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Register a new userGlobal fluctuations for 1D log-gas dynamics. (2) Covariance kernel and support
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Jeremie Unterberger M
Publication date
2018
fields
Physics
and research's language is
English
Authors
Jeremie Unterberger
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We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, $ dlambda_t^i=frac{1}{sqrt{N}} dW_t^i - V(lambda_t^i) dt+ frac{beta}{2N} sum_{j ot=i} frac{dt}{lambda^i_t-lambda^j_t}, qquad i=1,ldots,N, $ with $beta>1$, sometimes called generalized Dysons Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $beta$-ensemble, with sufficiently regular convex potential $V$. The limit $Ntoinfty$ is known to satisfy a mean-field Mc Kean-Vlasov equation. Fluctuations around this limit have been shown by the author to define a Gaussian process solving some explicit martingale problem written in terms of a generalized transport equation. We prove a series of results concerning either the Mc Kean-Vlasov equation for the density $rho_t$, notably regularity results and time-evolution of the support, or the associated hydrodynamic fluctuation process, whose space-time covariance kernel we compute explicitly.
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We study in this article the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, begin{equation} dlambda_t^i=frac{1}{sqrt{N}} dW_t^i - V(lambda_t^i) dt+ frac{beta}{2N} sum_{j ot=i} frac{dt}{lambda^i_t-lambda^j_t}, qquad i=1,ldots,N, end{equation} with $beta>1$, sometimes called generalized Dysons Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $beta$-ensemble, with sufficiently regular convex potential $V$. The limit $Ntoinfty$ is known to satisfy a mean-field Mac-Kean-Vlasov equation. We prove that, for suitable initial conditions, fluctuations around the limit are Gaussian and satisfy an explicit PDE. The proof is very much indebted to the harmonic potential case treated in Israelsson cite{Isr}. Our key argument consists in showing that the time-evolution generator may be written in the form of a transport operator on the upper half-plane, plus a bounded non-local operator interpreted in terms of a signed jump process. As an essential technical argument ensuring the convergence of the above scheme, we give an $N$-independent large-deviation type estimate for the probability that $sup_{tin[0,T]}max_{i=1,ldots,N} |lambda_t^i|$ is large, based on a multi-scale argument and entropic bounds.
Understanding the rich behavior that emerges from systems of interacting quantum particles, such as electrons in materials, nucleons in nuclei or neutron stars, the quark-gluon plasma, and superfluid liquid helium, requires investigation of systems that are clean, accessible, and have tunable parameters. Ultracold quantum gases offer tremendous promise for this application largely due to an unprecedented control over interactions. Specifically, $a$, the two-body scattering length that characterizes the interaction strength, can be tuned to any value. This offers prospects for experimental access to regimes where the behavior is not well understood because interactions are strong, atom-atom correlations are important, mean-field theory is inadequate, and equilibrium may not be reached or perhaps does not even exist. Of particular interest is the unitary gas, where $a$ is infinite, and where many aspects of the system are universal in that they depend only on the particle density and quantum statistics. While the unitary Fermi gas has been the subject of intense experimental and theoretical investigation, the degenerate unitary Bose gas has generally been deemed experimentally inaccessible because of three-body loss rates that increase dramatically with increasing $a$. Here, we investigate dynamics of a unitary Bose gas for timescales that are short compared to the loss. We find that the momentum distribution of the unitary Bose gas evolves on timescales fast compared to losses, and that both the timescale for this evolution and the limiting shape of the momentum distribution are consistent with universal scaling with density. This work demonstrates that a unitary Bose gas can be created and probed dynamically, and thus opens the door for further exploration of this novel strongly interacting quantum liquid.
We present measurements of the dynamical structure factor $S(q,omega)$ of an interacting one-dimensional (1D) Fermi gas for small excitation energies. We use the two lowest hyperfine levels of the $^6$Li atom to form a pseudo-spin-1/2 system whose s-wave interactions are tunable via a Feshbach resonance. The atoms are confined to 1D by a two-dimensional optical lattice. Bragg spectroscopy is used to measure a response of the gas to density (charge) mode excitations at a momentum $q$ and frequency $omega$. The spectrum is obtained by varying $omega$, while the angle between two laser beams determines $q$, which is fixed to be less than the Fermi momentum $k_textrm{F}$. The measurements agree well with Tomonaga-Luttinger theory.
In this work we analyze PT-symmetric double-well potentials based on a two-mode picture. We reduce the problem into a PT-symmetric dimer and illustrate that the latter has effectively two fundamental bifurcations, a pitchfork (symmetry-breaking bifurcation) and a saddle-center one, which is the nonlinear analog of the PT-phase-transition. It is shown that the symmetry breaking leads to ghost states (amounting to growth or decay); although these states are not true solutions of the original continuum problem, the systems dynamics closely follows them, at least in its metastable evolution. Past the second bifurcation, there are no longer states of the original continuum system. Nevertheless, the solutions can be analytically continued to yield a new pair of branches, which is also identified and dynamically examined. Our explicit analytical results for the dimer are directly compared to the full continuum problem, yielding a good agreement.
We propose a spatially and temporally nonlocal exchange-correlation (xc) kernel for the spin-unpolarized fluid phase of ground-state jellium, for use in time-dependent density functional and linear response calculations. The kernel is constructed to satisfy known properties of the exact xc kernel, to accurately describe the correlation energies of bulk jellium, and to satisfy frequency-moment sum rules at a wide range of bulk jellium densities. All exact constraints satisfied by the recent MCP07 kernel [A. Ruzsinszky, et al., Phys. Rev. B 101, 245135 (2020)] are maintained in the new tightly-constrained 2021 (TC21) kernel, while others are added.
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