The Maxwell-Boltzmann distribution is a hallmark of statistical physics in thermodynamic equilibrium linking the probability density of a particle's kinetic energies to the temperature of the system that also determines its configurational fluctuations. This unique relation is lost for Hot Brownian Motion, e.g., when the Brownian particle is constantly heated to create an inhomogeneous temperature in the surrounding liquid. While the fluctuations of the particle, in this case, can be described with an effective temperature, it is not unique for all degrees of freedom and suggested to be different at different timescales. In this work, we report on our progress to measure the effective temperature of Hot Brownian Motion in the ballistic regime. We have constructed an optical setup to measure the displacement of a heated Brownian particle with a temporal resolution of 10~ns giving a corresponding spatial resolution of about 23~pm for a 0.92~${\rm \mu m}$ PMMA particle in water. Using a gold-coated polystyrene (AuPS) particle of 2.15~${\rm \mu m}$ diameter we determine the mean squared displacement of the particle over more than 6 orders of magnitude in time. Our data recovers the trends for the effective temperature at long timescales, yet shows also clear effects in the region of hydrodynamic long-time tails.

中文翻译：

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测量单个受热粒子的 Maxwell-Boltzmann 分布

Maxwell-Boltzmann 分布是热力学平衡中统计物理学的一个标志，它将粒子动能的概率密度与系统的温度联系起来，系统的温度也决定了它的配置波动。热布朗运动失去了这种独特的关系，例如，当布朗粒子不断加热以在周围液体中产生不均匀的温度时。虽然在这种情况下粒子的波动可以用有效温度来描述，但它并不是所有自由度都是唯一的，并且建议在不同的时间尺度上有所不同。在这项工作中，我们报告了我们在弹道状态下测量热布朗运动有效温度的进展。我们构建了一个光学装置来测量加热布朗粒子的位移，时间分辨率为 10~ns，对于水中的 0.92~${\rm \mu m}$ PMMA 粒子，相应的空间分辨率约为 23~pm . 使用直径为 2.15~${\rm \mu m}$ 的镀金聚苯乙烯 (AuPS) 颗粒，我们确定了颗粒在超过 6 个数量级的时间内的均方位移。我们的数据恢复了长时间尺度上有效温度的趋势，但在水动力长时尾区域也显示出明显的影响。15~${\rm \mu m}$ 直径，我们确定了粒子在超过 6 个数量级的时间内的均方位移。我们的数据恢复了长时间尺度上有效温度的趋势，但在水动力长时尾区域也显示出明显的影响。15~${\rm \mu m}$ 直径，我们确定了粒子在超过 6 个数量级的时间内的均方位移。我们的数据恢复了长时间尺度上有效温度的趋势，但在水动力长时尾区域也显示出明显的影响。