Breakdown of hydrodynamics in a one-dimensional cold gas

The following model is studied analytically and numerically: point particles with masses <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>m</a:mi> <a:mo>,</a:mo> <a:mi>μ</a:mi> <a:mo>,</a:mo> <a:mi>m</a:mi> <a:mo>,</a:mo> <a:mo>⋯</a:mo> </a:mrow> </a:math> ( <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>m</b:mi> <b:mo>≥</b:mo> <b:mi>μ</b:mi> </b:mrow> </b:math> ) are distributed over the positive half-axis. Their dynamics is initiated by giving a positive velocity to the particle located at the origin; in its course, the particles undergo elastic collisions. We show that, for certain values of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>m</c:mi> <c:mo>/</c:mo> <c:mi>μ</c:mi> </c:mrow> </c:math> , starting from the initial state where the particles are equidistant, the system evolves in a hydrodynamic way: (i) the rightmost particle (blast front) moves as <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:msup> <d:mi>t</d:mi> <d:mi>δ</d:mi> </d:msup> </d:math> with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi>δ</e:mi> <e:mo><</e:mo> <e:mn>1</e:mn> </e:mrow> </e:math> ; (ii) recoiled particles behind the front enter the negative half-axis; and (iii) the splatter—the particles with locations <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>x</f:mi> <f:mo>≤</f:mo> <f:mn>0</f:mn> </f:mrow> </f:math> —moves in the ballistic way and eventually takes over the whole energy of the system. These results agree with those obtained in S. Chakraborti , , for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mi>m</g:mi> <g:mo>/</g:mo> <g:mi>μ</g:mi> <g:mo>=</g:mo> <g:mn>2</g:mn> </g:mrow> </g:math> , and random initial particle positions. At the same time, we explicitly found the collection of positive numbers <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mo>{</h:mo> <h:msub> <h:mi mathvariant="script">M</h:mi> <h:mi>i</h:mi> </h:msub> <h:mo>,</h:mo> <h:mi>i</h:mi> <h:mo>∈</h:mo> <h:mi mathvariant="bold">N</h:mi> <h:mo>}</h:mo> </h:mrow> </h:math> such that, for <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mi>m</k:mi> <k:mo>/</k:mo> <k:mi>μ</k:mi> <k:mo>=</k:mo> <k:msub> <k:mi mathvariant="script">M</k:mi> <k:mi>i</k:mi> </k:msub> </k:mrow> <k:mo>,</k:mo> <k:mo> </k:mo> <k:mrow> <k:mi>i</k:mi> <k:mo>≤</k:mo> <k:mn>700</k:mn> </k:mrow> </k:math> , the following holds: (a) the splatter is absent; (b) the number of simultaneously moving particles is at most three; and (c) the blast front moves in the ballistic way. However, if, similarly as in S. Chakraborti ., the particle positions are sampled from a uniformly distributed ensemble, for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>/</m:mo> <m:mi>μ</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi mathvariant="script">M</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:math> the system evolves in a hydrodynamic way.