Abstract:We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions $d \leqslant 3$. For $d > 1$ the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. More precisely, we prove the convergence of the relative partition function and of the reduced density matrices in the $L^r$-norm with optimal exponent $r$. Moreover, we prove the convergence in the $L^\infty$-norm of Wick-ordered reduced density matrices, which allows us to control correlations of Wick-ordered particle densities as well as the asymptotic distribution of the particle number. Our proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials and can, in turn, be expressed as a gas of interacting Brownian loops and paths. When the gas is confined by an external trapping potential, we control the decay of the reduced density matrices using excursion probabilities of Brownian bridges.

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中文翻译：

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正温度下量子玻色气体的平均场极限

摘要：我们证明了相互作用的量子玻色气体的大规范吉布斯态在平均场极限下收敛到非线性薛定谔方程的吉布斯测度，其中气体的密度变大，相互作用强度与反比成正比。密度。我们的结果在维度 $d \leqslant 3$ 中成立。对于 $d > 1$，负正则分布支持吉布斯测度，我们必须重新归一化交互。更准确地说，我们证明了在具有最优指数 $r$ 的 $L^r$-范数中的相对配分函数和约化密度矩阵的收敛性。此外，我们证明了 Wick 有序缩减密度矩阵的 $L^\infty$-范数的收敛性，这使我们能够控制 Wick 有序粒子密度的相关性以及粒子数的渐近分布。我们的证明是基于大规范吉布斯状态的功能积分表示，其中收敛到平均场极限正式来自于定义不明确的非高斯测量的无限维静止相位参数。我们通过引入白噪声型辅助场使这一论点变得严格，通过该辅助场，泛函积分表示为由时间相关的周期性随机势驱动的热方程的传播子，进而可以表示为相互作用的布朗循环和路径。当气体被外部俘获势限制时，

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