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

We consider overdamped Langevin dynamics for the attractive log gas on the torus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper T Superscript sans-serif d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">d</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathbb {T}}^\mathsf {d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif d greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">d</mml:mi> </mml:mrow> <mml:mo> ≥ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {d}\geq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif d equals 2"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">d</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {d}=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , this model coincides with a periodic version of the parabolic-elliptic Patlak-Keller-Segel model of chemotaxis. The attractive log gas (for our choice of units) is well-known to have a critical inverse temperature <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta Subscript normal c Baseline equals 2 sans-serif d"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> β </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">d</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta _{\mathrm {c}}={2\mathsf {d}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> corresponding to when the free energy is bounded from below. Moreover, it is well-known that the uniform distribution is always a stationary state regardless of the temperature. We identify another temperature threshold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta Subscript normal s"> <mml:semantics> <mml:msub> <mml:mi> β </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\beta _{\mathrm {s}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sharply corresponding to the nonlinear stability of the uniform distribution. We show that for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than beta Subscript normal s"> <mml:semantics> <mml:mrow> <mml:mi> β </mml:mi> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi> β </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta &gt;\beta _{\mathrm {s}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the uniform distribution does not minimize the free energy and moreover is nonlinearly unstable, while for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than beta Subscript normal s"> <mml:semantics> <mml:mrow> <mml:mi> β </mml:mi> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi> β </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta &gt;\beta _{\mathrm {s}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , it is stable. We also show that there exists <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta Subscript normal u"> <mml:semantics> <mml:msub> <mml:mi> β </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">u</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\beta _{\mathrm {u}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which uniqueness of equilibria holds for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than beta Subscript normal u"> <mml:semantics> <mml:mrow> <mml:mi> β </mml:mi> <mml:mo>&gt;</mml:mo>