Mathematik in den Naturwissenschaften Leipzig Large deviations for trapped interacting Brownian particles and paths
Mathematik in den Naturwissenschaften Leipzig Large deviations for trapped interacting Brownian particles and paths
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莱比锡自然科学中心的数学 捕获相互作用的布朗粒子和路径的大偏差
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发表时间：
2004
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通讯作者：
Wolfgang König
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作者：
Stefan Adams;J. Bru;Wolfgang König
我们介绍了两个概率模型N相互作用的布朗运动在一个陷阱中的Rd相互排斥的力量。这两个模型被定义在有限的时间间隔下的陷阱哈密顿量和两个各自的对相互作用哈密顿量的转换路径措施。第一对相互作用表现出粒子排斥性，而第二个施加路径排斥性。我们分析了这两个模型的极限发散时间与固定数量N的布朗运动。特别是，我们证明了大偏差原则的正常化占领措施。速率函数的极小值与某个相关算子有关，即N个相互作用的俘获粒子系统的汉密尔顿算子。更准确地说，在粒子排斥模型中，最小值是其基态，而在路径排斥模型中，最小值是其基态。在路径排斥的情况下，我们还讨论了Dirac型相互作用的情况下，这是严格定义的布朗交叉局部时间。我们证明了一个大偏差的结果为一个离散变量的模型。这项研究是一个贡献的N个被困的相互作用玻色子的量子系统的数学公式作为玻色-爱因斯坦凝聚的模型，著名的1995年实验的成功的动机搜索。最近，Lieb等人根据著名的Gross-Pitaevskii公式描述了基态的大N行为，其中涉及对势的散射长度。我们证明了基态的大N行为也可以用Gross-Pitaevskii公式来描述，但是对势的散射长度用它的积分来代替。Max-Planck Institute for Mathematics in the Sciences，Inselstraße 22-26，D-04103 Leipzig，德国，adams@mis.mpg.de 2School of Theoretical Physics，都柏林Institute for Advanced Studies，10，Burlington Road，都柏林4，爱尔兰3Fachbereich Mathematik und Informatik，Johannes-Gutenberg-Universität Mainz，Staudingerweg 9，D-55099 Mainz，德国，jbbru@mathematik.uni-mainz.de 4Mathematisches Institut，Universität Leipzig，Augustusplatz 10/11，D-04109 Leipzig，德国， koenig@math.uni-leipzig.de MSC 2000. 60 F10; 60 J65; 82 B10; 82 B26。
We introduce two probabilistic models for N interacting Brownian motions moving in a trap in Rd under mutually repellent forces. The two models are defined in terms of transformed path measures on finite time intervals under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The first pair interaction exhibits a particle repellency, while the second one imposes a path repellency. We analyse both models in the limit of diverging time with fixed number N of Brownian motions. In particular, we prove large deviations principles for the normalised occupation measures. The minimisers of the rate functions are related to a certain associated operator, the Hamilton operator for a system of N interacting trapped particles. More precisely, in the particle-repellency model, the minimiser is its ground state, and in the path-repellency model, the minimisers are its ground product-states. In the case of path-repellency, we also discuss the case of a Dirac-type interaction, which is rigorously defined in terms of Brownian intersection local times. We prove a large-deviation result for a discrete variant of the model. This study is a contribution to the search for a mathematical formulation of the quantum system of N trapped interacting bosons as a model for Bose-Einstein condensation, motivated by the success of the famous 1995 experiments. Recently, Lieb et al. described the large-N behaviour of the ground state in terms of the well-known Gross-Pitaevskii formula, involving the scattering length of the pair potential. We prove that the large-N behaviour of the ground product-states is also described by the Gross-Pitaevskii formula, however with the scattering length of the pair potential replaced by its integral. Max-Planck Institute for Mathematics in the Sciences, Inselstraße 22-26, D-04103 Leipzig, Germany, adams@mis.mpg.de 2School of Theoretical Physics, Dublin Institute for Advanced Studies, 10, Burlington Road, Dublin 4, Ireland 3Fachbereich Mathematik und Informatik, Johannes-Gutenberg-Universität Mainz, Staudingerweg 9, D-55099 Mainz, Germany, jbbru@mathematik.uni-mainz.de 4Mathematisches Institut, Universität Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany, koenig@math.uni-leipzig.de MSC 2000. 60F10; 60J65; 82B10; 82B26.