Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInvariant Gibbs dynamics for the dynamical sine-Gordon model
132
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this note, we study the hyperbolic stochastic damped sine-Gordon equation (SdSG), with a parameter $beta^2 > 0$, and its associated Gibbs dynamics on the two-dimensional torus. After introducing a suitable renormalization, we first construct the Gibbs measure in the range $0<beta^2<4pi$ via the variational approach due to Barashkov-Gubinelli (2018). We then prove almost sure global well-posedness and invariance of the Gibbs measure under the hyperbolic SdSG dynamics in the range $0<beta^2<2pi$. Our construction of the Gibbs measure also yields almost sure global well-posedness and invariance of the Gibbs measure for the parabolic sine-Gordon model in the range $0<beta^2<4pi$.
rate research
Read More
We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary multiplicative Gaussian chaos, we prove local well-posedness of SSG for any value of a parameter $beta^2 > 0$ in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and Shen (2016) and Chandra, Hairer, and Shen (2018), where the parameter is restricted to the subcritical range: $0 < beta^2 < 8 pi$. We also present a triviality result for the unrenormalized SSG.
We consider a parabolic sine-Gordon model with periodic boundary conditions. We prove a fundamental maximum principle which gives a priori uniform control of the solution. In the one-dimensional case we classify all bounded steady states and exhibit some explicit solutions. For the numerical discretization we employ first order IMEX, and second order BDF2 discretization without any additional stabilization term. We rigorously prove the energy stability of the numerical schemes under nearly sharp and quite mild time step constraints. We demonstrate the striking similarity of the parabolic sine-Gordon model with the standard Allen-Cahn equations with double well potentials.
We investigate the invariance of the Gibbs measure for the fractional Schrodinger equation of exponential type (expNLS) $ipartial_t u + (-Delta)^{frac{alpha}2} u = 2gammabeta e^{beta|u|^2}u$ on $d$-dimensional compact Riemannian manifolds $mathcal{M}$, for a dispersion parameter $alpha>d$, some coupling constant $beta>0$, and $gamma eq 0$. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case $gamma>0$, the measure is well-defined in the whole regime $alpha>d$ and $beta>0$ (Theorem 1.1 (i)), while in the focusing case $gamma<0$ its partition function is always infinite for any $alpha>d$ and $beta>0$, even with a mass cut-off of arbitrary small size (Theorem 1.1 (ii)). (ii) We then study the dynamics (expNLS) with random initial data of low regularity. We first use a compactness argument to prove weak invariance of the Gibbs measure in the whole regime $alpha>d$ and $0<beta < beta^star_alpha$ for some natural parameter $0<beta^star_alphasim (alpha-d)$ (Theorem 1.3 (i)). In the large dispersion regime $alpha>2d$, we can improve this result by constructing a local deterministic flow for (expNLS) for any $beta>0$. Using the Gibbs measure, we prove that solutions are almost surely global for $0<beta llbeta^star_alpha$, and that the Gibbs measure is invariant (Theorem 1.3 (ii)). (iii) Finally, in the particular case $d=1$ and $mathcal{M}=mathbb{T}$, we are able to exploit some probabilistic multilinear smoothing effects to build a probabilistic flow for (expNLS) for $1+frac{sqrt{2}}2<alpha leq 2$, locally for arbitrary $beta>0$ and globally for $0<beta ll beta^star_alpha$ (Theorem 1.5).
In this paper, we study the long-time dynamics and stability properties of the sine-Gordon equation $$f_{tt}-f_{xx}+sin f=0.$$ Firstly, we use the nonlinear steepest descent for Riemann-Hilbert problems to compute the long-time asymptotics of the solutions to the sine-Gordon equation whose initial condition belongs to some weighted Sobolev spaces. Secondly, we study the asymptotic stability of the sine-Gordon equation. It is known that the obstruction to the asymptotic stability of the sine-Gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. Combining the long-time asymptotics and a refined approximation argument, we analyze the asymptotic stability properties of the sine-Gordon equation in weighted energy spaces. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds.
We study the dissipative dynamics of one-dimensional fermions, described in terms of the sine-Gordon model in its massive boson or semi-classical limit, while keeping track of forward scattering processes. The system is prepared in the gapped ground state, and then coupled to environment through local currents within the Lindblad formalism. The heating dynamics of the system is followed using bosonization. The single particle density matrix exhibits correlations between the left and right moving particles. While the density matrix of right movers and left movers is translationally invariant, the left-right sector is not, corresponding to a translational symmetry breaking charge density wave state. Asymptotically, the single particle density matrix decays exponentially with exponent proportional to $-gamma t|x|Delta^2$ where $gamma$ and $Delta$ are the dissipative coupling and the gap, respectively. The charge density wave order parameter decays exponentially in time with an interaction independent decay rate. The second Renyi entropy grows linearly with time and is essentially insensitive to the presence of the gap.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
