On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum
We deal with the two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics),...
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| Datum: | 2008 |
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| Sprache: | English |
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Інститут фізики конденсованих систем НАН України
2008
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| Schriftenreihe: | Condensed Matter Physics |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/119137 |
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| Zitieren: | On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum / E. Lytvynov, P.T. Polara // Condensed Matter Physics. — 2008. — Т. 11, № 2(54). — С. 223-236. — Бібліогр.: 24 назв. — англ. |
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irk-123456789-1191372017-06-05T03:03:27Z On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum Lytvynov, E. Polara, P.T. We deal with the two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence of respective generators on a set of cylinder functions, in the L²-norm with respect to the invariant measure of the processes. The latter measure is supposed to be a Gibbs measure corresponding to a potential of pair interaction, in the low activity–high temperature regime. Our result generalizes that of [Random. Oper. Stoch. Equa., 2007, 15, 105], which was proved for a special Glauber (Kawasaki, respectively) dynamics. Ми розглядаємо такi два типи рiвноважних стохастичних динамiк нескiнченно-частинкових систем в континуумi: перестрибуючi частинки (динамiка Кавасакi), тобто динамiка, коли кожна частинка випадковим чином перескакує в просторi; динамiка типу народження-знищення (динамiка Глаубера), при якiй частинки не рухаються, а народжуються i знищуються випадковим чином. Ми доводимо, що для широкого класу динамiк Глаубера кожна така динамiка може бути одержана як скейлiнгова границя динамiки Кавасакi. Точнiше, ми доводимо збiжнiсть вiдповiдних генераторiв на множинi цилiндричних функцiй в нормi L² вiдносно вiдповiдної iнварiантної мiри процесу. Остання є мiрою Гiббса, що вiдповiдає потенцiалу парної взаємодiї в режимi мала активнiсть / високi температури. Наш результат узагальнює результат роботи [Finkelshtein D.L. et al., Random Oper. Stochastic Equations], одержаний для спецiальних типiв динамiк Глаубера i Кавасакi. 2008 Article On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum / E. Lytvynov, P.T. Polara // Condensed Matter Physics. — 2008. — Т. 11, № 2(54). — С. 223-236. — Бібліогр.: 24 назв. — англ. 1607-324X PACS: 02.50.Ey, 02.50.Ga DOI:10.5488/CMP.11.2.223 http://dspace.nbuv.gov.ua/handle/123456789/119137 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We deal with the two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum:
hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops
over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there
is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber
dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence
of respective generators on a set of cylinder functions, in the L²-norm with respect to the invariant measure
of the processes. The latter measure is supposed to be a Gibbs measure corresponding to a potential of pair
interaction, in the low activity–high temperature regime. Our result generalizes that of [Random. Oper. Stoch.
Equa., 2007, 15, 105], which was proved for a special Glauber (Kawasaki, respectively) dynamics. |
| format |
Article |
| author |
Lytvynov, E. Polara, P.T. |
| spellingShingle |
Lytvynov, E. Polara, P.T. On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum Condensed Matter Physics |
| author_facet |
Lytvynov, E. Polara, P.T. |
| author_sort |
Lytvynov, E. |
| title |
On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum |
| title_short |
On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum |
| title_full |
On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum |
| title_fullStr |
On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum |
| title_full_unstemmed |
On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum |
| title_sort |
on convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119137 |
| citation_txt |
On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum / E. Lytvynov, P.T. Polara // Condensed Matter Physics. — 2008. — Т. 11, № 2(54). — С. 223-236. — Бібліогр.: 24 назв. — англ. |
| series |
Condensed Matter Physics |
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2025-07-08T15:18:19Z |
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2025-07-08T15:18:19Z |
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1837092463783706624 |