Table of contents for issue 48, volume 52, Journal of Physics A: Mathematical and Theoretical

Dynamical mean field theory (DMFT) is a tool that allows one to analyze the stochastic dynamics of N interacting degrees of freedom in terms of a self-consistent 1-body problem. In this work, focusing on models of ecosystems, we present the derivation of DMFT through the dynamical cavity method, and we develop a method for solving it numerically. Our numerical procedure can be applied to a large variety of systems for which DMFT holds. We implement and test it for the generalized random Lotka–Volterra model, and show that complex dynamical regimes characterized by chaos and aging can be captured and studied by this framework.

For a system in contact with several reservoirs r at different inverse-temperatures , we describe how the Markov jump dynamics with the generalized detailed balance condition can be analyzed via a statistical physics approach of dynamical trajectories over a long time interval . The relevant intensive variables are the time-empirical density , that measures the fractions of time spent in the various configurations , and the time-empirical jump densities , that measure the frequencies of jumps from configuration to configuration when it is the reservoir r that furnishes or absorbs the corresponding energy difference ().

We develop a formalism to compute the statistics of the top eigenpair of weighted sparse graphs with finite mean connectivity and bounded maximal degree. Framing the problem in terms of optimisation of a quadratic form on the sphere and introducing a fictitious temperature, we employ the cavity and replica methods to find the solution in terms of self-consistent equations for auxiliary probability density functions, which can be solved by population dynamics. This derivation allows us to identify and unpack the individual contributions to the top eigenvector’s components coming from nodes of degree k. The analytical results are in perfect agreement with numerical diagonalisation of large (weighted) adjacency matrices, and are further cross-checked on the cases of random regular graphs and sparse Markov transition matrices for unbiased random walks.

We construct birational maps that satisfy the parametric set-theoretical Yang–Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related to integrable nonlinear Schrödinger type equations and study the refactorisation problems of the product of their associated Darboux matrices. Additionally, we study various algebraic properties of the derived maps, such as invariants and associated symplectic or Poisson structures, and we prove their complete integrability in the Liouville sense.

Searching for a signal-emitting source in a turbulent medium is challenging due to sporadic and intermittent measurements. Individual searching agents through sharing information and collaboration can achieve a very effective, flexible and noise-tolerant search in turbulent environments. In this work, by extending the single-agent infotaxis to multi-agent systems, a collaborative infotaxis strategy is proposed to synthesize the spatio-temporal sensing capabilities of a group of agents and optimize the search time. Each agent adjusts its coupling strength with conspecifics and incorporates the conspecific’s observations with dynamic heterogeneous weights according to cognition differences. The cognition difference between agents is measured by the dissimilarity of probability maps. When each agent determines its actions by maximizing the synthesized information of the group, efficient and flexible search behaviors of the group are manifested in various scenarios. The proposed strategy can provide an efficient approach to the investigation of the collaborative behaviors of biological species.

We study state conversion in a parity-time (PT) symmetry broken non-Hermitian two level system. We construct a theory and explain the underlying mechanism for state conversion and define adiabatic evolutions in non-Hermitian systems. The adiabatic theorem can be used if the initial state is an instantaneous eigenstate in Hermitian systems. We show that the system can adiabatically follow the modulationally stable instantaneous eigenstate sooner or later, regardless of the initial state if the system parameters are varied slowly in non-Hermitian systems. We discuss the topological feature of dynamical circling in the parameter space.

We study the mixing behavior of random Lindblad generators with no symmetries, using the dynamical map or propagator of the dissipative evolution. In particular, we determine the long-time behavior of a dissipative form factor, which is the trace of the propagator, and use this as a diagnostic for the existence or absence of a spectral gap in the distribution of eigenvalues of the Lindblad generator. We find that simple generators with a single jump operator are slowly mixing, and relax algebraically in time, due to the closing of the spectral gap in the thermodynamic limit. Introducing additional jump operators or a Hamiltonian opens up a spectral gap which remains finite in the thermodynamic limit, leading to exponential relaxation and thus rapid mixing. We use the method of moments and introduce a novel diagrammatic expansion to determine exactly the form factor to leading order in Hilbert space dimension N. We also present numerical support for our main results.

An infinite family of exact solutions of the two-photon Rabi model was found by investigating the differential algebraic properties of the Hamiltonian. This family corresponds to energy level crossings not covered by the Juddian class, which is given by elemetary functions. In contrast, the new states are expressible in terms of parabolic cylinder or Bessel functions. We discuss three approaches for discovering this hidden structure: factorization of differential equations, Kimura transformation, and a doubly-infinite, transcendental basis of the Bargmann space.

Thermodynamics of quantum systems out-of-equilibrium is very important for the progress of quantum technologies, however, the effects of many-body interactions and their interplay with temperature, different drives and dynamical regimes is still largely unknown. Here we present a systematic study of these interplays in the case of driven Hubbard chains subject to a variety of interaction (from non-interacting to strongly correlated) and dynamical regimes (from sudden quench to quasi-adiabatic), and discuss which effects all these ingredients have on the work extraction and entropy production. As treatment of many-body interacting systems is highly challenging, we introduce a simple approximation which includes, for the average quantum work, many-body interactions only via the initial state, while the dynamics is fully non-interacting. We demonstrate that this simple approximation is surprisingly good for estimating both the average quantum work and the related entropy production, even when many-body correlations are significant.