Condensed Matter

We analyse the linear stability of fixed points in large dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modelling, among others, ecosystems consisting of a large number of species that interact through a food-web. We develop an exact theory for the spectral distribution and the leading eigenvalue of the corresponding sparse Jacobian matrices. This theory reveals that the nature of local interactions have a strong influence on system's stability. In particular, we show that fixed points of dynamical systems defined on random graphs are always unstable if they are large enough, except if all interactions are of the predator-prey type. This qualitatively new feature for antagonistic systems is accompanied by a peculiar oscillatory behaviour of the dynamical response of the system after a perturbation, when the mean degree of the graph is small enough. Moreover, for antagonistic systems we also find that there exist a dynamical phase transition and critical mean degree above which the response becomes non-oscillatory.

The dynamics and the steady states of a point-like tracer particle immersed in a confined critical fluid are studied. The fluid is modeled field-theoretically in terms of an order parameter (concentration or density field) obeying dissipative or conservative equilibrium dynamics and (non-)symmetry-breaking boundary conditions. The tracer, which represents, e.g., a colloidal particle, interacts with the fluid by locally modifying its chemical potential or its correlation length. The coupling between tracer and fluid gives rise to a nonlinear and non-Markovian tracer dynamics, which is investigated here analytically and via numerical simulations for a one-dimensional system. From the coupled Langevin equations for the tracer-fluid system we derive an effective Fokker-Planck equation for the tracer by means of adiabatic elimination as well as perturbation theory within a weak-coupling approximation. The effective tracer dynamics is found to be governed by a fluctuation-induced (Casimir) potential, a spatially dependent mobility, and a spatially dependent (multiplicative) noise, the characteristics of which depend on the interaction and the boundary conditions. The steady-state distribution of the tracer is typically inhomogeneous. Notably, when detailed balance is broken, the driving of the temporally correlated noise can induce an effective attraction of the tracer towards a boundary.

Having established the fact that interacting classical kicked rotor systems exhibit long-lived prethermal phase with quasi-conserved average Hamiltonian before entering into chaotic heating regime, we use spatio-temporal fluctuation correlation of kinetic energy to probe the above dynamic phases. We remarkably find the diffusive transport of fluctuation in the prethermal regime reminding us the underlying hydrodynamic picture in a generalized Gibbs ensemble with a definite temperature that depends on the driving parameter and initial conditions. On the other hand, the heating regime, characterized by a diffusive growth of kinetic energy, can sharply localize the correlation at the fluctuation center for all time. Consequently, we attribute non-diffusive and non-localize structure of correlation to the crossover regime, connecting the prethermal phase to the heating phase, where the kinetic energy displays complicated growth structure. We understand these numerical findings using the notion of relative phase matching where prethermal phase (heating regime) refers to an effectively coupled (isolated) nature of the rotors.

We study fluctuating dynamics of a freely movable piston that separates an infinite cylinder into two regions filled with ideal gas particles at the same pressure but different temperatures. To investigate statistical properties of the time-averaged velocity of the piston in the long-time limit, we perturbatively calculate the large deviation function of the time-averaged velocity. Then, we derive an infinite number of effective Langevin equations yielding the same large deviation function as in the original model. Finally, we provide two possibilities for uniquely determining the form of the effective model.

A wave function exposed to measurements undergoes pure state dynamics, with deterministic unitary and probabilistic measurement induced state updates, defining a quantum trajectory. For many-particle systems, the competition of these different elements of dynamics can give rise to a scenario similar to quantum phase transitions. To access it despite the randomness of single quantum trajectories, we construct an n -replica Keldysh field theory for the ensemble average of the n -th moment of the trajectory projector. A key finding is that this field theory decouples into one set of degrees of freedom that heats up indefinitely, while n?? others can be cast into the form of pure state evolutions generated by an effective non-Hermitian Hamiltonian. This decoupling is exact for free theories, and useful for interacting ones. In particular, we study locally measured Dirac fermions in (1+1) dimensions, which can be bosonized to a monitored interacting Luttinger liquid at long wavelengths. For this model, the non-Hermitian Hamiltonian corresponds to a quantum Sine-Gordon model with complex coefficients. A renormalization group analysis reveals a gapless critical phase with logarithmic entanglement entropy growth, and a gapped area law phase, separated by a Berezinskii-Kosterlitz-Thouless transition. The physical picture emerging here is a pinning of the trajectory wave function into eigenstates of the measurement operators upon increasing the monitoring rate.

This work describes a simple agent model for the spread of an epidemic outburst, with special emphasis on mobility and geographical considerations, which we characterize via statistical mechanics and numerical simulations. As the mobility is decreased, a percolation phase transition is found separating a free-propagation phase in which the outburst spreads without finding spatial barriers and a localized phase in which the outburst dies off. Interestingly, the number of infected agents is subject to maximal fluctuations at the transition point, building upon the unpredictability of the evolution of an epidemic outburst. Our model also lends itself to test with vaccination schedules. Indeed, it has been suggested that if a vaccine is available but scarce it is convenient to select carefully the vaccination program to maximize the chances of halting the outburst. We discuss and evaluate several schemes, with special interest on how the percolation transition point can be shifted, allowing for higher mobility without epidemiological impact.

We study a self-organized critical system under influence of turbulent motion of the environment. The system is described by the anisotropic continuous stochastic equation proposed by Hwa and Kardar [{\it Phys. Rev. Lett.} {\bf 62}: 1813 (1989)]. The motion of the environment is modelled by the isotropic Kazantsev--Kraichnan "rapid-change" ensemble for an incompressible fluid: it is Gaussian with vanishing correlation time and the pair correlation function of the form ∝δ(t− t ′ )/ k d+ξ , where k is the wave number and ξ is an arbitrary exponent with the most realistic values ξ=4/3 (Kolmogorov turbulence) and ξ→2 (Batchelor's limit). Using the field-theoretic renormalization group, we find infrared attractive fixed points of the renormalization group equation associated with universality classes, i.e., with regimes of critical behavior. The most realistic values of the spatial dimension d=2 and the exponent ξ=4/3 correspond to the universality class of pure turbulent advection where the nonlinearity of the Hwa--Kardar (HK) equation is irrelevant. Nevertheless, the universality class where both the (anisotropic) nonlinearity of the HK equation and the (isotropic) advecting velocity field are relevant also exists for some values of the parameters ε=4−d and ξ . Depending on what terms (anisotropic, isotropic, or both) are relevant in specific universality class, different types of scaling behavior (ordinary one or generalized) are established.

Maxwell refrigerator as a device that can transfer heat from a cold to hot temperature reservoir making use of information reservoir was introduced by Mandal et al. \cite{Mandal2013a}. The model has a two state demon and a bit stream interacting with two thermal reservoirs simultaneously. We work out a simpler version of the refrigerator where the demon and bit system interact with the reservoirs separately and for a duration long enough to establish equilibrium. The efficiency, η , of the device when working as an engine as well as the coefficient of performance (COP) when working as a refrigerator are calculated. It is shown that the maximum efficiency matches that of a Carnot engine/refrigerator working between the same temperatures, as expected. The COP at maximum power decreases as 1 T h when T h > T c ≫ΔE ( k B =1 ), where T h and T c are the temperatures of the hot and cold reservoirs respectively and ΔE is the level spacing of the demon. η at maximum power of the device, when working as a heat engine, is found to be T h 0.779+ T h when T c ≪ΔE and T h ≫ΔE .

The canonical ensemble plays a crucial role in statistical mechanics in and out of equilibrium. For example, the standard derivation of the fluctuation theorem relies on the assumption that the initial state of the heat bath is the canonical ensemble. On the other hand, the recent progress in the foundation of statistical mechanics has revealed that a thermal equilibrium state is not necessarily described by the canonical ensemble but can be a quantum pure state or even a single energy eigenstate, as formulated by the eigenstate thermalization hypothesis (ETH). Then, a question raised is how these two pictures, the canonical ensemble and a single energy eigenstate as a thermal equilibrium state, are compatible in the fluctuation theorem. In this paper, we theoretically and numerically show that the fluctuation theorem holds in both of the long and short-time regimes, even when the initial state of the bath is a single energy eigenstate of a many-body system. Our proof of the fluctuation theorem in the long-time regime is based on the ETH, while it was previously shown in the short-time regime on the basis of the Lieb-Robinson bound and the ETH [Phys. Rev. Lett. 119, 100601 (2017)]. The proofs for these time regimes are theoretically independent and complementary, implying the fluctuation theorem in the entire time domain. We also perform a systematic numerical simulation of hard-core bosons by exact diagonalization and verify the fluctuation theorem in both of the time regimes by focusing on the finite-size scaling. Our results contribute to the understanding of the mechanism that the fluctuation theorem emerges from unitary dynamics of quantum many-body systems, and can be tested by experiments with, e.g., ultracold atoms.

Much has been learned about universal properties of the eigenstate entanglement entropy for one-dimensional lattice models, which is described by a Hermitian Hamiltonian. While very less of it has been understood for non-Hermitian systems. In the present work we study a non-Hermitian, non-interacting model of fermions which is invariant under combined PT transformation. Our models show a phase transition from PT unbroken phase to broken phase as we tune the hermiticity breaking parameter. Entanglement entropy of such systems can be defined in two different ways, depending on whether we consider only right (or equivalently only left) eigenstates or a combination of both left and right eigenstates which form a complete set of bi-orthonormal eigenstates. We demonstrate that the entanglement entropy of the ground state and also of the typical excited states show some unique features in both of these phases of the system. Most strikingly, entanglement entropy obtained taking a combination of both left and right eigenstates shows an exponential divergence with system size at the transition point. While in the PT -unbroken phase, the entanglement entropy obtained from only the right (or equivalently left) eigenstates shows identical behavior as of an equivalent Hermitian system which is connected to the non-Hermitian system by a similarity transformation.