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This book opens with a self-contained introduction to path integrals in Euclidean quantum mechanics and statistical mechanics, and moves on to cover lattice field theory, spin systems, gauge theories and more. Each chapter ends with illustrative problems.

The Toda chain is the prime example of a classical integrable system with strictly local conservation laws. Relying on the Dumitriu–Edelman matrix model, we obtain the generalized free energy of the Toda chain and thereby establish a mapping to the one-dimensional log-gas with an interaction strength of order 1 /  N . The (deterministic) local density of states of the Lax matrix is identified as the object, which should evolve according to generalized hydrodynamics.

We compare bipartite (Euclidean) matching problems in classical and quantum mechanics. The quantum case is treated in terms of a quantum version of the Wasserstein distance. We show that the optimal quantum cost can be cheaper than the classical one. We treat in detail the case of two particles: the equal mass case leads to equal quantum and classical costs. Moreover, we show examples with different masses for which the quantum cost is strictly cheaper than the classical cost.

We rigorously prove that the S = 1 / 2 anisotropic Heisenberg chain (XYZ chain) with next-nearest-neighbor interaction, which is anticipated to be non-integrable, is indeed non-integrable in the sense that this system has no nontrivial local conserved quantity. Our result covers some important models including the Majumdar–Ghosh model, the Shastry–Sutherland model, and many other zigzag spin chains as special cases. These models are shown to be non-integrable while they have some solvable energy eigenstates. In addition to this result, we provide a pedagogical review of the proof of non-integrability of the S = 1 / 2 XYZ chain with Z magnetic field, whose proof technique is employed in our result.

A genetic algorithm (GA) is an evolutionary algorithm inspired by the natural selection and biological processes of reproduction of the fittest individual. GA is one of the most popular optimization algorithms that is currently employed in a wide range of real applications. Initially, the GA fills the population with random candidate solutions and develops the optimal solution from one generation to the next. The GA applies a set of genetic operators during the search process: selection, crossover, and mutation. This article aims to review and summarize the recent contributions to the GA research field. In addition, the definitions of the GA essential concepts are reviewed. Furthermore, the article surveys the real-life applications and roles of GA. Finally, future directions are provided to develop the field.

Using the framework of nonlinear fluctuating hydrodynamics (NFH), we examine equilibrium spatio-temporal correlations in classical ferromagnetic spin chains with nearest neighbor interactions. In particular, we consider the classical XXZ-Heisenberg spin chain (also known as Lattice Landau–Lifshitz or LLL model) evolving deterministically and chaotically via Hamiltonian dynamics, for which energy and z -magnetization are the only locally conserved fields. For the easy-plane case, this system has a low-temperature regime in which the difference between neighboring spin’s angular orientations in the XY plane is an almost conserved field. According to the predictions of NFH, the dynamic correlations in this regime exhibit a heat peak and propagating sound peaks, all with anomalous broadening. We present a detailed molecular dynamics test of these predictions and find a reasonably accurate verification. We find that, in a suitable intermediate temperature regime, the system shows two sound peaks with Kardar-Parisi-Zhang (KPZ) scaling and a heat peak where the expected anomalous broadening is less clear. In high temperature regimes of both easy plane and easy axis case of LLL, our numerics show clear diffusive spin and energy peaks and absence of any sound modes, as one would expect. We also simulate an integrable version of the XXZ-model, for which the ballistic component instead moves with a broad range of speeds rather than being concentrated in narrower peaks around the sound speed.

In this paper, the bipartite consensus tracking control problem is investigated for a class of nonlinear fractional-order multi-agent systems (FOMASs) with unknown dynamics, actuator faults, and input nonlinearities. Based on the adaptive backstepping technique, an adaptive bipartite consensus tracking control framework is constructed for FOMASs, where both cooperative and competitive relationships among agents are implemented. Furthermore, a fault compensation mechanism is proposed to relax the restriction on the number of actuators that can fail, and allow the existence of different types of input nonlinearities for each actuator. In addition, an improved adaptive self-triggered control mechanism that can be dynamically adjusted depending on the bipartite consensus error is extended to FOMASs to save network resources. Then, by means of the fractional-order Lyapunov stability criterion, it is theoretically proved that the proposed control scheme ensures that all signals of the closed-loop systems are bounded and drives the bipartite consensus error into a desired neighborhood of the origin. Finally, simulation results are provided to confirm the effectiveness of the proposed control scheme.

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.

We rigorously justify the mean-field limit of an N -particle system subject to Brownian motions and interacting through the Newtonian potential in R 3 . Our result leads to a derivation of the Vlasov–Poisson–Fokker–Planck (VPFP) equations from the regularized microscopic N -particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and the mean-field trajectories is bounded by N - 1 3 + ε ( 1 63 ≤ ε < 1 36 ) with a blob size of N - δ ( 1 3 ≤ δ < 19 54 - 2 ε 3 ) up to a probability of 1 - N - α for any α > 0 . Moreover, we prove the convergence rate between the empirical measure associated to the regularized particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates rely on the randomness coming from the initial data and from the Brownian motions.