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It is demonstrated that an isolated generic quantum many-body system does relax to a state well described by the standard statistical mechanical prescription. The thermalization happens at the level of individual eigenstates, allowing the computation of thermal averages from knowledge of any eigenstate in the microcanonical energy window. An understanding of the temporal evolution of isolated many-body quantum systems has long been elusive. Recently, meaningful experimental studies 1 , 2 of the problem have become possible, stimulating theoretical interest 3 , 4 , 5 , 6 , 7 . In generic isolated systems, non-equilibrium dynamics is expected 8 , 9 to result in thermalization: a relaxation to states in which the values of macroscopic quantities are stationary, universal with respect to widely differing initial conditions, and predictable using statistical mechanics. However, it is not obvious what feature of many-body quantum mechanics makes quantum thermalization possible in a sense analogous to that in which dynamical chaos makes classical thermalization possible 10 . For example, dynamical chaos itself cannot occur in an isolated quantum system, in which the time evolution is linear and the spectrum is discrete 11 . Some recent studies 4 , 5 even suggest that statistical mechanics may give incorrect predictions for the outcomes of relaxation in such systems. Here we demonstrate that a generic isolated quantum many-body system does relax to a state well described by the standard statistical-mechanical prescription. Moreover, we show that time evolution itself plays a merely auxiliary role in relaxation, and that thermalization instead happens at the level of individual eigenstates, as first proposed by Deutsch 12 and Srednicki 13 . A striking consequence of this eigenstate-thermalization scenario, confirmed for our system, is that knowledge of a single many-body eigenstate is sufficient to compute thermal averages—any eigenstate in the microcanonical energy window will do, because they all give the same result.

No turning back According to the laws of fluid motion, when a simple fluid or suspension of particles is slowly stirred then unstirred — imagine a spoon in a jar of honey — all parts of the system should miraculously return to their starting points. This is a consequence of the time-reversible equations of motion, at least for two-dimensional flows. But in more complex flows, such as those in three-dimensional or rigorously stirred systems, this delicate effect is destroyed. An investigation of a slowly sheared suspension of solid particles now reveals the microscopic processes behind this transition to irreversible behaviour. Beyond a concentration-dependent threshold strain, irreversibility sets in as a result of chaotic collisions between the particles. Systems governed by time reversible equations of motion often give rise to irreversible behaviour 1 , 2 , 3 . The transition from reversible to irreversible behaviour is fundamental to statistical physics, but has not been observed experimentally in many-body systems. The flow of a newtonian fluid at low Reynolds number can be reversible: for example, if the fluid between concentric cylinders is sheared by boundary motion that is subsequently reversed, then all fluid elements return to their starting positions 4 . Similarly, slowly sheared suspensions of solid particles, which occur widely in nature and science 5 , are governed by time reversible equations of motion. Here we report an experiment showing precisely how time reversibility 6 fails for slowly sheared suspensions. We find that there is a concentration dependent threshold for the deformation or strain beyond which particles do not return to their starting configurations after one or more cycles. Instead, their displacements follow the statistics of an anisotropic random walk 7 . By comparing the experimental results with numerical simulations, we demonstrate that the threshold strain is associated with a pronounced growth in the Lyapunov exponent (a measure of the strength of chaotic particle interactions). The comparison illuminates the connections between chaos, reversibility and predictability.

We report measurements of synchronization in two nanomechanical beam oscillators coupled by a mechanical element. We charted multiple regions of frequency entrainment or synchronization by their corresponding Arnold's tongue diagrams as the oscillator was driven at subharmonic and rational commensurate frequencies. Demonstration of multiple synchronized regions could be fundamentally important to neurocomputing with mechanical oscillator networks and nanomechanical signal processing for microwave communication.

Pulling rank in the city Many distributions, such as the size of cities, companies or the Internet, follow scaling laws that imply an element of stability. A new approach to this type of analysis suggests that a much more turbulent dynamics is at work, but that it is largely hidden when observations focus on a single instant of time. The 'rank clock' is a way of visualizing the behaviour of a system — city size distributions in this case — over long time periods. Tested on three very different city systems over very different time periods, the clocks show that civilizations and cities rise and fall in size many times and on many scales, ruling out universal rank-size scaling at the micro-level and associated models of growth by proportionate effect. But clocks can track significant changes, such as the rise and fall of Rome and the impact of the Industrial Revolution. This paper introduces a method, termed the 'rank clock', to visualize the dynamical behaviour of city size distributions over long periods of time. The clocks show that cities and civilizations rise and fall in size at many times and on many scales, ruling out universal rank-size scaling at the micro-level and associated models of growth by proportionate effect. Many objects and events, such as cities, firms and internet hubs, scale with size 1 , 2 , 3 , 4 in the upper tails of their distributions. Despite intense interest in using power laws to characterize such distributions, most analyses have been concerned with observations at a single instant of time, with little analysis of objects or events that change in size through time (notwithstanding some significant exceptions 5 , 6 , 7 ). It is now clear that the evident macro-stability in such distributions at different times can mask a volatile and often turbulent micro-dynamics, in which objects can change their position or rank-order rapidly while their aggregate distribution appears quite stable. Here I introduce a graphical representation termed the ‘rank clock’ to examine such dynamics for three distributions: the size of cities in the US from ad  1790, the UK from ad  1901 and the world from 430  bc . Our results destroy any notion that rank–size scaling is universal: at the micro-level, these clocks show cities and civilizations rising and falling in size at many times and on many scales. The conventional model explaining such scaling on the basis of growth by proportionate effect cannot replicate these micro-dynamics, suggesting that such models and explanations are considerably less general than has hitherto been assumed.

An improved prescribed performance control using a backstepping technique and adaptive fuzzy is proposed for a strict feedback nonlinear dynamic system. A new virtual variable was defined to generate the virtual control that forces the tracking errors to fall within prescribed boundaries, and an adaptive fuzzy system was used to obtain required approximation performances. A strict feedback controller and adaptive laws for estimating the unknown non-linear function were designed to avoid a singularity problem and calculation of the explosive number of terms generated by the error transformations of conventional error constraint method and the recursive steps of traditional backstepping control. Lyapunov stability analysis confirmed the boundedness and convergence of the closed-loop system. The prescribed error constraint performance of the proposed control scheme was validated by applying it to control the position of a second-order non-linear system and a robot manipulator.

Quantum degenerate Fermi gases provide a remarkable opportunity to study strongly interacting fermions. In contrast to other Fermi systems, such as superconductors, neutron stars or the quark-gluon plasma of the early Universe, these gases have low densities and their interactions can be precisely controlled over an enormous range. Previous experiments with Fermi gases have revealed condensation of fermion pairs. Although these and other studies were consistent with predictions assuming superfluidity, proof of superfluid behaviour has been elusive. Here we report observations of vortex lattices in a strongly interacting, rotating Fermi gas that provide definitive evidence for superfluidity. The interaction and therefore the pairing strength between two 6 Li fermions near a Feshbach resonance can be controlled by an external magnetic field. This allows us to explore the crossover from a Bose–Einstein condensate of molecules to a Bardeen–Cooper–Schrieffer superfluid of loosely bound pairs. The crossover is associated with a new form of superfluidity that may provide insights into high-transition-temperature superconductors. Fermionic superfluidity A clear signature for superfluidity — the frictionless flow seen in some liquids at temperatures close to absolute zero — is the formation of a lattice of quantum vortices in a rotating system. This ‘smoking gun’ has been observed for the first time in an ultracold gas of lithium-6 atoms, confirming the prediction that these quantum gases are superfluids. This system could be a useful model for studies on high-temperature superconductivity and exotic matter such as quark–gluon plasma or neutron stars.

Heisenberg's principle states that the product of uncertainties of position and momentum should be no less than the limit set by Planck's constant, Planck's over 2pi/2. This is usually taken to imply that phase space structures associated with sub-Planck scales (<

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