Large Errors in Kinetic Temperature Measurements Using Particle Tracking Velocimetry

TL;DR

The paper addresses inaccuracies in kinetic-temperature measurements obtained from Particle Tracking Velocimetry (PTV) caused by finite camera spatial resolution. It introduces a controlled simulation that prescribes a Maxwellian velocity distribution at a known one-dimensional temperature $T_x$, then applies pixel discretization and the standard two-frame PTV reconstruction to isolate resolution-induced errors from acceleration and tracking mismatch. Results show that spatial discretization can produce large errors in $T_{\mathrm{meas}}$, ranging from tens of percent at $T_x \sim 10\,\mathrm{eV}$ to thousands of percent at $T_x \sim 0.1\,\mathrm{eV}$, with higher frame rates potentially increasing the error due to pixel locking. The work provides a practical protocol to estimate and mitigate these lower-bound errors for dusty-plasma-like experiments and highlights the need for frame-interval adjustments to maintain velocity resolution.

Abstract

We report on random errors in kinetic temperature measurements due to finite spatial resolution in particle tracking velocimetry. Using simulated data, we isolate the error caused by finite spatial resolution from other sources of uncertainty, such as particle acceleration and particle mismatch. A sample of particle velocities is generated from a Maxwellian distribution at a prescribed kinetic temperature. Particle positions are assigned randomly and discretized to match a prescribed spatial resolution. Velocities are reconstructed using the two-frame tracking method, and the resulting kinetic temperature is calculated and compared to the true kinetic temperature. Results show that under typical experimental conditions, the uncertainty in particle positions propagates into large errors in the velocity distribution and the measured kinetic temperature. We find that this might introduce errors ranging from tens of percent at high kinetic temperatures ($\sim 10$~eV) to thousands of percent at low temperatures ($\sim 0.1$~eV).

Figures (3)

• Figure 1: Comparison of true and measured velocity distributions. (a) Histogram of a random velocity sample drawn from Eq. (\ref{['eqMax']}) (Methodology Step 1). (b) Histogram of velocities reconstructed from the same sample using the PTV method (Methodology Step 5) for resolution $R = 24.39\,\mu\mathrm{m/px}$ and a frame rate of $99$ fps. The solid line in both panels represents the theoretical Maxwellian distribution. Note that in panel (b), the velocities take on only eight discrete values. This is a phenomenon peculiar to PTV called "pixel locking" feng2007accurate. The discrepancy between the Maxwellian distribution and the shape of the histogram in (b) illustrates the error introduced by finite spatial resolution. Results are shown for a true kinetic temperature of 10 eV.
• Figure 2: Histogram of discrepancies between true particle velocities $v_{xi}$ and velocities reconstructed using PTV $v_{x i,\mathrm{meas}}$. The figure illustrates that roughly half of the reconstructed velocities deviate from the true values by 300% or more. These large microscopic errors inevitably propagate into the macroscopic calculation of the kinetic temperature. The data correspond to the same random sample used in Fig. \ref{['fvelHist']}, drawn from a Maxwellian distribution at a kinetic temperature of 10 eV. Discrete values of the discrepancies are the manifestation of the "pixel locking" feng2007accurate.
• Figure 3: Contour plots of the fractional error in the kinetic temperature measured by PTV, as a function of camera frame rate and spatial resolution. Results are shown for three true kinetic temperatures: (a) 10 eV, (b) 1.0 eV, and (c) 0.10 eV. The error arises solely from finite spatial resolution. The red cross ($\times$) marks the operating point for our first experimental condition ($\nu=99~\mathrm{fps}$, $R=24.39~\mu\mathrm{m/px}$), and the red diamond ($\diamond$) marks the second condition ($\nu=294~\mathrm{fps}$, $R=30.69~\mu\mathrm{m/px}$). The white contour lines represent isolines of constant fractional error. The uncertainty increases with frame rate and decreases with coarser spatial resolution (larger $R$). Note the large uncertainties for typical frame rates and resolutions (center of each panel), especially in the low-temperature case (panel (c)).