Long-tailed dissipationless hydromechanics: Weak thermalization and ergodicity breaking

We analyze the dynamic properties of dissipationless generalized Langevin equations in the presence of fluid inertial kernels possessing power-law tails, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>k</a:mi> <a:mrow> <a:mo>(</a:mo> <a:mi>t</a:mi> <a:mo>)</a:mo> </a:mrow> <a:mo>∼</a:mo> <a:msup> <a:mi>t</a:mi> <a:mrow> <a:mo>−</a:mo> <a:mi>κ</a:mi> </a:mrow> </a:msup> </a:mrow> </a:math> . For <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>κ</b:mi> <b:mo>&gt;</b:mo> <b:mn>1</b:mn> </b:mrow> </b:math> no thermalization occurs, and particle motion is ballistic. Since the long-term statistical properties of velocity depend on the initial preparation of the system, this case falls in the realm of nonergodic behavior as momentum transfer is concerned. Conversely, new phenomena arise for <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mn>0</c:mn> <c:mo>&lt;</c:mo> <c:mi>κ</c:mi> <c:mo>&lt;</c:mo> <c:mn>1</c:mn> </c:mrow> </c:math> . In this case, a form of weak thermalization appears in the presence of thermal/hydrodynamic fluctuations and attractive potentials. However, the absence of dissipation clearly emerges once an external constant force is applied: an asymptotic settling velocity cannot be achieved as the expected value of the particle velocity diverges.