We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat external potential field, in contact with a thermal bath. For an infinite system size, the particles may escape the force field and diffuse freely at large length scales. The partition function diverges and hence the standard canonical ensemble fails. This is replaced with tools stemming from infinite ergodic theory. Boltzmann-Gibbs statistics, even though not normalized, still describes integrable observables, like energy and occupation times. The Boltzmann infinite density is derived heuristically using an entropy maximization principle, as well as via a first-principles calculation using an eigenfunction expansion in the continuum of low-energy states. A generalized virial theorem is derived, showing how the virial coefficient describes the delay in the diffusive spreading of the particles, found at large distances. When the process is non-recurrent, e.g. diffusion in three dimensions with a Coulomb-like potential, we use weighted time averages to restore basic canonical relations between time and ensemble averages.

我们研究与热浴接触的渐近平坦外部势场中粒子的过度阻尼随机动力学。对于无限的系统大小，粒子可能会逃逸力场，并在较大的长度尺度上自由扩散。分区函数发散，因此标准规范集合失败。取而代之的是源自无限遍历理论的工具。Boltzmann-Gibbs统计数据尽管未进行归一化，但仍描述了可积分的可观测量，例如能量和占用时间。玻尔兹曼无限密度是使用熵最大化原理通过启发式推导得出的，也可以通过使用低能量状态连续体中本征函数展开的第一原理计算得出。推导了一个广义的病毒定理，展示了病毒系数如何描述在远距离处发现的颗粒扩散扩散的延迟。当该过程是非周期性的，例如在具有库仑势的三维空间中扩散时，我们使用加权时间平均值来恢复时间与整体平均值之间的基本规范关系。