What a field equation is
A field is a quantity spread over a space: the temperature across a room, the pressure along a pipe. A field equation says, at every point, how the value there relates to the values around it. Maxwell's equations do this for electricity and magnetism. Tell the equation where the edges of the region are and what happens there, and it returns the field. Nobody picks its shape.
Statistics never had one
A distribution is also spread over a space: the range of values your data can take. But statistics does not solve for it. It picks a family, such as the normal, the gamma or a mixture, and fits a few numbers, or it smooths the data with a kernel of a chosen width. Either way, part of the shape comes from the analyst, and the answer moves when the choice does.
Here, the distribution is a solution
The working papers write down a field equation whose solution is a distribution. You state the range the data live on: the whole line, a half line, or a bounded interval. The equation does the rest and returns the shape. The textbook shapes, the bell curve, the gamma and the beta, appear as its simplest solutions, one for each kind of range, rather than as assumptions.
Nothing restricts the shape beyond that. A distribution with two humps, a long tail or a hard edge is as much a solution as a bell curve; the sample decides how much detail it supports.
After that, it is arithmetic
The equation comes with its own coordinates, so a distribution becomes a short list of numbers. Comparing two distributions, finding the path from one to the other, and putting an error bar on the result all become sums over those numbers. There is no simulation and no resampling, and running the comparison twice gives the same answer both ways round.
Comparing whole populations
Some data live in hundreds of dimensions. The outline of a cell, traced at a hundred points, is one such object, and a study might have a hundred cells per group. That is far too little data to estimate the full distribution, but you can still compare two groups. One number says whether they differ. Four more say where: center, spread, skew and tails. The weights on those numbers come from the same kind of equation, not from a tuning choice.
In the working paper, on NIST's cell image database, this finds three samples of one cell line to be a single population and sets a second cell line sixteen times farther away. It agrees with NIST's published analysis, and it located two errors in our own earlier analysis of the same cells.
Where physics comes in
The working paper derives the equation from the geometry of space and time, which is why one equation serves both statistics and physics. When a distribution is physical, a cloud of trapped atoms for instance, the numbers that describe it also give its physical quantities: its pressure, its energy and the forces acting on it. The physics experiments on this site run the same kind of equation on molecules, confined electrons and fluid flow.
What is classical, and what is new
Much of the machinery is old and well tested. The distances are classical, and the comparison used for whole populations belongs to a family Giné studied in 1975. What this work adds is the equation that chooses the coordinates and the weights, and the observation that several familiar ways of measuring the distance between distributions are one construction with different weights.
What is not done yet
Comparing shape populations, the NIST cells and the cold-atom image are worked out in the papers and are coming to the experiments page. Generating new samples works wherever the distribution can be fitted. For shapes, in hundreds of dimensions, it is still a prototype: it reproduces the coarse variation in a population but falls short on mid-scale detail, and the cause is still open. A direct power comparison against other tests on the NIST cells is set up but not yet run. Reading physical quantities from a distribution is worked out in the paper but is not yet an experiment on this site.