What is “Order” and how is it Measured?
Consider a system in a general state of thermodynamic non-equilibrium; i.e., not necessarily at equilibrium and, in fact, possibly far from equilibrium. What is the level of order in the system?
Let the system have generally K dimensions, defined by continuous coordinates x=(x1 ,…,xK) and a probability density function (pdf) p(x). Let also the system have a well defined surface S, where L is the longest chord connecting two of its surface points.
Whereas the concept of disorder has long been quantified in physics as the Boltzmann/Shannon entropy H, until recently the polar opposite concept of order R was not so quantified. Instead R was regarded as the opposite of disorder, in the sense that the order should go up when the disorder goes down. On this basis, Ockham’s razor would favor H; the so-called “negentropy”
H = Σn Pn log Pn (1)
Notice that this uses absolute probabilities Pn . But recall that our system is described by a density function p(x), not absolute probabilities. Moreover, attempting to use (1) in the continuous limit Pn –> p( xn )dx does not work because the term log dx is unbounded.
Notice also in this regard that Ockham’s razor cannot by itself prove anything physical. For example, on the same basis other choices than the negentropy exist, such as a cross entropy, or 1=H or even exp(-H). All do go up when disorder H goes down. Clearly there are too many possible answers. Hence “order” must be defined independent of the concept of disorder H. It must be defined on its own physical terms, i.e. as the result of a distinct physical effect. Hence we have to seek a physical route to finding the measure of order for our continuous system.
We let that effect be “coarse graining.” Mathematically, this is any process that degrades a given system by replacing its intrinsic signal values with their mathematical projection, such as naturally occur in C.T. scan imagery. Physically, coarse graining a system thereby deletes or reduces the presence of the highest level of detail in its pdf p(x). Consequently, on the ultimate, quantum level, coarse graining demarks the transition from the finest, quantum level to a classical universe.
Hence the concept of order was defined on the basis that it decreases or stays constant under any coarse graining operation,
δR ≤ 0 for δt > 0, (2)
where δt is the positive time interval over which the coarse graining operation took place.
From this requirement it was found in Refs.  and  below that the order R in the system obeys
R = (1/8)L2I, where I=∫dx|grad p|2/p or I = ∫dx|grad q|2, where q = √p (3)
is the probability amplitude function and I is the Fisher information. Cencov’s famous inequality was used as well. The length L is the largest chord length connecting two points on the surface of the system. Notice that this expression (3) for the order R has no mathematical resemblance to the negentropy H defined by (1).
It is interesting that, along with the Fisher order R of (3), the Kullback-Liebler entropy
Hkl = Σn Pn log (Pn / Qn ) (4)
also satisfies the coarse graining requirement (2). However Hkl has an arbitrary ‘reference’ probability law Qn and, so, has ambiguous meaning as a measure of order. Since measure R given by (3) likewise obeys requirement (2), the ambiguity could be removed if a unique law Qn could be found that makes the form (4) go over into the order (3). In fact it is known that that law is Qn = infinitesimally displaced law Pn, or Q(x) = P(x+Δx), Δx tiny. Thus a continuous system has, in effect, a single measure of order R, and it has the form (3).
Recall that (2) is a system requirement, in particular that it be coarse grained. Thus, both the order R and information I are properties of the system, rather than of its data for example. This property of the Fisher order supplements its meaning as a measure of the accuracy in data from the system. The latter is the famous statement of the Cramer-Rao error bound, that the minimum mean-squared error in estimating a parameter from data obeys e2min = 1/I . Thus, Fisher information is both a system measure and a data measure. Finally, because of coarse-graining effect (2) and the proportionality (3) between R and I, Fisher information defines the direction of time. See also . Thus it is an ‘entropy,’ in that general sense.
The following properties of expression (3) for R were found and discussed in  and :
(a) It increases with system extension L, indicating increased order due to mere repetition of details (as in an apartment building, where each subsequent storey monotonically adds to the level of structural order.).
(b) It is unitless, and hence permits meaningful comparisons of order for different types of scenario, such as a bacterial culture and a hydrogen atom.
(c) It has the fractal property of being invariant under linear system stretch y=ax, a=const.
(d) The order R increases as the square of the number n of oscillations in each direction for a K-dimensional sinusoid pdf p(x), independent of their amplitude and independent of their spacing (by (c) preceding). Use of this p(x) in Eq. (3) gives a general level of order R = (nπK)2/2 where n = number of oscillations and K = its number of dimensions. Notice that R turns out to be sensitive to the sheer number of structural details that are present in the system, not to how densely packed or spread apart they are. In this regard R resembles the Kolmogoroff-Chaitin complexity, as well as being a measure of order. An example from  follows:
The above pdf has a level of order R = (nπK)2/2 = 78.96, since this sinusoidal pattern has K = 2 dimensions (x,y) and n = 5 ripples in each dimension. See other examples in  and .
(e) As in the above example, it increases as the square of the number of dimensions K in the system. Hence string theories of dimensions K = 11, 21, or etc. admit of extra- high levels of order and complexity.
(f) It was found in  that living cells undergoing severe distortion, say by crushing, can still maintain their level of order; suggesting that life could well have originated deep within the earth under severe pressure.
(g) A healthy living cell can, and does, develop increased order R over time because it is of finite extension L  and open to the environment. The price paid, via the 2nd law of thermodynamics, is waste heat and other random products that are shipped outside the cell. Thus, the cell can increase order, and live, at the expense of increasing disorder elsewhere. Essential is the existence of its finite boundary, the cell membrane, making possible the separation of structure with increased order (on the inside) from structure with decreased order (on the outside).
(h) But what of that ultimate system, the universe? The 2nd law states that it is losing order, but only provided it is closed to outside influences. But is it? If it has an “outside” (like the cell above), then perhaps its order inside can likewise increase, at the expense of increased entropy outside. In fact in an astronomical application , it was found that, contrary to intuition, the universe is not losing order due to its relentless Hubble expansion.
In a further paper  the level of order in a flat-space universe was computed, as level R = 1060 . Also taking into account space curvature, levels of order R = 1090 and 10120 were found  for, respectively, an observer at rest with respect to the origin of coordinates and one that is freely falling. The huge value 10120 arises, in particular, because the curvature of the space observed by the free falling observer is greatest, giving a greater value of length L in Eq. (3). The value 10120 is also about the size of the estimated maximum level of the discrete negentropy measure (1), in a black-hole universe. This agreement seems to indicate that, in the universe overall, there is about as much discrete as continuous structure.
- B.R. Frieden and R.J.Hawkins, “Quantifying system Order for full and partial coarse graining,” Phys. Rev. E 82, 066117, 1-8 (2010)
- B.R. Frieden and R.A. Gatenby, “Order in a Multiply-Dimensioned System,” Phys. Rev. E 84, 011128, 1-9 (2011)
- B. R. Frieden and H. C. Rosu, “Fisher’s arrow of time in quantum cosmology,” Mod. Phys. Lett. A 13, 39 (1998)
- B.R. Frieden and R.A. Gatenby, “Information Dynamics in Living Systems: Prokaryotes, Eukaryotes, and Cancer,” PLoS ONE 6(7): e22085. doi:10.1371/journal.pone.0022085 (2011)
- B.R. Frieden, A. Plastino and A.R. Plastino, “Effect upon universal Order of Hubble expansion,” Physica A doi: 10.1016/j.physa.2011.08.005 (2011); Physica A 391 410-413 (2012)
- B.R. Frieden, A. Plastino and A.R. Plastino, “Fisher order measure and Petri’s Universe model,” Physica A (to be published, 2011)
- B.R. Frieden and M. Petri, “Motion-dependent levels of order in a relativistic universe,” Physical Review E (to be published, 2012)