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تسجيل مستخدم جديدتجميع في نموذج ستوشاستيكي لغاز ذو بعد واحد
Clustering in a stochastic model of one-dimensional gas
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تأليف
Vladislav V. Vysotsky
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نقدَّيُّنا لتحليلٍ كميٍّ للتجمُّع في نموذجٍ احتماليٍّ للغازِ ذو بعدٍ واحد. في الزمنِ الصفريِّ، يتكوّن الغازُ من $n$ جسيماتٍ متشابهاتٍ موزَّعةٍ عشوائيًّا على الخطِ الحقيقيِّ وبسرعاتٍ صفريَّة. تبدأ الجسيماتُ في الحركةِ تحتَ تأثيرِ الجذبِ المتبادل. عندما تتصادم الجسيماتُ، تلتصقَ ببعضها البعضَ يشكِّلُون جسيمةً جديدةً، والتي تسمى التجمُّع، والتي تحدَّدُ شئونَها الجساميَّةَ والسرعةَ بما يطابقُ قوانينَ الاحتفاظ. نحنُ مهتمونُ بسلوكِ الأصعبِ كما يصلُ إلى الأبدِ من $K_n(t)$ عندما يصلُ $n$ إلى الأبدِ، حيثُ يشيرُ $K_n(t)$ إلى عددِ التجمُّعاتِ في النظامِ الذي يحتوي على $n$ جسيماتٍ أصليَّة. نتائجُنا الرئيسيَّةُ هي قانونُ الحدِّ الوظيفيِّ لـ $K_n(t)$. دليلُ ثبوتِها يستندُ إلى خاصيةِ التحديدِ المكتشفةِ لعمليَّةِ التجمُّع، التي تشيرُ إلى أنَّ سلوكَ كلِّ جسيمةٍ يعتمدُ بشكلٍ أساسيِّ على حركةِ الجسيماتِ المجاورة.
We give a quantitative analysis of clustering in a stochastic model of one-dimensional gas. At time zero, the gas consists of $n$ identical particles that are randomly distributed on the real line and have zero initial speeds. Particles begin to move under the forces of mutual attraction. When particles collide, they stick together forming a new particle, called cluster, whose mass and speed are defined by the laws of conservation. We are interested in the asymptotic behavior of $K_n(t)$ as $nto infty$, where $K_n(t)$ denotes the number of clusters at time $t$ in the system with $n$ initial particles. Our main result is a functional limit theorem for $K_n(t)$. Its proof is based on the discovered localization property of the aggregation process, which states that the behavior of each particle is essentially defined by the motion of neighbor particles.
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