Brownian motion · Your observations
What does your recording actually identify?
Load the positions of a particle you tracked, and see which diffusion estimate they support and on what assumptions. No recording? bm-07 generates a synthetic one to download.
From your observations to a conditional estimate
Local CSV · reference host calculationYour data stay in this tab. Nothing is uploaded or stored, and nothing goes into a share link. Loading a file does not show that it is a real measurement, or that the model fits it.
Track a real particle under a microscope, load its positions, and the lab estimates how fast it spreads. It also says what that estimate assumes, and what a recording cannot tell you on its own, such as whether camera errors or a slow drift of the stage have crept in.
The lab takes the positions of one or more particles at equal time steps, in micrometres and seconds, and estimates the diffusion coefficient from the steps: along each axis D̂ is the average of Δ2/(2τ), Section 4's λx2 = 2Dt read backwards. Before it gives an interval it checks what that rule assumes: steps independent of one another, no steady drift, and positions without camera error. It tests for drift and for correlation between neighbouring steps, and when those checks fail it says which estimate still applies, as the camera laboratory explains. A recording alone fixes D, not the number of molecules: that also needs the temperature, the viscosity and the particle's radius, each measured separately. Loading a file does not show that it is a real measurement, or that the model fits it.
Take bm-07's downloadable synthetic trajectory as an example: 50 steps of τ = 1 s in two directions, 100 squared steps. Their average divided by 2τ is D̂ = 8.83 × 10−13 m2/s. Because each step is a Gaussian draw, 100 D̂/D follows a chi-square law with 100 degrees of freedom, so the 95 percent interval runs from 100 D̂/129.6 to 100 D̂/74.2, that is from 6.81 to 11.89 × 10−13 m2/s. That interval holds only if the steps are independent and centred. A steady drift at speed u moves every step by uτ as well, adding u2τ2 to each squared step: at u = 0.1 μm/s that is 0.01 μm2, about half a percent of this trajectory's step variance of 1.77 μm2, and it shows up as an average step different from zero. Camera error adds 2σ2 to every step's variance and makes neighbouring steps negatively correlated, by −σ2; a negative correlation between neighbours is the sign of it, and then the estimator that uses disjoint pairs of frames is the one to trust. To go from D to N, Section 5 needs a·N = RT/(6πηD): at 293.15 K in water of viscosity 1 mPa·s, this D gives a·N = 1.465 × 1017 m/mol, and only a radius measured independently turns that into N.
Modern qualifications
Einstein expected displacements to be measured rather than speeds, and closed Section 5 hoping that a researcher would soon decide the question. Svedberg in 1906 and Henri in 1908, with the ultramicroscope and the cinematograph, made early attempts; Perrin's measurements from 1908 gave the first convincing confirmation. The format the lab reads, one row per frame, is modern, as are its drift and correlation checks and the estimators that allow for camera noise (Berglund 2010; Vestergaard, Blainey and Flyvbjerg 2014).
A small, explicit data format
The following is a format illustration, not experimental evidence. A row is a position; successive rows for the same track produce non-overlapping displacements. Choose time and position units in the form rather than relying on a filename or coordinate magnitude.
track,time,x,y A,0,0,0 A,1,0.4,-0.2 A,2,0.1,0.3
Choose ideal independent increments or camera-aware disjoint frame pairs. The camera method includes an explicitly known Gaussian localization noise scale and uniform exposure, rather than silently ignoring them. Pairs stay within the same track; unmatched final frames remain in the observations and are listed in the result and export. Unknown noise, selection bias and unsupported observation models still receive no interval.
An interval is not an authenticity certificate
Conditional coverage describes a procedure under a model, not a posterior probability for a realized interval. The camera method models the declared noise and exposure while holding their values, timing and spatial calibration exact. Uncertainty in those inputs, selection, confinement, correlated tracking errors and different particle properties are not included. A successful calculation does not establish that the recording satisfies the model. A negative estimate or an empty physical confidence set is retained as a diagnostic, never silently replaced by a plausible positive answer.