English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider overdamped Langevin dynamics for the attractive log gas on thetorus $\mathbb{T}^\mathsf{d}$, for $\mathsf{d}\geq 1$. In dimension$\mathsf{d}=2$, this model coincides with a periodic version of theparabolic-elliptic Patlak-Keller-Segel model of chemotaxis. The attractive loggas (for our choice of units) is well-known to have a critical inversetemperature $\beta_{\mathrm{c}}={2\mathsf{d}}$ corresponding to when the freeenergy is bounded from below. Moreover, it is well-known that the uniformdistribution is always a stationary state regardless of the temperature. Weidentify another temperature threshold $\beta_{\mathrm{s}}$ sharplycorresponding to the nonlinear stability of the uniform distribution. We showthat for $\beta>\beta_{\mathrm{s}}$, the uniform distribution does not minimizethe free energy and moreover is nonlinearly unstable, while for$\beta<\beta_{\mathrm{s}}$, it is stable. We also show that there exists$\beta_{\mathrm{u}}$ for which uniqueness of equilibria holds for$\beta<\beta_{\mathrm{u}}$.  We establish a uniform-in-time rate for entropic propagation of chaos for arange of $\beta<\beta_{\mathrm{s}}$. To our knowledge, this is the first suchresult for singular attractive interactions and affirmatively answers aquestion of Bresch et al. arXiv:2011.08022. The proof of the convergence isthrough the modulated free energy method, relying on a modulated logarithmicHardy-Littlewood-Sobolev (mLHLS) inequality. Unlike Bresch et al., we show thatsuch an inequality holds without truncation of the potential -- the avoidanceof the truncation being essential to a uniform-in-time result -- forsufficiently small $\beta$ and provide a counterexample to the mLHLS inequalitywhen $\beta>\beta_{\mathrm{s}}$. As a byproduct, we show that it is impossibleto have a uniform-in-time rate of propagation of chaos if$\beta>\beta_{\mathrm{s}}$.

The attractive log gas: stability, uniqueness, and propagation of chaos