Colloidal free expansion setup to illustrate diffusion involving small numbers of particles. (a) Schematic of experimental setup. (b) Several snapshots from the experiment. (c) Normalized histogram of particle positions during the experiment. The solution to the diffusion equation for the microfluidic “free expansion” experiment is superposed for comparison.
Schematic of the simple dog-flea model. (a) State of the system at time ; (b) a particular microtrajectory in which two fleas jump from the dog on the left and one flea jumps from the dog on the right; (c) occupancies of the dogs at time .
Schematic of the distribution of fluxes for different times as the system approaches equilibrium.
Schematic of which trajectories are potent and which are impotent. The shaded region corresponds to the impotent trajectories for which and are either equal or approximately equal and hence make relatively small change in the macrostate. The unshaded region corresponds to potent trajectories.
Illustration of the potency of the microtrajectories associated with different distributions of particles on the two dogs. The total number of particles .
Illustration of the notion of bad actors. Bad actors are the microtrajectories that contribute net particle motion that has the opposite sign from the macroflux.
The fraction of all possible trajectories that go against the direction of the macroflux for . The fraction of bad actors is highest at .
Illustration of Newton’s law of viscosity. The fluid is sheared with a constant stress. The fluid velocity decreases continuously from its maximum value at the top of the fluid to zero at the bottom. There is thus a gradient in the velocity which can be related to the shear stress in the fluid.
The fraction of potent trajectories as a function of for , and and . The minimum value of the potency does not occur at , but at . This value of also corresponds to its equilibrium value given by .
Trajectory multiplicity for the case where and . Each entry corresponds to the total number of trajectories for the particular values of and .
Trajectory multiplicity for and when the system is far from macroscopic equilibrium. In this case .
Trajectory multiplicity for and when the system is at macroscopic equilibrium.
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