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Register a new userDelta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
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We revisit the Lieb-Liniger model for $n$ bosons in one dimension with attractive delta interaction in a half-space $mathbb{R}^+$ with diagonal boundary conditions. This model is integrable for arbitrary value of $b in mathbb{R}$, the interaction parameter with the boundary. We show that its spectrum exhibits a sequence of transitions, as $b$ is decreased from the hard-wall case $b=+infty$, with successive appearance of boundary bound states (or boundary modes) which we fully characterize. We apply these results to study the Kardar-Parisi-Zhang equation for the growth of a one-dimensional interface of height $h(x,t)$, on the half-space with boundary condition $partial_x h(x,t)|_{x=0}=b$ and droplet initial condition at the wall. We obtain explicit expressions, valid at all time $t$ and arbitrary $b$, for the integer exponential (one-point) moments of the KPZ height field $bar{e^{n h(0,t)}}$. From these moments we extract the large time limit of the probability distribution function (PDF) of the scaled KPZ height function. It exhibits a phase transition, related to the unbinding to the wall of the equivalent directed polymer problem, with two phases: (i) unbound for $b>-frac{1}{2}$ where the PDF is given by the GSE Tracy-Widom distribution (ii) bound for $b<-frac{1}{2}$, where the PDF is a Gaussian. At the critical point $b=-frac{1}{2}$, the PDF is given by the GOE Tracy-Widom distribution.
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We study the out-of-equilibrium properties of a classical integrable non-relativistic theory, with a time evolution initially prepared with a finite energy density in the thermodynamic limit. The theory considered here is the Non-Linear Schrodinger equation which describes the dynamics of the one-dimensional interacting Bose gas in the regime of high occupation numbers. The main emphasis is on the determination of the late-time Generalised Gibbs Ensemble (GGE), which can be efficiently semi-numerically computed on arbitrary initial states, completely solving the famous quench problem in the classical regime. We take advantage of known results in the quantum model and the semiclassical limit to achieve new exact results for the momenta of the density operator on arbitrary GGEs, which we successfully compare with ab-initio numerical simulations. Furthermore, we determine the whole probability distribution of the density operator (full counting statistics), whose exact expression is still out of reach in the quantum model.
Supercritical fluids play a significant role in elucidating fundamental aspects of liquid matter under extreme conditions. They have been extensively studied at pressures and temperatures relevant to various industrial applications. However, much less is known about the structural behaviour of supercritical fluids and no structural crossovers have been observed in static compression experiments in any temperature and pressure ranges beyond the critical point. The structure of supercritical state is currently perceived to be uniform everywhere on the pressure-temperature phase diagram, and to change only in a monotonic way even moving around the critical point, not only along isotherms or isobars. Conversely, we observe structural crossovers for the first time in a deeply supercritical sample through diffraction measurements in a diamond anvil cell and discover a new thermodynamic boundary on the pressure-temperature diagram. We explain the existence of these crossovers in the framework of the phonon theory of liquids using molecular dynamics simulations. The obtained results are of prime importance since it implies a global reconsideration of the mere essence of the supercritical phase. Furthermore, this discovery may pave the way to new unexpected applications and to the exploration of exotic behaviour of confined fluids relevant to geo- and planetary sciences.
In this paper we provide a detailed description of the eigenvalue $ E_{D}(x_0)leq 0$ (respectively $ E_{N}(x_0)leq 0$) of the self-adjoint Hamiltonian operator representing the negative Laplacian on the positive half-line with a Dirichlet (resp. Neuman) boundary condition at the origin perturbed by an attractive Dirac distribution $-lambda delta(x-x_0)$ for any fixed value of the magnitude of the coupling constant. We also investigate the $lambda$-dependence of both eigenvalues for any fixed value of $x_0$. Furthermore, we show that both systems exhibit resonances as poles of the analytic continuation of the resolvent. These results will be connected with the study of the ground state energy of two remarkable three-dimensional self-adjoint operators, studied in depth in Albeverios monograph, perturbed by an attractive $delta$-distribution supported on the spherical shell of radius $r_0$.
The present paper considers some classical ferromagnetic lattice--gas models, consisting of particles that carry $n$--component spins ($n=2,3$) and associated with a $D$--dimensional lattice ($D=2,3$); each site can host one particle at most, thus implicitly allowing for hard--core repulsion; the pair interaction, restricted to nearest neighbors, is ferromagnetic, and site occupation is also controlled by the chemical potential $mu$. The models had previously been investigated by Mean Field and Two--Site Cluster treatments (when D=3), as well as Grand--Canonical Monte Carlo simulation in the case $mu=0$, for both D=2 and D=3; the obtained results showed the same kind of critical behaviour as the one known for their saturated lattice counterparts, corresponding to one particle per site. Here we addressed by Grand--Canonical Monte Carlo simulation the case where the chemical potential is negative and sufficiently large in magnitude; the value $mu=-D/2$ was chosen for each of the four previously investigated counterparts, together with $mu=-3D/4$ in an additional instance. We mostly found evidence of first order transitions, both for D=2 and D=3, and quantitatively characterized their behaviour. Comparisons are also made with recent experimental results.
A two-dimensional lattice gas model is proposed. The ground state of this model with a fixed density is neither periodic nor quasi-periodic. It also depends on system size in an irregular manner. On the other hand, it is ordered in the sense that the entropy density is zero in the thermodynamic limit. The existence of a thermodynamic transition associated with such irregularly ordered ground states is conjectured from a duality relation for a thermodynamic function. This conjecture is supported by a phenomenological argument and numerical experiments.
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