2026/02/09 by Mathis Guéneau, Satya N. Majumdar, Grégory Schehr
Physics and Astronomy · Engineering · #Micro and Nano Robotics #Advanced Thermodynamics and Statistical Mechanics #Particle Dynamics in Fluid Flows
paper · pdf · doi:10.1103/mlkm-vgbd
We derive the exact nonequilibrium steady state of a run-and-tumble particle (RTP) in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>d</a:mi> </a:math> dimensions confined in an isotropic harmonic trap <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>V</b:mi> <b:mrow> <b:mo>(</b:mo> <b:mi mathvariant="bold">r</b:mi> <b:mo>)</b:mo> </b:mrow> <b:mo>=</b:mo> <b:mi>μ</b:mi> <b:msup> <b:mi>r</b:mi> <b:mn>2</b:mn> </b:msup> <b:mo>/</b:mo> <b:mn>2</b:mn> </b:mrow> </b:math> , with <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>r</d:mi> <d:mo>=</d:mo> <d:mo>∥</d:mo> <d:mi mathvariant="bold">r</d:mi> <d:mo>∥</d:mo> </d:mrow> </d:math> . Rotational invariance reduces the problem to the stationary single-coordinate marginal <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:msub> <f:mi>p</f:mi> <f:mi>X</f:mi> </f:msub> <f:mrow> <f:mo>(</f:mo> <f:mi>x</f:mi> <f:mo>)</f:mo> </f:mrow> </f:mrow> </f:math> , from which the radial distribution <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:msub> <g:mi>p</g:mi> <g:mi>R</g:mi> </g:msub> <g:mrow> <g:mo>(</g:mo> <g:mi>r</g:mi> <g:mo>)</g:mo> </g:mrow> </g:mrow> </g:math> and the full joint stationary density follow by explicit integral transforms. We first focus on a generalized trapped RTP in one dimension, where post-tumble velocities are drawn from an arbitrary distribution <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mi>W</h:mi> <h:mo>(</h:mo> <h:mi>v</h:mi> <h:mo>)</h:mo> </h:mrow> </h:math> . Using a Kesten-type recursion, we represent its stationary position in terms of a stick-breaking (or Dirichlet) process, yielding closed-form expressions for its distribution and its moments. Specializing <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mi>W</i:mi> <i:mo>(</i:mo> <i:mi>v</i:mi> <i:mo>)</i:mo> </i:mrow> </i:math> to the projected velocity law of an isotropic RTP, we reconstruct <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:msub> <j:mi>p</j:mi> <j:mi>R</j:mi> </j:msub> <j:mrow> <j:mo>(</j:mo> <j:mi>r</j:mi> <j:mo>)</j:mo> </j:mrow> </j:mrow> </j:math> and the full joint distribution of all the coordinates in <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mi>d</k:mi> <k:mo>=</k:mo> <k:mn>1</k:mn> <k:mo>,</k:mo> <k:mn>2</k:mn> <k:mo>,</k:mo> <k:mn>3</k:mn> </k:mrow> </k:math> . In <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:mi>d</l:mi> <l:mo>=</l:mo> <l:mn>1</l:mn> </l:mrow> </l:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> , the radial law simplifies to a beta distribution, while in <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mrow> <n:mi>d</n:mi> <n:mo>=</n:mo> <n:mn>3</n:mn> </n:mrow> </n:math> , we derive closed-form expressions for <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mrow> <o:msub> <o:mi>p</o:mi> <o:mi>R</o:mi> </o:msub> <o:mrow> <o:mo>(</o:mo> <o:mi>r</o:mi> <o:mo>)</o:mo> </o:mrow> </o:mrow> </o:math> and the stationary joint distribution <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:mi>P</p:mi> <p:mo>(</p:mo> <p:mi>x</p:mi> <p:mo>,</p:mo> <p:mi>y</p:mi> <p:mo>,</p:mo> <p:mi>z</p:mi> <p:mo>)</p:mo> </p:mrow> </p:math> , which differ from a beta distribution. In all cases, we characterize a persistence-controlled shape transition at the turning surface <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"> <q:mrow> <q:mi>r</q:mi> <q:mo>=</q:mo> <q:msub> <q:mi>v</q:mi> <q:mn>0</q:mn> </q:msub> <q:mo>/</q:mo> <q:mi>μ</q:mi> </q:mrow> </q:math> , where <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"> <r:msub> <r:mi>v</r:mi> <r:mn>0</r:mn> </r:msub> </r:math> is the self-propulsion speed. We further include thermal noise characterized by a diffusion coefficient <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"> <s:mrow> <s:mi>D</s:mi> <s:mo>></s:mo> <s:mn>0</s:mn> </s:mrow> </s:math> , showing that the stationary law is a Gaussian convolution of the <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"> <t:mrow> <t:mi>D</t:mi> <t:mo>=</t:mo> <t:mn>0</t:mn> </t:mrow> </t:math> result, which regularizes turning-point singularities and controls the crossover between persistence- and diffusion-dominated regimes as <u:math xmlns:u="http://www.w3.org/1998/Math/MathML"> <u:mrow> <u:mi>D</u:mi> <u:mo>→</u:mo> <u:mn>0</u:mn> </u:mrow> </u:math> and <v:math xmlns:v="http://www.w3.org/1998/Math/MathML"> <v:mrow> <v:mi>D</v:mi> <v:mo>→</v:mo> <v:mi>∞</v:mi> </v:mrow> </v:math> , respectively. All analytical predictions are systematically validated against numerical simulations.