To study the dynamical distributions in diffusion, we devised a microfluidics experiment. Using the techniques of soft lithography, chip fabrication,^{18} and the Sylgard 184 Silicone elastomer kit (Dow Corning Corporation), we made a microfluidics chamber having approximate dimensions of 400 μm by 100 μm, partitioned into two regions (see ). The cross section of this chamber is a segment of a circular disc, with a maximum depth of 10 μm (see ). The chamber is filled on one side with a solution containing about 200 colloidal, green fluorescent polystyrene particles 0.29 μm in diameter (Duke Scientific, Cat. No. G300) (see ). The beads are at an optimized concentration so that the interactions are negligible^{19} while at the same time permitting sufficient statistics over a wider range of Δ*N* and *N*.

At time *t* = 0, we open a microfluidic gate (i.e., a partition), allowing particles to diffuse from one side to the other, taking periodic snapshots under an Olympus IX71 inverted microscope. (We performed the same experiment under equilibrium conditions where the initial concentration was uniform across the whole chamber (results not shown, see ^{ref 20}).) We take three snapshots of the beads in the chamber every time interval of Δ*t* = 10 s, for 6 h. Since there is a possibility that some particles temporarily overlap and/or are out of focus in a single snapshot, taking three snapshots of each minimizes that error to 1–2%, which corresponds to 2–4 particles out of the 200. The snapshots are taken using fluorescence microscopy with a SONY DFW-V500 camera. (During the time when no snapshots are taken, a shutter prevents the experimental chamber from being exposed to the incident light, to prevent photobleaching and heating the chamber.) We then determine the particle positions at each snapshot using a computerized centroid tracking algorithm.^{25}

The time-dependent particle density is determined by dividing the chamber into a number of equal-sized bins of value Δ*x* each along the longest dimension of 400 μm and by computing the number of particles in each bin as a function of time. Although the microfluidic chamber is three-dimensional, it can be shown that, in the case of weak particle–particle and particle–wall interactions, the problem can be collapsed to a one-dimension diffusion problem. Therefore, we bin only along the *x*-axis, the direction of the concentration gradient.

As expected from the equations presented in the previous section,

eqs 3–

7, the results presented below are independent of the choice of the bin size for bin sizes that are reasonable (i.e., clearly bins with a size compared to the entire chamber are not useful). Indeed, different values for the bin size were used for the data analysis, all producing results that agree with the ones shown in section III. However, the choice of the bin size affects the statistics for each combination of

*N*_{1} and

*N*_{2} as well as the range of

*N* and Δ

*N* themselves. If the bins are too wide, then there will not be enough statistics and the range of values for

*N* will not include small numbers (since it will be rare to have one or two particles in a single bin). On the other hand, if the bin size is too small, we may not have a sufficient range of values for

*N* and Δ

*N* (since small bins will rarely have more than a few particles). Also, if the bins are too small, a particle may jump across multiple bins within the time interval, Δ

*t*. Therefore, the optimal choice of the bin size was made on the basis of the bead's expected mean excursion within the time interval, Δ

*t*, which is x

. This is the only relevant microscopic length scale. Here,

*D* is the diffusion coefficient for an individual bead given by the Stokes formula.

^{26}For a bead of 0.29 μm in diameter suspended in water at room temperature, the Stokes formula gives a diffusion coefficient, *D*, of approximately 1.5 μm^{2}/s. This value, within experimental error, is equal to the one we obtain by fitting our data of the concentration profile at different times to the one-dimensional diffusion equation using *D* as our fitting parameter (i.e., *D* = 1.3 ± 0.27 μm^{2}/s). This gives a bin size of Δ*x* ≈ 5 μm. By observing all the consecutive bin pairs for all the frames taken, we were able to obtain, on average, about 5000 points for each combination of *N*_{1} and *N*_{2}. Given the bead concentration in the microfluidics channel, *N*_{1} and *N*_{2} ranged from 0 to 6. The choice of bin size determines the value of the jump probability, *p*, as discussed in ^{ref 21}.

We can find the flux at a plane

*i* at a specific time interval from the computed particle distribution statistics as a function of position

*x* and time

*t* mentioned above. Since the microfluidic chamber is isolated, the total number of particles stays the same from one frame to the next. As a result of this conservation in particle number, the flux at plane

*i* + 1,

*J*_{i+1}, that is, the plane that separates bins

*i* and

*i* + 1, can be easily evaluated by using the continuity equation:

where

*N*_{i} is the number of particles in bin

*i*. Since the microfluidic chamber is isolated, from our boundary conditions, the flux

*J*_{0} (flux at

*x* = 0) is zero at all times. Combined with

eq 9, we obtain

*J*_{1}(

*t*). Thus, from the analysis of these images, we obtain complete sets of the values of {

*N*_{i}(

*t*)} and {

*J*_{i}(

*t*)} in all of the bins and at all times of observation. Then, for each pair of consecutive bins with specific values of

*N*_{1} and

*N*_{2}, we construct the histogram of

*J* values. Upon normalization, the histogram becomes the flux probability distribution,

*P*(

*J*).