{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,28]],"date-time":"2026-10-28T03:21:38Z","timestamp":1793157698902,"version":"4.5.0"},"reference-count":95,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[2026,7,22]],"date-time":"2026-07-22T00:00:00Z","timestamp":1784678400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100016394","name":"Association Neurofibromatoses et Recklinghause","doi-asserted-by":"publisher","award":["ANR-23-CE30-0020-01"],"award-info":[{"award-number":["ANR-23-CE30-0020-01"]}],"id":[{"id":"10.13039\/501100016394","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. E"],"accepted":{"date-parts":[[2026,6,18]]},"abstract":"<jats:p>\n                    We derive the exact nonequilibrium steady state of a run-and-tumble particle (RTP) in\n                    <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <a:mi>d<\/a:mi>\n                    <\/a:math>\n                    dimensions confined in an isotropic harmonic trap\n                    <b:math xmlns:b=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <b:mrow>\n                        <b:mi>V<\/b:mi>\n                        <b:mrow>\n                          <b:mo>(<\/b:mo>\n                          <b:mi mathvariant=\"bold\">r<\/b:mi>\n                          <b:mo>)<\/b:mo>\n                        <\/b:mrow>\n                        <b:mo>=<\/b:mo>\n                        <b:mi>\u03bc<\/b:mi>\n                        <b:msup>\n                          <b:mi>r<\/b:mi>\n                          <b:mn>2<\/b:mn>\n                        <\/b:msup>\n                        <b:mo>\/<\/b:mo>\n                        <b:mn>2<\/b:mn>\n                      <\/b:mrow>\n                    <\/b:math>\n                    , with\n                    <d:math xmlns:d=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <d:mrow>\n                        <d:mi>r<\/d:mi>\n                        <d:mo>=<\/d:mo>\n                        <d:mo>\u2225<\/d:mo>\n                        <d:mi mathvariant=\"bold\">r<\/d:mi>\n                        <d:mo>\u2225<\/d:mo>\n                      <\/d:mrow>\n                    <\/d:math>\n                    . Rotational invariance reduces the problem to the stationary single-coordinate marginal\n                    <f:math xmlns:f=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <f:mrow>\n                        <f:msub>\n                          <f:mi>p<\/f:mi>\n                          <f:mi>X<\/f:mi>\n                        <\/f:msub>\n                        <f:mrow>\n                          <f:mo>(<\/f:mo>\n                          <f:mi>x<\/f:mi>\n                          <f:mo>)<\/f:mo>\n                        <\/f:mrow>\n                      <\/f:mrow>\n                    <\/f:math>\n                    , from which the radial distribution\n                    <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <g:mrow>\n                        <g:msub>\n                          <g:mi>p<\/g:mi>\n                          <g:mi>R<\/g:mi>\n                        <\/g:msub>\n                        <g:mrow>\n                          <g:mo>(<\/g:mo>\n                          <g:mi>r<\/g:mi>\n                          <g:mo>)<\/g:mo>\n                        <\/g:mrow>\n                      <\/g:mrow>\n                    <\/g:math>\n                    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\n                    <h:math xmlns:h=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <h:mrow>\n                        <h:mi>W<\/h:mi>\n                        <h:mo>(<\/h:mo>\n                        <h:mi>v<\/h:mi>\n                        <h:mo>)<\/h:mo>\n                      <\/h:mrow>\n                    <\/h:math>\n                    . 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\n                    <i:math xmlns:i=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <i:mrow>\n                        <i:mi>W<\/i:mi>\n                        <i:mo>(<\/i:mo>\n                        <i:mi>v<\/i:mi>\n                        <i:mo>)<\/i:mo>\n                      <\/i:mrow>\n                    <\/i:math>\n                    to the projected velocity law of an isotropic RTP, we reconstruct\n                    <j:math xmlns:j=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <j:mrow>\n                        <j:msub>\n                          <j:mi>p<\/j:mi>\n                          <j:mi>R<\/j:mi>\n                        <\/j:msub>\n                        <j:mrow>\n                          <j:mo>(<\/j:mo>\n                          <j:mi>r<\/j:mi>\n                          <j:mo>)<\/j:mo>\n                        <\/j:mrow>\n                      <\/j:mrow>\n                    <\/j:math>\n                    and the full joint distribution of all the coordinates in\n                    <k:math xmlns:k=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <k:mrow>\n                        <k:mi>d<\/k:mi>\n                        <k:mo>=<\/k:mo>\n                        <k:mn>1<\/k:mn>\n                        <k:mo>,<\/k:mo>\n                        <k:mn>2<\/k:mn>\n                        <k:mo>,<\/k:mo>\n                        <k:mn>3<\/k:mn>\n                      <\/k:mrow>\n                    <\/k:math>\n                    . In\n                    <l:math xmlns:l=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <l:mrow>\n                        <l:mi>d<\/l:mi>\n                        <l:mo>=<\/l:mo>\n                        <l:mn>1<\/l:mn>\n                      <\/l:mrow>\n                    <\/l:math>\n                    and\n                    <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <m:mrow>\n                        <m:mi>d<\/m:mi>\n                        <m:mo>=<\/m:mo>\n                        <m:mn>2<\/m:mn>\n                      <\/m:mrow>\n                    <\/m:math>\n                    , the radial law simplifies to a beta distribution, while in\n                    <n:math xmlns:n=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <n:mrow>\n                        <n:mi>d<\/n:mi>\n                        <n:mo>=<\/n:mo>\n                        <n:mn>3<\/n:mn>\n                      <\/n:mrow>\n                    <\/n:math>\n                    , we derive closed-form expressions for\n                    <o:math xmlns:o=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <o:mrow>\n                        <o:msub>\n                          <o:mi>p<\/o:mi>\n                          <o:mi>R<\/o:mi>\n                        <\/o:msub>\n                        <o:mrow>\n                          <o:mo>(<\/o:mo>\n                          <o:mi>r<\/o:mi>\n                          <o:mo>)<\/o:mo>\n                        <\/o:mrow>\n                      <\/o:mrow>\n                    <\/o:math>\n                    and the stationary joint distribution\n                    <p:math xmlns:p=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <p:mrow>\n                        <p:mi>P<\/p:mi>\n                        <p:mo>(<\/p:mo>\n                        <p:mi>x<\/p:mi>\n                        <p:mo>,<\/p:mo>\n                        <p:mi>y<\/p:mi>\n                        <p:mo>,<\/p:mo>\n                        <p:mi>z<\/p:mi>\n                        <p:mo>)<\/p:mo>\n                      <\/p:mrow>\n                    <\/p:math>\n                    , which differ from a beta distribution. In all cases, we characterize a persistence-controlled shape transition at the turning surface\n                    <q:math xmlns:q=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <q:mrow>\n                        <q:mi>r<\/q:mi>\n                        <q:mo>=<\/q:mo>\n                        <q:msub>\n                          <q:mi>v<\/q:mi>\n                          <q:mn>0<\/q:mn>\n                        <\/q:msub>\n                        <q:mo>\/<\/q:mo>\n                        <q:mi>\u03bc<\/q:mi>\n                      <\/q:mrow>\n                    <\/q:math>\n                    , where\n                    <r:math xmlns:r=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <r:msub>\n                        <r:mi>v<\/r:mi>\n                        <r:mn>0<\/r:mn>\n                      <\/r:msub>\n                    <\/r:math>\n                    is the self-propulsion speed. We further include thermal noise characterized by a diffusion coefficient\n                    <s:math xmlns:s=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <s:mrow>\n                        <s:mi>D<\/s:mi>\n                        <s:mo>&gt;<\/s:mo>\n                        <s:mn>0<\/s:mn>\n                      <\/s:mrow>\n                    <\/s:math>\n                    , showing that the stationary law is a Gaussian convolution of the\n                    <t:math xmlns:t=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <t:mrow>\n                        <t:mi>D<\/t:mi>\n                        <t:mo>=<\/t:mo>\n                        <t:mn>0<\/t:mn>\n                      <\/t:mrow>\n                    <\/t:math>\n                    result, which regularizes turning-point singularities and controls the crossover between persistence- and diffusion-dominated regimes as\n                    <u:math xmlns:u=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <u:mrow>\n                        <u:mi>D<\/u:mi>\n                        <u:mo>\u2192<\/u:mo>\n                        <u:mn>0<\/u:mn>\n                      <\/u:mrow>\n                    <\/u:math>\n                    and\n                    <v:math xmlns:v=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <v:mrow>\n                        <v:mi>D<\/v:mi>\n                        <v:mo>\u2192<\/v:mo>\n                        <v:mi>\u221e<\/v:mi>\n                      <\/v:mrow>\n                    <\/v:math>\n                    , respectively. All analytical predictions are systematically validated against numerical simulations.\n                  <\/jats:p>","DOI":"10.1103\/mlkm-vgbd","type":"journal-article","created":{"date-parts":[[2026,6,18]],"date-time":"2026-06-18T16:27:42Z","timestamp":1781800062000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":3,"title":["Exact stationary state of a\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mi>d<\/mml:mi>\n                    <\/mml:math>\n                    -dimensional run-and-tumble particle in a harmonic potential"],"prefix":"10.1103","volume":"114","author":[{"given":"Mathis","family":"Gu\u00e9neau","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01bf9rw71","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institut f\u00fcr Physik komplexer Systeme"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9844-6980","authenticated-orcid":true,"given":"Satya N.","family":"Majumdar","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00w67e447","id-type":"ROR","asserted-by":"publisher"}],"name":"LPTMS"},{"id":[{"id":"https:\/\/ror.org\/02feahw73","id-type":"ROR","asserted-by":"publisher"}],"name":"CNRS"},{"id":[{"id":"https:\/\/ror.org\/03xjwb503","id-type":"ROR","asserted-by":"publisher"}],"name":"Universit\u00e9 Paris-Saclay"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gr\u00e9gory","family":"Schehr","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02mph9k76","id-type":"ROR","asserted-by":"publisher"}],"name":"Laboratoire de Physique Th\u00e9orique et Hautes Energies"},{"id":[{"id":"https:\/\/ror.org\/02en5vm52","id-type":"ROR","asserted-by":"publisher"}],"name":"Sorbonne Universit\u00e9"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2026,7,22]]},"reference":[{"key":"mlkm-vgbdCc1R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.85.1143","type":"journal-article"},{"key":"mlkm-vgbdCc2R1","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/aa6bc5","type":"journal-article"},{"key":"mlkm-vgbdCc3R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2017.12.137","type":"journal-article"},{"key":"mlkm-vgbdCc4R1","volume-title":"Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences","author":"F. 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