulting (digitised) trajectories as outputs, we study simplicity bias in the logistic map, Gauss map, sine map, Bernoulli map, and tent map. We find that the logistic map, Gauss map, and sine map all exhibit simplicity bias upon sampling of map initial values and parameter values, but the Bernoulli map and tent map do not. The simplicity bias upper bound on the output pattern probability is used to make a priori predictions regarding the probability of output patterns. In some cases, the predictions are surprisingly accurate, given that almost no details of the underlying dynamical systems are assumed. More generally, we argue that studying probability-complexity relationships may be a useful tool when studying patterns in dynamical systems.

Abstract:

We study the phase diagram of magic-angle twisted symmetric trilayer graphene in the presence of uniaxial heterostrain and interlayer displacement field. For experimentally reasonable strain, our mean-field analysis finds robust Kekulé spiral order whose doping-dependent ordering vector is incommensurate with the moiré superlattice, consistent with recent scanning tunneling microscopy experiments, and paralleling the behavior of closely related twisted bilayer graphene (TBG) systems. Strikingly, we identify a possibility absent in TBG: the existence of commensurate Kekulé spiral order even at zero strain for experimentally realistic values of the interlayer potential in a trilayer. Our studies also reveal a complex pattern of charge transfer between weakly and strongly dispersive bands in strained trilayer samples as the density is tuned by electrostatic gating, that can be understood intuitively in terms of the "cascades"in the compressibility of magic-angle TBG.

Abstract:

A Marangoni surfer is an object embedded in a gas–liquid interface, propelled by gradients in surface tension. We derive an analytical theorem for the lower bound on the viscous dissipation by a Marangoni surfer in the limit of small Reynolds and capillary numbers. The minimum dissipation can be expressed with the reciprocal difference between drag coefficients of two passive bodies of the same shape as the Marangoni surfer, one in a force-free interface and the other in an interface with surface incompressibility. The distribution of surface tension that gives the optimal propulsion is given by the surface tension of the solution for the incompressible surface and the flow is a superposition of both solutions. For a surfer taking the form of a thin circular disk, the minimum dissipation is 16μaV2, giving a Lighthill efficiency of 1/3. This places the Marangoni surfers among the hydrodynamically most efficient microswimmers

Abstract:

We study the competition between Haar-random unitary dynamics and measurements for unstructured systems of qubits. For projective measurements, we derive various properties of the statistical ensemble of Kraus operators analytically, including the purification time and the distribution of Born probabilities. The latter generalizes the Porter–Thomas distribution for random unitary circuits to the monitored setting and is log-normal at long times. We also consider weak measurements that interpolate between identity quantum channels and projective measurements. In this setting, we derive an exactly solvable Fokker–Planck equation for the joint distribution of singular values of Kraus operators, analogous to the Dorokhov–Mello–Pereyra–Kumar (DMPK) equation modelling disordered quantum wires. We expect that the statistical properties of Kraus operators we have established for these simple systems will serve as a model for the entangling phase of monitored quantum systems more generally.