User:Thermochap/Sandbox

Wide angle view of TEM diffraction tilting.

This page is like an electronic notebook for playing with some equations that I think are cool. It's still not quite as flexible as a physical notebook with numbered and lined pages, but at this point I would say that it is complementary in that regard. The page also serves as a place for storing information on HTML, TeX, and Wikimedia editing markup that might come in handy for displaying equations elsewhere.

Note: Although many of these equations can be used to generate beautiful graphics, on these pages the visualizations will mostly be left to YOUR imagination. Feel free to suggest interesting things to plot, nonetheless.

Did you know?

• ...that Kikuchi lines, formed in diffraction patterns by diffusely scattered electrons, are useful tools in electron microscopy of crystalline and nanocrystalline materials? (from Wikipedia's main page on 4/28/2008)

transcription/translation notes

Consider this short double-helix region with the template-strand on bottom, somewhere after the upsteam or leftward promoter-region (e.g. which in bacteria offers a binding site to the RNA polymerase sigma-subunit to start transcription of DNA to mRNA), and well before the downstream or rightward terminator-section's inverted-repeats cause transcription to stop:

3'-GACTAGACTGACTACGCATACGATCA-5' (non-template "coding" strand)
5'-CTGATCTGACTGATGCGTATGCTAGT-3' (template strand)

Transcription of the template-strand proceeds from the sugar-phosphate backbone's 5' (upstream phosphate) through the 1' base connections to the 3' (downstream sugar-ring) end, yielding a messenger RNA strand from this section which (now grouped into three-base codons starting from the upstream or 5' side) looks like:

5'-GAC UAG ACU GAC UAC GCA UAC GAU CA-3' (transcribed messenger RNA)

where as usual we've used C⇒G, G⇒C, T⇒A, and A⇒U on the template strand to get a copy of the non-template "coding" strand with T⇒U.

Let's also assume that on this mRNA strand this section starts somewhere after an upstream or leftward ribosome binding site (like AGGAGGU) and the subsequent (AUG ⇒ Methionine) start codon, but before a stop codon (UGA, UAG or UAA) that signals the end of translation to protein:

AsparticAcid-STOP. (resulting protein segment)

Oops. As you can see there was a stop codon in this string, so the resulting protein ends here.

Editing tips

Note: The markup to create the following items will be visible in the editing window, if you click on [edit] in the upper right corner of this section.

Here's a sample reference that can be cited more than once. Multiple formats for automatically numbered footnotes are available, although only one of these formats is used below.

You can put stuff into a non-wrapping box 
by starting each line with a space.  This 
might be useful for program code (note the
equally-spaced font) as well as to 
highlight an equation or two.

And then there is the question of inter-wikilinks. For example, w:Apple is the article on the (English) Wikipedia for Apple, while wikt:Apple would be the analogous one on Wiktionary. There is probably a list of prefixes, e.g. for wikiversity, wikibooks, wikicommons, et cetera.

Don't forget that you can insert fractions into text like this: 7⁄8. You can also put TeX math into text like this: ${\frac {7}{8}}$.

An example of TeX math in its own indented line is this equation from Miller index for the normal to the (hkl) plane of a hexagonal crystal in terms of both covariant reciprocal-lattice basis one-forms or co-vectors a*, b*, c*, and contravariant direct-lattice basis vectors a, b, c:

$(hkl)=h{\vec {a^{*}}}+k{\vec {b^{*}}}+l{\vec {c^{*}}}={\frac {2}{3a^{2}}}(2h+k){\vec {a}}+{\frac {2}{3a^{2}}}(h+2k){\vec {b}}+{\frac {1}{c^{2}}}(l){\vec {c}}$.  

Here of course a*=(b×c)/V_c, b*=(c×a)/V_c, and c*=(a×b)/V_c where the unit cell volume is V_c=a·(b×c).

Two fun equations from scattering theory are the kinematic equation from reciprocal lattice for single-scattering of a coherent beam (e.g. of electrons, photons, or neutrons) by a cluster of atoms,

$I[{\vec {g}}]=\sum _{j=1}^{N}\sum _{k=1}^{N}f_{j}[{\vec {g}}]f_{k}[{\vec {g}}]e^{2\pi i{\vec {g}}\cdot {\vec {r}}_{jk}}$,

and its angle-averaged sister from powder diffraction, the Deybe equation for scattering from a random collection of such clusters:

$I_{powder}[g]=\sum _{j=1}^{N}\sum _{k=1}^{N}f_{j}[g]f_{k}[g]{\frac {\sin[2\pi gr_{jk}]}{2\pi gr_{jk}}}$.

A pretty example from proper acceleration is this equation for relativistic coordinate acceleration in terms of proper (physical) acceleration A as well as the various geometric accelerations that arise from the coordinate system's affine connection:

${\frac {dU^{\lambda }}{d\tau }}=A^{\lambda }-\Gamma ^{\lambda }{}_{\mu \nu }U^{\mu }U^{\nu }$.

Experimenting with equations

clustering algorithms

In laTeX markup, that last equation (circled in those 17 Feb 2019 notes) might look like:

$\langle CU\rangle \equiv {\tfrac {\sum _{k=1}^{K}CU[C_{k}]}{K}}={\tfrac {\sum _{k=1}^{K}p[C_{k}]\sum _{i=1}^{I}p[{\hat {A}}|C_{k}]\sum _{j=1}^{J}p[A_{i}=V_{ij}|C_{k}]^{2}-\sum _{i=1}^{I}p[{\hat {A}}]\sum _{j=1}^{J}p[A_{i}=V_{ij}]}{K}}$.

As you can see with a look at the markup in the editor, I've tried to put limits on the sums, and used \tfrac instead of \frac to get the whole line to fit across a single page in word.

In mediawiki markup, it might instead look like:

⟨CU⟩ ≡ (Σ_k=1,Kp[C_k]Σ_i=1,Ip[A*|C_k]Σ_j=1,Jp[A_i=V_ij|C_k]² - Σ_i=1,Ip[A*]Σ_j=1,Jp[A_i=V_ij]²)/K.

In the latter I've added the bold to show that there are actually 5 different probability functions involved. Also, I think that there are ways in wiki markup to "put a hat" on the A, but it seemed easier here just to use an asterisk (A*) instead. Thermochap (talk) 14:50, 18 February 2019 (UTC)

Note that for the wiki markup example, cut-and-paste into google docs, followed by conversion to word, might do a better job than direct pasting into word. Thermochap (talk) 16:00, 18 February 2019 (UTC)

columnar growth on a cylinder

Circular darkfield aperture (left) simulation of 2nm metal particles on a nano-cylinder (right)

These equations stem from an investigation of lattice fringe visibility in nanoparticles grown on the surface of a cylindrical fiber or tube. Look for more on the resulting geometry problems here in the days ahead. In the meantime...

How would you solve this equation for φ_local? Here φ_local is the local azimuth of a lattice fringe's tilt axis with respect to a nanotube on which it resides, ζ is the angle between that lattice plane's normal and a reference or growth plane, θ is the angle between the growth plane and the viewing direction, and φ_view is the azimuth of the projected fringe and the tube axis. If you have a closed-form solution, by all means post it in this section!

$\phi _{view}=\arctan \left[{\frac {\cos[\zeta ]\sin[\theta ]+\cos[\theta ]\sin[\zeta ]\sin[\phi _{local}]}{\cos[\phi _{local}]\sin[\zeta ]}}\right]$

It would also be nice to solve this equation for beam angle from face-on, set equal to π/2, for θ:

${\frac {\pi }{2}}=\arctan \left[{\frac {\sqrt {(\cos[\phi ]\sin[\zeta ])^{2}+(\cos[\zeta ]\sin[\theta ]+\cos[\theta ]\sin[\phi ]\sin[\zeta ])^{2}}}{\cos[\theta ]\cos[\zeta ]-\sin[\theta ]\sin[\phi ]\sin[\zeta ]}}\right]$.

Note: Because of branch cuts associated with inverse trig functions, the solutions might be a bit messy. Suggestions invited, nonetheless.

On the subject of Miller and direct lattice indices for planes and zones in hexagonal crystal and graphene (hk0) layers, do the curious relationships below make sense to you?

${\vec {a^{*}}}=(100)\parallel \langle {210\rangle }\perp ({\bar {1}}20)\parallel \langle {010\rangle }={\vec {b}}$ ${\vec {b^{*}}}=(010)\parallel \langle {120\rangle }\perp (2{\bar {1}}0)\parallel \langle {100\rangle }={\vec {a}}$ $(1{\bar {1}}0)\parallel \langle {1{\bar {1}}0\rangle }\perp (110)\parallel \langle {110\rangle }$

In particular, do you know of an easy way to confirm the assertions about directions being parallel?

fringe visibility / bend contour rocking curves

Bend contour and lattice fringe visibility as a function of specimen thickness and beam tilt.

Speaking of the visibility of lattice fringes as a function of specimen orientation in a transmission electron microscope, the expression for fringe visibility half-angle looks like:

$\alpha _{max}=\sin ^{-1}\left(d{\frac {\Gamma }{t}}+{\frac {\lambda }{2d}}\left\{1-\left(d{\frac {\Gamma }{t}}\right)^{2}\right\}\right)$.

In the expression above, d is the spacing of the lattice planes, t is the crystal thickness, λ is the wavelength of the electrons, and Γ is a “visibility factor” on the order of 1 that empirically accounts for the signal-to-noise ratio in the method used to detect fringes.

Replacing Γ/t with deviation parameter or excitation error s in the above equation, one can solve for deviation parameter in terms of off-edge angle α to get:

$s[\alpha ]=\left|{\frac {{\sqrt {1-2(\lambda /d)\sin {\alpha }+(\lambda /d)^{2}}}-1}{\lambda }}\right|$.

The Fourier shape transform of a spherical crystal in three dimensions, in terms of spatial frequency g and sphere diameter t, may be written:

$\Xi [g]={\frac {\sin {\pi gt}-\pi gt\cos {\pi gt}}{2\pi ^{2}g^{3}}}$.

Consider a lattice plane canted by angle α radians from the edge-on position along the electron-beam direction. Since all reciprocal lattice spots will be convolved with the shape transform, the intensity of Bragg scattering from the brightfield (unscattered beam) image can be estimated by adding amplitudes (in the coherent kinematic scattering case) from both sides of the lattice plane by adding Ξ[s] values for deviation s[α] evaluated at ±α. The result is illustrated in the figure at left.

An interesting related problem: What's the area of the orientation sphere subtended by the intersection between two such bands, say each of half-angle α, since this relates to the probability of encountering lattice cross-fringes in a collection of randomly-oriented crystals. Consider first the case of equal bands intersecting at right angles. We are interested in σ₂, twice the area σ₁ of a single intersection, because great-circle bands intersect twice on opposing sides of the orientation unit sphere.

For α<π/4 a lovely equation for the exact value of the intersection solid angle has been discovered by Dr. Jon Bailey, to wit:

$\sigma _{1}=8\sum \limits _{n=1}^{\infty }{\frac {(-1)^{n+1}\sin[\alpha ]^{2n}}{n!}}F[n]\prod \limits _{k=1}^{n}{({\frac {3}{2}}-k)}=4\alpha ^{2}+{\frac {2}{9}}\alpha ^{6}+{\frac {8}{45}}\alpha ^{8}+$...,

where hypergeometric function F[n] is

$F[n]\equiv \sum \limits _{m=0}^{n-1}{\frac {(n-1)!}{m!(n-1-m)!}}{\frac {1}{2m+1}}$.

For α>π/4, upper and lower visibility zones connect leaving only four circular "cross-free caps". Hence σ₁ then becomes 2π(2sin[α]-1).

These observations show that an excellent approximation for small angles is σ₂ ~ 2(2α)², i.e. twice the area one would calculate for a flat square of side 2α. The approximation error (from above) is to first order (4/9)α⁶, which is still below 0.5% when bandwidth 2α is a radian.

When crossing bands have half-angles of α₁ and α₂, and they intersect at an angle of φ radians, the flat polygon estimate for the area of an intersection plus its complement (on the other side of the sphere) becomes:

$\sigma _{2}\cong {\frac {2(2\alpha _{1})(2\alpha _{2})}{\sin[\phi ]}}$.

Intersection areas of multiple bands, e.g. for analysis of fringe visibility in 10-crystal icosahedral twins, remain an active area of inquiry. If you like thinking about geometry problems like this, leave a note here.

particle outflows due to stellar radiation

Most of the lighter elements of which we are made were formed by nuclear processes in red giant stars. Some of these elements (e.g. carbon) likely condensed in the atmospheres of these stars, with the resulting particles ejected into the interstellar medium by radiation pressure from that star. Recent laboratory work on microscopic particles that were formed in the atmosphere of such stars has raised questions about the details of these condensation and ejection processes. The equations here offer some preliminary insights (hopefully) into the latter.

Before worrying about interaction cross-section changes as particle size moves below the wavelength of the radiation, let's consider the process of radiation pressure ejection assuming simple geometric cross-sections. For an initially stationary particle, begin by balancing gravity and radiation pressure forces:

$G{\frac {Mm}{R^{2}}}={\frac {L}{c}}{\frac {\pi r^{2}}{4\pi R^{2}}}$.

Here M is the mass of the star, R is the distance from the star (which cancels out in the equation), L is the star's luminosity (radiative energy output per unit time), and of course G and c are the gravitation and lightspeed constants respectively. Assuming that particle mass m relates to density ρ via m = (4/3)πr³ρ, the foregoing equation yields a critical radius for balancing these two forces which is around a quarter micron for 3 g/cc particles in our solar system, and perhaps 50 times that for particles in the neighborhood of the asymptotic giant branch stars which manufactured the carbon found in micron-sized presolar graphite onions from the meteorite Murchison.

Since these particles are much smaller in size than the critical radius (but still not much smaller than the wavelength of the stellar radiation), we might try ignoring gravity in estimating the resultant acceleration. This yields:

$a={\frac {\Sigma F}{m}}\sim {\frac {L}{c}}{\frac {\pi r^{2}}{4\pi R^{2}}}{\frac {1}{{\frac {4}{3}}\pi r^{3}\rho }}={\frac {3L}{16\pi cR^{2}r\rho }}\sim {\frac {3\sigma T^{4}}{4cr\rho }}$.

Here σ is the constant that figures in the Stefan-Boltzmann law, which for a star presumably looks something like L = 4πR²εσT⁴. The reason that stellar radius R doesn't appear on the right side of the acceleration equation above is thus because the ratio between stellar luminosity and photosphere area is roughly proportional to T⁴. This suggests that radiative acceleration at the photosphere surface may be similar for all stars of comparable surface temperature, and in the absence of other factors on the order of 5 m/s² for the above-mentioned grains.

What are the primary factors that lessen the effect of this radiation pressure just outside the surface of such a star? Collision with gas molecules are one possibility. Do gas densities and velocities make this a significant factor? Where can we find models of molecule and particle number density and size distribution as a function of distance away from the center of an asymptotic giant branch star after 2nd dredge up? To consider gas molecule transport, of course, as well as radiation effects on much smaller particles, we can no longer ignore the wavelength dependence of the interaction cross-section. Rayleigh and Mie scattering equations may come in handy here. Suggestions are invited for getting a quantitative handle on these effects.

some KL divergence stuff

informatiion unit table

Table of selected units for information-uncertainty, entropy, surprisal, Gibbs availability, heat capacity, etc.

unit	#choices	base	bits	nats	bytes	J/K	kcal/K
bit	2#bits	2	ln2[2]=1	ln[2]≅0.693	ln256[2]=0.125	kln[2]≅9.57×10-24	k"ln[2]≅2.29×10-27
nat	e#nats	e	ln2[e]≅1.443	ln[e]=1	ln256[e]≅0.180	k≅1.38×10-23	k"≅3.30×10-27
byte	256#bytes	256	ln2[256]=8	ln[256]≅5.545	ln256[256]=1	kln[256]≅7.66×10-23	k"ln[256]≅1.83×10-26
J/K	eS/k	e1/k≅101022.5	1/(kln[2])≅1.04×1023	1/k≅7.24×1022	1/(kln[256])≅1.31×1022	kln[e1/k]=1	k"ln[e1/k]≅2.39×10-4
kcal/K	eS/k"	e1/k"≅101026.1	1/(k"ln[2])≅4.37×1026	1/k"≅3.03×1026	1/(k"ln[256])≅5.47×1025	kln[e1/k"]≅4187	k"ln[e1/k"]=1

measures of useful information

Boltzmann's version of S = k ln[W].

There are generally three interconvertable forms: as a fraction or probability p between 0 and 1, as a reciprocal-probablity or multiplicity w between 1 and +∞, and as a log-multiplicity or surprisal in information units between 0 and +∞.

Surprisal:

$0\leq s_{\text{p}}\equiv k\ln \left[w_{\text{p}}\right]\equiv k\ln \left[{\frac {1}{p}}\right]\leq +\infty$,

where 0 ≤ p ≤ 1 and 1 ≤ w_p ≤ +∞.

When k=1/ln[2] then surprisal units are bits. Probability p = 1/2^sBits, so that there are N bits of surprisal in throwing all heads on the first toss of N coins. If k is 1 then the units are nats, while if k is Boltzmann's constant 1.38×10^-23 then surprisal is in Joules/Kelvin.

Evidence for a True-False Assertion: $e_{\text{p}}=s_{\text{1-p}}-s_{\text{p}}$.

When k=1/ln[2], evidence is in bits and we can say that "odds-ratio" = 2^eBits. Since independent bits of evidence add, thresholds for various levels of proof might be established that are as easy to get a feel for as are the odds of throwing all heads on a handful of coins.

Entropy, uncertainty, or average surprisal:

$0\leq S_{\text{p/p}}\leq S_{\text{q/p}}\equiv k\ln \left[W_{\text{q/p}}\right]\equiv \langle {s_{\text{q}}\rangle }_{\text{p}}=k\sum _{i=1}^{N}p_{i}\ln \left[{\frac {1}{q_{i}}}\right]\leq +\infty$,

where operating probability-set Σ_ip_i = 1 and expected or reference probability-set Σ_iq_i = 1. Although written for discrete probability-sets, this expression also naturally adapts to continuous and quantum mechanical probability-sets.

Note that the upper limit on S_p/p is ln₂[N}. Also the fact that S_q/p ≥ S_p/p, i.e. that measurements using the wrong model q are always likely to be more surprised by observational data than those using the operating-model p, underlies maximum-likelyhood parameter-estimation (curve-fitting) and Bayesian model-selection as well as the positivity of the subsystem-correlation and thermodynamic-availability (e.g. ordered-energy) measures discussed below.

Kullback-Leibler divergence or net surprisal of q from p:

$0\leq I_{\text{q/p}}\equiv k\ln \left[M_{\text{q/p}}\right]\equiv \langle {s_{\text{q}}-s_{\text{p}}\rangle }_{\text{p}}=k\sum _{i=1}^{N}p_{i}\ln \left[{\frac {p_{i}}{q_{i}}}\right]\leq +\infty$,

where q is the expected probability-set and p is the operating probability-set. This quantity is also the negative of the Shannon-Jaynes entropy, useful for Bayesian maximum-entropy inference about systems with average-value constraints when prior-information is available.

• In ecology and related fields, unbiased estimates for the KL divergence of "model from reality" are useful in ranking models against experimental data with help from Akaike Information Criterion applied to residuals that the models fail to explain.

• In communications theory, in clade analysis, and in quantum computing the KL divergence of "uncorrelated from correlated" measures the mutual information associated with fidelity, inheritance, and entanglement.

• In thermodynamics, the KL divergence of "ambient from actual" measures distance to equilibrium or Gibbs availability, and when multiplied by ambient temperature: available work.

Here we note that the foregoing are special cases of KL divergence as a measure of useful information. Each is, however, typically applied on only one level of organization at a time. In addition to offering some new and surprising applications, we discuss how some of the most interesting future applications may be more explicit about their relationship to correlations on multiple levels.

Some expansions

KL-divergence can also be written as an infinite sum of moments (or averaged powers) for the dimensionless deviation from ambient δ_i=(p_i/q_i-1) using the McLaurin expansion for Log[1+δ] as:

${\frac {I_{\text{q/p}}}{k}}=\left\langle \ln \left[{\frac {p}{q}}\right]\right\rangle _{q}\equiv \sum _{i=1}^{N}p_{i}\ln \left[1+\delta _{i}\right]=\sum _{j=1}^{\infty }{\frac {(-1)^{j-1}}{j}}\left\langle \delta ^{j}\right\rangle _{\text{p}}=\langle {\delta \rangle }_{\text{p}}-{\frac {\langle {\delta ^{2}\rangle }_{\text{p}}}{2}}+{\frac {\langle {\delta ^{3}\rangle }_{\text{p}}}{3}}-...$.

Here the subscript p on the angle brackets denotes an average over the actual (p_i) rather than the ambient (q_i) probabilities. Low-order moments would dominate the infinite sum on the right if δ_i<1 ∀ i∈{1,N}, i.e. if the reference probability q_i is less than half the operating probability p_i for any value of i. High values of q_i/p_i cause less trouble in this context.

KL divergence converges more rapidly in moments of δ averaged over ambient probability q:

${\frac {I_{\text{q/p}}}{k}}=\left\langle {\frac {p}{q}}\ln \left[{\frac {p}{q}}\right]\right\rangle _{q}\equiv \sum _{i=1}^{N}q_{i}(1+\delta _{i})\ln \left[1+\delta _{i}\right]=\sum _{j=2}^{\infty }{\frac {(-1)^{j}}{j(j-1)}}\left\langle \delta ^{j}\right\rangle _{\text{q}}={\frac {\langle {\delta ^{2}\rangle }_{\text{q}}}{2}}-{\frac {\langle {\delta ^{3}\rangle }_{\text{q}}}{6}}+{\frac {\langle {\delta ^{4}\rangle }_{\text{q}}}{12}}-...$.

This latter equality uses the fact that the ambient average of δ is zero, and shows that for δ values much less than 1 that KL-divergence is approximately half the ambient average of δ² i.e. that mutual information for small deviations goes as the ambient mean square deviation or variance from expected.

It might be interesting to examine these convergences further if q is a uniform distribution i.e. if q_i = 1/N ∀ i∈{1,N}. Then convergence is assured as long as p_i < 2/N ∀ i∈{1,N}, in which case I_q/p ≈ ½kN²⟨(p-q)²⟩_q.

Multiple subsystems

Mutual information for M=2

Mutual information, a special case of KL-divergence, is a measure of correlation between identical subsystems with second-law teeth which finds application to a wide variety of digital systems in the code-based sciences e.g. of molecular codes, data compression, and quantum computing. Put simply, it's the information about the state of system A that you obtain on learning the state of system B and vice-versa, or in mathematical terms the amount by which uncertainty about the whole is less than the sum of uncertainties about the parts i.e. I_AB=S_A+S_B-S_AB.

For simplicity one might model N states for each of two identical subsystems so that joint probabilities are written p_ij where Σ_iΣ_jp_ij=1 for both i,j ∈ {1,N}. The reference-probability for mutual information is the product r_ij≡p_ip_j of marginal probabilities p_i=Σ_jp_ij and p_j=Σ_ip_ij. In bits this mutual information

$0\leq I_{\text{r/p}}\equiv \langle {s_{\text{r}}-s_{\text{p}}\rangle }_{\text{p}}=\sum _{i=1}^{N}\sum _{j=1}^{N}p_{ij}\ln _{2}\left[{\frac {p_{ij}}{p_{i}p_{j}}}\right]\leq \ln _{2}[N]$

is a multi-moment measure of subsystem correlation, since completely independent subsystems will have p_ij=p_ip_j≡r_ij. The upper limit arises if S_AB = S_A+S_B-I_AB goes to zero when S_A is zero and S_B is at its maximum value of ln₂[N], i.e. when knowledge of the state of A along with the mutual information I_AB fully specifies the system state in the absence of any knowledge about B.

Total correlation for M>2

Generalizing mutual information to more than two subsystems is not as simple as one might expect. As in the work above we stick with the KL-divergence strategy because the surprisal-based correlations that it measures are always positive, and because of its 2nd law connections. In this context total-correlation (sum of marginal uncertainties minus the joint uncertainty) and the associated connected-correlations breakdown are tools of complex-system correlation-analysis which show promise for providing insight into the evolution and maintenance of layered correlation-heirarchies as well.

As above, total correlation in bits for M identical systems may be written as

$0\leq I_{\text{r/p}}=\sum _{i=1}^{N}\sum _{j=1}^{N}...\sum _{k=1}^{N}p_{ij...k}\ln _{2}\left[{\frac {p_{ij...k}}{p_{i}p_{j}...p_{k}}}\right]=\sum _{m=2}^{M}I_{\text{r/p}}^{(m)}\leq (M-1)\ln _{2}[N]$

where reference-probability r is a product of M marginals, namely r_ij...k≡p_ip_j...p_k, and each joint-probability has M indices represented by the letters i, j, ..., k.

The Bialek work cited above describes how this total correlation can be broken into order-m terms for m=2 to M, so that I_r/p=Σ_mI_r/p^(m) where the order-m correlation terms are also positive. In fact, we conjecture the following upper limits as well:

$0\leq I_{\text{r/p}}^{(m)}\leq (M-m+1)\ln _{2}[N]{\text{ for all }}m\in \{2,M\}$.

Even for very simple cases (like when M=3 and N=2), calculation of the I_r/p^(m) can be a bit messy.

Various probability sets

As mentioned above, the general joint probability may be written p_ij...k while the independent-subsystem reference probability-set may be written as r_ij...k≡p_ip_j...p_k. The uniform probability assignment for M independent subsystems each with N states is u_ij...k=1/N^M so that Σ_iΣ_j...Σ_ku_ij...k=1, with marginal probabilities equal to 1/N. For N=2 and M=3, u_ijk=1/8, while u_i=½, u_j=½, and u_k=½ so that product of marginals r_ijk≡u_iuju_k=1/8 as well, confirming subsystem independence.

Aside: The difference between KL divergence of general set q with respect to p and the total correlation of p might be written:

$I_{\text{q/p}}-I_{\text{r/p}}=\langle {s_{\text{q}}-s_{\text{r}}\rangle }_{\text{p}}=\sum _{i=1}^{N}\sum _{j=1}^{N}...\sum _{k=1}^{N}p_{ij...k}\ln _{2}\left[{\frac {p_{i}p_{j}...p_{k}}{q_{ij...k}}}\right]$.

Does this in general obey any useful inequalities?

ideal gas applications

Pressure versus volume plot of available work from a mole of Argon gas relative to ambient, calculated as T_o times KL divergence.

For a monatomic ideal gas at fixed temperature one can write the available work with respect to ambient as:

$W=\Delta G=NkT_{o}\Theta \left[{\frac {V}{V_{o}}}\right]$.

where Θ[x]≡x-1-lnx≥0 and

$V_{o}={\frac {NkT_{o}}{P_{o}}}$.

Here as usual ambient values for absolute temperature, volume and pressure are represented by T_o, V_o, and P_o respectively.

More generally, we can write the thermodynamic availability in information units as

$\Delta I={\frac {W}{T_{o}}}=Nk\left(\Theta \left[{\frac {V}{V_{o}}}\right]+{\frac {3}{2}}\Theta \left[{\frac {T}{T_{o}}}\right]\right)$.

Thus a collection of N gas atoms whose volume V and temperature T differs from ambient is a potential source of net surprisal and hence available work. This is true regardless of whether the collection of atoms is either hotter/cooler than ambient, or more/less dense.

quantifying multilayer correlation

Consider an L×n system having L layers each with n possible states per layer. Let M be the multiplicity of correlated layers, where 0≤M≤L-1. For integer values of correlated-layer multiplicity M, the simplest matrix of all n^L probabilities has n^L-M of those probabilities equal to p_max=1/n^L-M with the remaining n^L-n^L-M probabilities equal to zero. Total state uncertainty is S_tot = (L-M) ln₂n, and the marginal probability of any given layer j={0,L-1} is p_j = 1/n with associated uncertainty of S_j = ln₂n. Finally, the mutual information is I_net = ΣS_j - S_tot = M ln₂n. Thus I_net is one of a larger class of net surprisal (KL-divergence) functions that measure an increase in thermodynamic availability, often thanks to reversible thermalization of energy available for doing work in the world around.

The simplest example of this in turn might be for L=3 and n=2. For instance, imagine level 0 substrate states {land, sea}, first level locomotion states {legs, fins}, and second level color states {brown,blue}. In this case the uncorrelated M=0 probability set (all mixtures of states from each level) has these eight probability assignments:

$\left({\begin{array}{ll}\left\{p_{000},p_{001}\right\}&\left\{p_{010},p_{011}\right\}\\\left\{p_{100},p_{101}\right\}&\left\{p_{110},p_{111}\right\}\end{array}}\right)=\left({\begin{array}{ll}\left\{{\frac {1}{8}},{\frac {1}{8}}\right\}&\left\{{\frac {1}{8}},{\frac {1}{8}}\right\}\\\left\{{\frac {1}{8}},{\frac {1}{8}}\right\}&\left\{{\frac {1}{8}},{\frac {1}{8}}\right\}\end{array}}\right)$.

The first entry in the array above is the probablity of a brown land dweller with legs, the second is that of a blue land dweller with legs, rightward is that of a brown land dweller with fins et cetera until the last entry: the probability of a blue sea dweller with fins. In this uncorrelated case, all eight different types of organism are found with equal probability e.g. blue land dwellers with fins are as likely as brown sea dwellers with legs. Hence the total uncertainty ln₂8 = 3 bits equals the sum of the three 1 bit marginal uncertainties, and the mutual information is zero.

For the M=1 set, legs always go with land and fins always with sea while colors remain random. This has a probability array that looks like:

$\left({\begin{array}{ll}\left\{{\frac {1}{4}},{\frac {1}{4}}\right\}&\{0,0\}\\\{0,0\}&\left\{{\frac {1}{4}},{\frac {1}{4}}\right\}\end{array}}\right)$.

In this partially correlated case, total state uncertainty is only 2 bits (four types of organism) even though the marginal uncertainties still sum to 3 bits i.e. you still have one bit of uncertainty about the state of any randomly picked organism on a given niche level. Hence the mutual information is now 3-2=1 bit.

In the fully-correlated M=2 set, land, legs, and brown color always go together as do sea, fins and blue. The probability array now looks like:

$\left({\begin{array}{ll}\left\{{\frac {1}{2}},0\right\}&\{0,0\}\\\{0,0\}&\left\{0,{\frac {1}{2}}\right\}\end{array}}\right)$.

In this fully correlated case, total state uncertainty is only 1 bit (i.e. there are only two types of organism), the marginal uncertainties still sum to 3 bits, and therefore the mutual information is now 3-1=2 bits. Thus here as elsewhere the niche multiplicity and mutual information (hence KL-divergence) go hand in hand.

Note that both non-substrate levels (i.e. locomotion and color) relate to individual fitness. In that sense they do NOT represent different niche layers if separate layers point inward or outward with respect to distinct physical boundary types (like molecule surfaces, cell membranes, metazoan skins, or code-pool edges). Nonetheless the example does illustrate some elements of correlated-state math in a multi-scale system.

niche layers for metazoans

For comparison, a table of properties for a true 7×n multilayer model is provided below. The first two rows refer to the various layers, while the remaining rows refer instead to various values of inter-layer correlation. In those top rows the self and pair columns look inward and outward (respectively) with respect to metazoan skins. The family and hierarchy columns similarly focus in/out with respect to molecule code-pool boundaries, while the culture and profession columns predicate themselves instead on the physical boundary between idea-pools.

Matchups with respect to skin and code-pool boundaries, in a 7-layer model with n states per layer:

layer names	substrate	self <=	=> pair	family <=	=> hierarchy	culture <=	=> profession
some relevant properties	land, air, water	nutrition, fitness, learning	friend, partner, mentor	ancestors, offspring, inlaws	community, employer, government	tradition, arts/sports, religion	specialty, archives, field/study
niche-layer multiplicity	0	1	2	3	4	5	6
pmax	1/n7	1/n6	1/n5	1/n4	1/n3	1/n2	1/n
Stotal	7 ln2[n]	6 ln2[n]	5 ln2[n]	4 ln2[n]	3 ln2[n]	2 ln2[n]	ln2[n]
pmarginal	1/n	1/n	1/n	1/n	1/n	1/n	1/n
ΣSmarginal	7 ln2[n]	7 ln2[n]	7 ln2[n]	7 ln2[n]	7 ln2[n]	7 ln2[n]	7 ln2[n]
Inet=ΣSm-St	0	ln2[n]	2 ln2[n]	3 ln2[n]	4 ln2[n]	5 ln2[n]	6 ln2[n]

A small but important subset of bio and social science issues, relevant mainly to the health of single-species animal communities, are put into an integrative context with this layered-niche network approach. Included in these, for example, are the quantitative relevance of gene and meme pool diversity, the fidelity of molecule and idea code expression and processes associated with the parallel evolution of such codes, the mechanisms and impact of industrial QA, strictures against nepotism in the workplace, the success of work toward decoupling merit systems and politics, and the separate boundaries used by culture (e.g. church) and hierarchy (e.g. state) in comparison to the common boundary shared by professional observation (e.g. science) and cultural belief.

Even this simple model makes room for variability in the assignment of state structures since there's no upper limit on the number of states per layer. Storyboard models for exploring some of the problems above still remain to be written. What applications do you think might be most important to explore?

the layered niche-network simplex

In developing operational measures for the niche-layer multiplicity associated with individual metazoans, one might consider the fractional resource allocation f_i in time and/or available work to each of six layers (directed inward/outward from skin, gene-pool and meme-pool) via:

$k\ln M_{i}=\sum _{j=1}^{6}f_{ij}s[f_{ij}]$ where $\sum _{j=1}^{6}f_{ij}=1$ and $0\leq f_{ij}\leq 1$.

This above conversion between the f_ij and M_i is one standard way to determine an effective number of accessible states from a set of unequal probabilities. If we put the intermediate quantity into bits, we might also write it as:

$M_{i}=2^{C_{i}}=e^{\sum _{j=1}^{6}f_{ij}\ln[{\frac {1}{f_{ij}}}]}$ where $C_{i}=\sum _{j=1}^{6}f_{ij}\ln _{2}[{\frac {1}{f_{ij}}}]$.

This ad hoc estimate of niche-layer multiplicity for individual organisms might prove useful in ecosystem settings with many species, since it recognizes the extent to which such organisms are able to buffer correlations on multiple scales. The effect of stresses that prevent functioning on more than one level might thus be quantified. Moreover in place of group selection in the traditional sense, physical boundary-focused selection mechanisms that consider code-pool boundaries as well as metazoan skins may guide the development of varied symbioses, including guilds defined as functional groupings between species, just as idea codes passed on to apprentices (thereby expanding a boundary) may have helped to define and support craft-guilds e.g. for shoemakers or bakers in medieval times.

The niche-layer multiplicity associated with a single-species community might further be defined as a harmonic average of the M_i values. The harmonic average M is written

$M=\left(\prod _{i=1}^{N}M_{i}\right)^{\frac {1}{N}}=2^{\left({\frac {1}{N}}\sum _{i=1}^{N}\sum _{j=1}^{6}f_{ij}\ln _{2}[{\frac {1}{f_{ij}}}]\right)}$.

In terms of C_i in bits, for a community of N individuals this could also be written as:

$M=2^{\langle {C\rangle }}$ where $\langle {C\rangle }={\frac {1}{N}}\sum _{i=1}^{N}C_{i}$.

More recent work suggests that a linear average (rather than harmonic) might be more appropriate, since for simple systems (see above) the mutual information in a correlated multi-layer system is proportional to multiplicity M rather than to ln₂M. This linear average is often written as:

$\langle {M\rangle }={\frac {1}{N}}\sum _{i=1}^{N}M_{i}={\frac {1}{N}}\sum _{i=1}^{N}2^{C_{i}}={\frac {1}{N}}\sum _{i=1}^{N}2^{\left(\sum _{j=1}^{6}f_{ij}\ln _{2}[{\frac {1}{f_{ij}}}]\right)}$. 

In either case the multiplicity values (M_i, M, and <M>) fall between 1 and 6, while the various C values above fall between 0 and ln₂[6] ~ 2.58 bits.

digital darkfield and the sinc wavelet

Although mutual information is a multi-moment measure of correlation, second-moment measures lie at the heart of exciting pattern-recognition tools in mathematical harmonic analysis (like wavelet theory) which lie intermediate between direct and reciprocal space. A new frontier for their application lies in detective work on atom-resolution images. The notes here connect these new developments to electron-optical "analog computing" strategies that have been in use for decades.

Using Cartesian coordinates in 2D and a common bandwidth (or frequency scale parameter) Δf for each direction, the windowlet transform of an input function s[x,y] (e.g. a deBroglie wave at the 2D exit surface of a physical specimen) is represented by:

$\psi [{\vec {x}},{\vec {f_{o}}},\Delta f]=(\Delta f)^{2}\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }s[{\vec {x'}}]h[\Delta f({\vec {x'}}-{\vec {x}}),{\frac {\vec {f_{o}}}{\Delta f}}]d^{2}{\vec {x'}}$.

Note that the output ψ is generally complex even if the input signal s is real. Here x={x,y} is a direct-space shift vector, and f_o is a frequency or reciprocal-space shift vector. Suppose that the kernel h[ξ,η] can be further written, in terms of dimensionless 2D variables ξ in direct-space and η in reciprocal space, as the product wg of a window function w[ξ] and the Fourier modulation function g[ξ,η] = e^-2πiξ·η. In that case the Fourier transform of the windowlet becomes:

$\Psi [{\vec {f}},{\vec {f_{o}}},\Delta f]=S[{\vec {f}}]W[{\frac {{\vec {f}}-{\vec {f_{o}}}}{\Delta f}}]$.

Here of course W and S are the Fourier transforms, respectively, of w and s.

Finally a square Shannon (or Littlewood-Paley) window in frequency space (this corresponds to a square aperture in the back-focal or transverse-momentum plane of an electron optical system) might be written as W[η_x,η_y] = rect[η_x] rect[η_y] where

$rect[\eta ]={\begin{cases}1&{\mbox{if }}|\eta |<{\frac {1}{2}}\\0&{\mbox{if }}|\eta |\geq {\frac {1}{2}}\end{cases}}$.

In direct space, the shape-transform of this aperture is a 2D sinc-window of the form:

$w[\xi _{x},\xi _{y}]={\frac {\sin[\pi \xi _{x}]}{\pi \xi _{x}}}{\frac {\sin[\pi \xi _{y}]}{\pi \xi _{y}}}$.

Although square or rectangular sinc-windowlets are useful when one wishes to comprehensively tile reciprocal space (wavelets per se typically use a hierarchy of 2×2 tilings), circular back-focal plane apertures are more common experimentally. These give rise in direct space to Airy (Bessel function) rather than sinc windows. The convergence properties of Airy windows are more isotropic.

When |f_ox|<Δf/2 and |f_oy|<Δf/2, microscopists who use electron-optical methods to do this calculation for an exit wave think of the map of intensity ψ*ψ as a brightfield image (exclusively so when illumination is parallel) because it includes the unscattered beam or DC peak. Regions in the image without scatterers are therefore bright.

Conversely when the frequency-space aperture excludes the unscattered beam, microscopists call the ψ*ψ map a darkfield image. Only objects with periodicity in bandwidth Δf, centered on the frequency shift vector f_o of this square aperture, light up in the ψ*ψ field. The corresponding phase maps of arg[ψ] = -i ln[ψ/|ψ|], on the other hand, contain very precise information on vector periodicity-displacement and hence strain tensors in projection. For example, maps of picometer-scale strain in atomic lattices have been made via these transforms, using microscopes with point-resolutions only in the 0.2 nanometer range.

The above windowlet is a direct-reciprocal inversion (in 2D) of the short-time Fourier transform. It has zero mean and is square integrable. Its transform also meets (at least for parallel illumination) the wavelet-decomposition admissibility condition in the darkfield case i.e. when the frequency window does not include the zero-frequency point or DC peak. The weak point of this decomposition is that support in direct space is not compact but instead exhibits sinc-system convergence, proportional to 1/x as x→∞ i.e. at the slow end of the Haar to sinc continuum.

roughness spectra & radial covariance

Speaking of harmonic analysis, variations in the vertical height h of a surface can be decomposed into a variance σ_h² with units of distance-squared plus a DC or mean-value component (left hand side of the equation below). Parseval's theorem and the Fourier transform H then lets one write this as a sum of azimuthally-averaged vertical roughness values R_n² for a given lateral spatial frequency-index n:

$\sigma _{h}^{2}+\langle {h\rangle }^{2}=\langle {h^{2}\rangle }={\frac {1}{N^{2}}}\sum _{j=0}^{N-1}\sum _{k=0}^{N-1}|h_{jk}|^{2}=\sum _{l=0}^{N-1}\sum _{m=0}^{N-1}|H_{lm}|^{2}\simeq \sum _{n=0}^{\frac {N}{2}}R_{n}^{2}$.

Here n corresponds to the dimensioned spatial frequency f=n/W, where W is the lateral field width of an N×N image, the Nyquist critical (maximum non-aliased) frequency in the data is N/(2W), and

$R_{n}^{2}\equiv 2\pi n\langle {|H_{n}|^{2}\rangle }$

where the azimuthal average in angle brackets is over all Fourier coefficients whose frequency indices l and m squared add more closely to n² than to some other integer squared. If one then plots log[R] as a function of log[period] where period=1/f, the DC components (n=0) vanish from consideration and a plot over log distance units on both axes can be used to tell the story of topography variation over many orders of magnitude in size.

Thus one can compare a plowed field, a redwood forest, a polished silicon wafer, and a stormy ocean on the same plot. How do vertical variations of these surfaces compare on lateral size scales of a nanometre, a micrometre, a millimetre and a metre? For example, where will they plot on each of these size scales with respect to the "line of hemispheres" i.e. the locus of spectra from equant objects (like hemispherical bumps of radius r) whose vertical variations peak out near R=r at lateral size scales near 1/f=r. Vertical variations over kilometre-scale separations laterally, for all except the silicon wafer, might be interesting as well.

The foregoing connects us to a number of pragmatic issues. One of these is the implicit periodic-boundary assumption of this discrete transform, which often gives rise to "edge effects" e.g. in power spectra of experimental data. Elegant ways around these remain to be discovered, although many different work-arounds have been discussed.

A book-keeping issue that comes up when writing programs is the discrete transform's normalization, below defined for the 2D transform using constants a and b. The inverse transform is obtained by changing the sign of both a and b, and of course exchanging h and H as well.

$H_{lm}\equiv {\frac {1}{N^{(1-a)}}}\sum _{j=0}^{N-1}\sum _{k=0}^{N-1}h_{jk}e^{b2\pi i{\frac {(jl+km)}{N^{2}}}}$.

The roughness analysis above uses {a,b}={-1,±1} so that the forward transform looks like an average over all pixels, although the signal processing convention is {a,b} = {1,-1}. More symmetric unitary transform conventions use {a,b} = {0,±1}. In one dimension the comparable equation looks like:

$H_{m}\equiv {\frac {1}{N^{\frac {(1-a)}{2}}}}\sum _{k=0}^{N-1}h_{k}e^{b2\pi i{\frac {km}{N^{2}}}}$.

Can you guess how a similar transformation in three dimensions would be set up?

sample bias and error in the mean

Here's an application example for sample autocovariance: An unbiased estimate of squared error in the mean of variable A, sampled at N known locations on an n-dimensional manifold x, may be written as:

$\langle ({\overline {A}}-\mu _{A})^{2}\rangle ={\frac {1}{N^{2}}}\sum _{i=1}^{N}\sum _{j=1}^{N}\sigma _{AA}[t_{ij}]={\frac {\sigma _{A}^{2}}{N}}+{\frac {1}{N^{2}}}\sum _{i=1}^{N}\sum _{j\neq i}^{N}\sigma _{AA}[t_{ij}]$,

where σ_AA[t_ij] is the autocovariance for variable A sampled at points i and j, σ_A is the population standard deviation, and the bar over A refers to the sample average. If we define sample bias coefficient ρ as the dimensionless average of A's auto-correlation over all sample point pairs, i.e. as

$\rho ={\frac {1}{N(N-1)}}\sum _{i=1}^{N}\sum _{j\neq i}^{N}{\frac {\sigma _{AA}[t_{ij}]}{\sigma _{A}^{2}}}$,

then this becomes

$\langle ({\overline {A}}-\mu _{A})^{2}\rangle =(1+(N-1)\rho ){\frac {\sigma _{A}^{2}}{N}}\simeq {\begin{cases}\rho \sigma _{A}^{2}&{\mbox{if }}\rho \gg {\frac {1}{N-1}}\\{\frac {1}{N}}\sigma _{A}^{2}&{\mbox{if }}|\rho |\ll {\frac {1}{N-1}}\\0&{\mbox{if }}\rho \leq -{\frac {1}{N-1}}\end{cases}}$.

The above relation can also be written in terms of expected value for sample variance (taken as a population):

$s_{AA}\equiv s_{A}^{2}\equiv {\frac {1}{N}}\sum _{i=1}^{N}(A_{i}-{\overline {A}})^{2}$,

if we can show that an unbiased estimator for σ_A² is:

$\sigma _{A}^{2}=\left({\frac {N}{N-1}}\right){\frac {\langle s_{A}^{2}\rangle }{1-\rho }}$.

This follows by expanding the definition of s_A² to show that:

$\langle s_{A}^{2}\rangle =\sigma _{A}^{2}-\langle ({\overline {A}}-\mu _{A})^{2}\rangle$.

As a result:

$\langle ({\overline {A}}-\mu _{A})^{2}\rangle ={\frac {1+(N-1)\rho }{(N-1)(1-\rho )}}\langle s_{A}^{2}\rangle \simeq {\begin{cases}{\frac {\rho }{1-\rho }}\langle s_{A}^{2}\rangle &{\mbox{if }}\rho \gg {\frac {1}{N-1}}\\{\frac {1}{N-1}}\langle s_{A}^{2}\rangle &{\mbox{if }}|\rho |\ll {\frac {1}{N-1}}\\0&{\mbox{if }}\rho \leq -{\frac {1}{N-1}}\end{cases}}$.

A useful "cube-cube containment estimate" of ρ for a 3D specimen with grain size d and sample size D is:

$\rho \simeq {\begin{cases}1-{\frac {D}{d}}+{\frac {2}{7}}\left({\frac {D}{d}}\right)^{2}&{\mbox{if }}D\leq d\\{\frac {2}{7}}\left({\frac {d}{D}}\right)^{5}-\left({\frac {d}{D}}\right)^{4}+\left({\frac {d}{D}}\right)^{3}&{\mbox{if }}D>d\end{cases}}$.

An exact "sphere-sphere containment model", which is non-physical because one cannot fill space completely with sphere of a single size, instead looks like:

$\rho \simeq {\begin{cases}1-{\frac {27}{35}}{\frac {D}{d}}+{\frac {2}{21}}\left({\frac {D}{d}}\right)^{3}&{\mbox{if }}D\leq d\\{\frac {2}{21}}\left({\frac {d}{D}}\right)^{6}-{\frac {27}{35}}\left({\frac {d}{D}}\right)^{4}+\left({\frac {d}{D}}\right)^{3}&{\mbox{if }}D>d\end{cases}}$.

one map two clocks

One starting point is Minkowski's flat-space metric equation as definition of the traveler time δτ elapsed between event points on the traveler's world-line separated by map distance δx and map time δt on a single reference map-frame of interconnected yardsticks and synchronized clocks:

$(c\underbrace {\delta \tau } _{\text{traveler time}})^{2}=(c\underbrace {\delta t} _{\text{map time}})^{2}-(\underbrace {\delta x} _{\text{map distance}})^{2}.\,$.

Dividing this by (cδt)² yields an expression relating Lorentz factor γ≡δt/δτ and coordinate velocity v≡δx/δt. Dividing (cδτ)² yields an expression relating Lorentz factor γ and proper-velocity w≡δx/δτ.

$\gamma \equiv {\frac {dt}{d\tau }}={\sqrt {1+\left({\frac {w}{c}}\right)^{2}}}={\frac {1}{\sqrt {1-({\frac {v}{c}})^{2}}}}=\cosh[\eta ]\equiv {\frac {e^{\eta }+e^{-\eta }}{2}}$.

All three of these velocity parameters also relate to hyperbolic velocity angle or rapdity:

$\eta =\sinh ^{-1}[{\frac {w}{c}}]=\tanh ^{-1}[{\frac {v}{c}}]=\pm \cosh ^{-1}[\gamma ]$.

Next define proper-acceleration as the change in proper-velocity w≡dx/dτ per unit time felt by the traveler e.g. in moving from frame C to C'. The unidirectional proper-velocity addition rule (Lorentz-factors γ≡dt/dτ multiply while coordinate-velocities v≡dx/dt add) then says that from map-frame B's point of view:

$\alpha \equiv {\frac {\delta w_{C'C}}{\delta \tau }}=\gamma {\frac {\delta w_{C'C}}{\delta t}}=\gamma \left(\gamma _{C'B}\gamma _{\text{BC}}{\frac {\delta v_{\text{CB}}}{\delta t}}\right)=\gamma ^{3}{\frac {\delta v}{\delta t}}=\gamma ^{3}a$.

If we think of energy as E ≡ γmc² one obtains a (1+1)D work-energy relation for mass m objects of the form:

${\frac {1}{m}}{\frac {\delta E}{\delta x}}={\frac {\delta (\gamma mc^{2})}{m\delta x}}=c^{2}{\frac {\delta \gamma }{\delta x}}=c^{2}{\frac {\delta \gamma }{\delta v}}{\frac {\delta v}{\delta x}}=c^{2}{\frac {\gamma ^{3}v}{c^{2}}}{\frac {\delta v}{\delta x}}=\gamma ^{3}{\frac {\delta x}{\delta t}}{\frac {\delta v}{\delta x}}=\gamma ^{3}{\frac {\delta v}{\delta t}}=\gamma ^{3}a=\alpha$.

A similar argument for momentum p ≡ mw = γmv relates the (1+1)D frame-variant force dp/dt to acceleration as well:

${\frac {1}{m}}{\frac {\delta p}{\delta t}}={\frac {\delta (mw)}{m\delta t}}={\frac {\delta w}{\delta t}}={\frac {\delta (\gamma v)}{\delta t}}=\gamma ^{3}{\frac {\delta v}{\delta t}}=\gamma ^{3}a=\alpha$.

From this follow three integrals of (1+1)D constant proper-accelerated motion that reduce to the familiar equations of constant coordinate-acceleration at low speeds:

$\alpha ={\frac {\Delta w}{\Delta t}}=c{\frac {\Delta \eta }{\Delta \tau }}=c^{2}{\frac {\Delta \gamma }{\Delta x}}{\text{ }}{\stackrel {v\ll c}{\rightarrow }}{\text{ }}a={\frac {\Delta v}{\Delta t}}={\frac {1}{2}}{\frac {\Delta (v^{2})}{\Delta x}}$.

These integrals also survive in the (3+1)D case, albeit with certain directional corrections required:

$\alpha =\gamma _{\perp }{\frac {\Delta w_{\parallel }}{\Delta t}}=c{\frac {\Delta \eta _{\parallel }}{\Delta \tau }}={\frac {c^{2}}{\gamma _{\perp }}}{\frac {\Delta \gamma }{\Delta x_{\parallel }}}{\text{ }}{\stackrel {v\ll c}{\rightarrow }}{\text{ }}a={\frac {\Delta v_{\parallel }}{\Delta t}}={\frac {1}{2}}{\frac {\Delta (v^{2})}{\Delta x_{\parallel }}}$.

some four-vector rearrangements

The four-vector equation in the editing tips section can be broken into timelike and spacelike parts. If w is the proper velocity dx/dτ, γ is as usual dt/dτ, we might the write the following:

$c^{2}\gamma {\frac {d\gamma }{dt}}=\gamma {\vec {v}}\cdot {\frac {d{\vec {w}}}{dt}}=\gamma {\vec {v}}\cdot ({\vec {a_{o}}}+{\vec {a_{g}}})$.

$\gamma {\frac {d{\vec {w}}}{dt}}=\gamma ({\vec {a_{o}}}+{\vec {a_{g}}})$.

In the above equations, a_o is an acceleration due to proper forces and a_g is presumably due to geometric forces. At low speeds they track the familiar coordinate acceleration vector. For unidirectional motion at any speed, a_o's magnitude tracks proper acceleration's magnitude. How do these quantities relate to the 4-vector terms above more generally?

If we multiply the above equations by mass m and divide by γ=dt/dτ, one obtains:

${\frac {dE}{dt}}={\vec {v}}\cdot {\frac {d{\vec {p}}}{dt}}=m{\vec {v}}\cdot {\frac {d{\vec {w}}}{dt}}$ (timelike) and ${\frac {d{\vec {p}}}{dt}}=m{\frac {d{\vec {w}}}{dt}}$ (spacelike).

The map frame rate of change of proper velocity dw/dt can, in turn, be broken down into proper and geometric force components based on the original 4-vector equation as follows:

${\frac {d{\vec {w}}}{dt}}={\vec {a_{o}}}+{\vec {a_{g}}}={\frac {1}{m}}(\Sigma {\vec {f_{o}}}+\Sigma {\vec {f_{g}}})$

Are f_o and f_g here similarly proper and geometric force components seen from the map-frame coordinate system, which respectively sum to cause the observed motion? If so, how do these components relate to the 4-vector components above, and the frame invariant proper force F_o=mα seen by the object?

In short, we might therefore write...

${\frac {dE}{dt}}={\vec {v}}\cdot {\frac {d{\vec {p}}}{dt}}$ (timelike) and ${\frac {d{\vec {p}}}{dt}}=\Sigma {\vec {f_{o}}}+\Sigma {\vec {f_{g}}}=m({\vec {a_{o}}}+{\vec {a_{g}}})$ (spacelike).

some geometric force rearrangements

Coordinate acceleration a_rot associated with an object from the perspective of a rotating frame adds to the object's physical or proper acceleration a_o a series of geometric terms:

${\vec {a}}_{rot}={\vec {a}}_{o}-{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}})-2{\vec {\omega }}\times {\vec {v}}_{rot}-{\frac {d{\vec {\omega }}}{dt}}\times {\vec {r}}$.

The first "centrifugal acceleration" term depends only on the radial position r and not velocity of our object, the second "Coriolis acceleration" term depends only on the object's velocity in the rotating frame v_rot but not its position, and the third "Euler acceleration" term depends only on position and the rate of change of the frame's angular velocity ω.

For an object observed at low speed from the vantage point of an accelerating frame, the coordinate acceleration observed depends on the acceleration of the frame. If the object is being accelerated in the same way as the frame, it appears to have no acceleration at all.

${\vec {a}}_{acc}={\vec {a}}_{o}-{\vec {a}}_{frame}$

In the Schwarzschild shell-frame case, we might similarly write:

${\vec {a}}_{shell}={\vec {a}}_{o}-{\sqrt {\frac {r}{r-r_{s}}}}{\frac {GM}{r^{2}}}{\hat {r}}$

where Schwarzschild radius r_s=2GM/c². Thus for r>>r_s, an upward proper force of magnitude GMm/r² is needed to prevent one from accelerating downward. At the earth's surface this becomes:

${\vec {a}}_{shell}={\vec {a}}_{o}-g{\hat {r}}$

where g is the downward 9.8 m/s² acceleration due to gravity, and r-hat is a unit vector in the radially outward direction from the center of the gravitating body. Thus here an outward proper force of mg is needed to keep one from accelerating downward.

Note: The foregoing results follow if one first calculates the Christoffel symbols:

$\left({\begin{array}{llll}\left\{\Gamma _{t,t}^{t},\Gamma _{t,r}^{t},\Gamma _{t,\theta }^{t},\Gamma _{t,\phi }^{t}\right\}&\left\{\Gamma _{r,t}^{t},\Gamma _{r,r}^{t},\Gamma _{r,\theta }^{t},\Gamma _{r,\phi }^{t}\right\}&\left\{\Gamma _{\theta ,t}^{t},\Gamma _{\theta ,r}^{t},\Gamma _{\theta ,\theta }^{t},\Gamma _{\theta ,\phi }^{t}\right\}&\left\{\Gamma _{\phi ,t}^{t},\Gamma _{\phi ,r}^{t},\Gamma _{\phi ,\theta }^{t},\Gamma _{\phi ,\phi }^{t}\right\}\\\left\{\Gamma _{t,t}^{r},\Gamma _{t,r}^{r},\Gamma _{t,\theta }^{r},\Gamma _{t,\phi }^{r}\right\}&\left\{\Gamma _{r,t}^{r},\Gamma _{r,r}^{r},\Gamma _{r,\theta }^{r},\Gamma _{r,\phi }^{r}\right\}&\left\{\Gamma _{\theta ,t}^{r},\Gamma _{\theta ,r}^{r},\Gamma _{\theta ,\theta }^{r},\Gamma _{\theta ,\phi }^{r}\right\}&\left\{\Gamma _{\phi ,t}^{r},\Gamma _{\phi ,r}^{r},\Gamma _{\phi ,\theta }^{r},\Gamma _{\phi ,\phi }^{r}\right\}\\\left\{\Gamma _{t,t}^{\theta },\Gamma _{t,r}^{\theta },\Gamma _{t,\theta }^{\theta },\Gamma _{t,\phi }^{\theta }\right\}&\left\{\Gamma _{r,t}^{\theta },\Gamma _{r,r}^{\theta },\Gamma _{r,\theta }^{\theta },\Gamma _{r,\phi }^{\theta }\right\}&\left\{\Gamma _{\theta ,t}^{\theta },\Gamma _{\theta ,r}^{\theta },\Gamma _{\theta ,\theta }^{\theta },\Gamma _{\theta ,\phi }^{\theta }\right\}&\left\{\Gamma _{\phi ,t}^{\theta },\Gamma _{\phi ,r}^{\theta },\Gamma _{\phi ,\theta }^{\theta },\Gamma _{\phi ,\phi }^{\theta }\right\}\\\left\{\Gamma _{t,t}^{\phi },\Gamma _{t,r}^{\phi },\Gamma _{t,\theta }^{\phi },\Gamma _{t,\phi }^{\phi }\right\}&\left\{\Gamma _{r,t}^{\phi },\Gamma _{r,r}^{\phi },\Gamma _{r,\theta }^{\phi },\Gamma _{r,\phi }^{\phi }\right\}&\left\{\Gamma _{\theta ,t}^{\phi },\Gamma _{\theta ,r}^{\phi },\Gamma _{\theta ,\theta }^{\phi },\Gamma _{\theta ,\phi }^{\phi }\right\}&\left\{\Gamma _{\phi ,t}^{\phi },\Gamma _{\phi ,r}^{\phi },\Gamma _{\phi ,\theta }^{\phi },\Gamma _{\phi ,\phi }^{\phi }\right\}\end{array}}\right)$

for the far-coordinate Schwarzschild metric (c dτ)² = (1-r_s/r)(c dt)² - (1/(1-r_s/r))dr² - r² dθ² - (r sin[θ])² dφ², where r_s is the Schwarzschild radius 2GM/c². The resulting array of coefficients becomes:

$\left({\begin{array}{llll}\left\{0,{\frac {r_{s}}{2r(r-r_{s})}},0,0\right\}&\left\{{\frac {r_{s}}{2r(r-r_{s})}},0,0,0\right\}&\{0,0,0,0\}&\{0,0,0,0\}\\\left\{{\frac {r_{s}c^{2}(r-r_{s})}{2r^{3}}},0,0,0\right\}&\left\{0,{\frac {r_{s}}{2r(r_{s}-r)}},0,0\right\}&\{0,0,r_{s}-r,0\}&\left\{0,0,0,(r_{s}-r)\sin ^{2}(\theta )\right\}\\\{0,0,0,0\}&\left\{0,0,{\frac {1}{r}},0\right\}&\left\{0,{\frac {1}{r}},0,0\right\}&\{0,0,0,-\cos(\theta )\sin(\theta )\}\\\{0,0,0,0\}&\left\{0,0,0,{\frac {1}{r}}\right\}&\{0,0,0,\cot(\theta )\}&\left\{0,{\frac {1}{r}},\cot(\theta ),0\right\}\end{array}}\right)$.

From this you can obtain the shell-frame proper acceleration by setting coordinate acceleration to zero and thus requiring that proper acceleration cancel the geometric acceleration of a stationary object i.e. $A^{\lambda }=\Gamma ^{\lambda }{}_{\mu \nu }U^{\mu }U^{\nu }$ = {0,GM/r²,0,0}. This does not solve the problem yet, since Schwarzschild coordinates in curved spacetime are book-keeper coordinates but not those of a local observer. The magnitude of the above proper acceleration 4-vector, namely α=Sqrt[1/(1-r_s/r)]GM/r², is however precisely what we want i.e. the upward frame-invariant proper acceleration needed to counteract the downward geometric acceleration felt by dwellers on the surface of a planet.

A special case of the above Christoffel symbol set is the flat-space spherical coordinate set obtained by setting r_s or M above to zero:

$\left({\begin{array}{llll}\left\{0,0,0,0\right\}&\left\{0,0,0,0\right\}&\{0,0,0,0\}&\{0,0,0,0\}\\\left\{0,0,0,0\right\}&\left\{0,0,0,0\right\}&\{0,0,-r,0\}&\left\{0,0,0,-r\sin ^{2}(\theta )\right\}\\\{0,0,0,0\}&\left\{0,0,{\frac {1}{r}},0\right\}&\left\{0,{\frac {1}{r}},0,0\right\}&\{0,0,0,-\cos(\theta )\sin(\theta )\}\\\{0,0,0,0\}&\left\{0,0,0,{\frac {1}{r}}\right\}&\{0,0,0,\cot(\theta )\}&\left\{0,{\frac {1}{r}},\cot(\theta ),0\right\}\end{array}}\right)$.

From this we can obtain, for example, the centripetal proper acceleration needed to cancel the centrifugal geometric acceleration of an object moving at constant angular velocity ω=dφ/dτ at the equator where θ=π/2. Forming the same 4-vector sum as above for the case of dθ/dτ and dr/dτ zero yields nothing more than the classical acceleration for rotational motion given above, i.e. $A^{\lambda }=\Gamma ^{\lambda }{}_{\mu \nu }U^{\mu }U^{\nu }$ = {0,-r(dφ/dτ)²,0,0} so that a_o=ω²r. Coriolis effects also reside in these connection coefficients, and similarly arise from coordinate-frame geometry alone.

Inserting collapsible animations

This is a useful trick, that I think I first encountered on the kinematics page, which allows one to make animations (or static figures for that matter as well) available only on demand. It also has potential for illustrating how the concepts on a given page can be used to address sample challenges. The simpler the solution, the better the correlation between page concepts and the problem at hand.

The perspective of a linearly-accelerated frame might be illustrated with specific calculated examples. For instance:

Worked example: Driving from block to block

Map and car frame perspectives of physical (red) and fictitious (blue) forces for a car driving from one stop sign to the next. In this illustration the car accelerates after a stop sign until midway up the block, at which point the driver is immediately off the accelerator and onto the brake so as to make the next stop. Leaving the stop sign In the following analyses, only forces forward or backward in the direction of travel will be considered. Both analyses predict that the forward force of the car on the passenger will equal the passenger mass m times the acceleration of the car. Thus if the acceleration of an 80 kg passenger is 1 gee (9.8 m/s2), then the forward force needed to do it is nearly 800 Newtons. Map frame during speed up $F_{road}=\Sigma F_{map}=ma_{map}=ma_{seat}.$ Here from the map frame perspective the passenger is seen to be continually accelerated forward, with an acceleration equal to the unbalanced forward force Froad of the car on the passenger, divided by the passenger's mass m. The fact that the road force on the passenger is unbalanced means that in this frame there are no other forces in the direction of travel on the passenger. Car frame during speed up $F_{road}-ma_{seat}=\Sigma F_{car}=ma_{car}=0.$ From the car frame perspective the passenger may be said to experience a forward road force canceled by a backward g-force, which results in no acceleration at all from the perspective of that frame. Unlike the forward road force exerted by the car on the passenger, the backward g-force acts on every bit of the passenger. Moreover the g-force magnitude is proportional to the passenger's mass so that, if allowed to cause acceleration, the acceleration would be mass-independent. Approaching the stop sign Map frame while braking $-F_{brake}=\Sigma F_{map}=ma_{map}=-ma_{belt}.$ Car frame while braking $-F_{brake}+ma_{belt}=\Sigma F_{car}=ma_{car}=0.$

The rotating frame perspective might be illustrated with specific calculated examples. For instance:

Worked example: Constant speed slingshot

Map and spin frame perspectives of forces associated with a stone released after being spun around on a massless rope. Forces on the stone include the inward centripetal (red) force seen in both frames, as well as the geometric (blue) force seen in the spin frame. Before the stone is released, the blue geometric force is purely centrifugal (pointing radially outward), while after release the geometric force is a sum of centrifugal and Coriolis components. Note that the centrifugal component (light blue) is always radial, while the Coriolis component (green) is always perpendicular to spin frame velocity. Also seen in both frames is the force on the rope's anchor point (magenta) caused by Newton's 3rd Law action-reaction to the centripetal force on the stone. Before projectile launch The following alternate analyses of motion before the stone is released consider only forces acting in the radial direction. Both analyses predict that string tension T=mv2/r. For example if the radius of the sling is r=1 metre, the velocity of the stone in the map frame is v=25 metres per second, and the stone's mass m=0.2 kilogram, then the tension in the string will be 125 Newtons. Map frame story before launch $-T_{centripetal}=\Sigma F_{radial}=ma_{radial}=-m{\frac {v^{2}}{r}}.$ Here the stone is seen to be continually accelerated inward so as to follow a circular path of radius r. The inward radial acceleration of aradial=v2/r is caused by a single unbalanced centripetal force T. The fact that the tension force is unbalanced means that, in this frame, the centrifugal (radially-outward) force on the stone is zero. Spin frame story before launch $m{\frac {v^{2}}{r}}-T_{centripetal}=\Sigma F_{rot}=ma_{rot}=0.$ From the spin frame perspective the stone may be said to experience balanced inward centripetal (T) and outward centrifugal (mv2/r) forces, which result in no acceleration at all from the perspective of that frame. Unlike the centripetal force, the frame-dependent centrifugal force acts on every bit of the circling stone much as gravity acts on every ounce of you. Moreover the centrifugal force magnitude is proportional to the stone's mass so that, if allowed to cause acceleration, the acceleration would be mass-independent. After projectile launch After the stone is released, in the spin frame both centripetal and Coriolis forces act in a delocalized way on all parts of the stone with accelerations that are independent of the stone's mass. By comparison in the map frame, after release no forces are acting on the projectile at all. The foregoing three properties: (i) ability to be eliminated by a suitable choice of frame, (ii) accelerations that are independent of mass, and (iii) delocalized action on every bit of mass are common to forces that arise from the connection term in a coordinate system's covariant derivative. Such geometric or improper forces include g-forces, Euler forces and gravity as well as Coriolis and centrifugal. They are in this way distinct from physical or proper forces that result e.g. from direct contact or electrostatic interaction.

Blockquotes, and code repair

Aside: The following paragraphs were used to temporarily replace a consensus introduction after it was unceremoniously deleted. How does such activity relate to mechanisms for regulation and repair of molecular codes similarly offered up for community access in eukaryotic cell interiors? This also illustrates use of Wikipedia's blockquote qualifier:

When an object is constrained to move in circular motion, the outward radial force seen to be acting on that object from its rotating vantage point is known as the centrifugal force (from Latin centrum "center" and fugere "to flee").

Because this force arises from the connection term in the accelerated coordinate system's covariant derivative, it may be referred to as a geometric or fictitious force (as distinct from a proper or physical force) even though its consequences from the perspective of that frame are very real. Such geometric forces allow one to apply Newton's laws locally in accelerated frames, and they act on every ounce of an object's being rather than e.g. via direct contact or electrostatic repulsion.

Centrifugal force should not be confused with the inward-acting centripetal force that causes a moving object to follow a circular path. The proper reaction to this centripetal force, exerted by such revolving objects on their surroundings, was in earlier times also called centrifugal although this use is less common today.

Who has these interests?

Complex-system informatics, nanoscience, materials astronomy, and the thermo-chapters of introductory physics?

• /size-scale awareness
• /TEMprep

Other wiki userlinks of potential interest

• wikibooks
• wikiversity
• creative commons

Other useful wikicode

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• <ref>{{cite book |last= |first= |authorlink= |coauthors= |title= |year= |publisher= |quote= | url= |isbn= }}</ref>
• <ref>{{cite news |first= |last= |authorlink= |coauthors= |title= |url= |quote= |publisher=[[New York Times]] |date= |accessdate=2024-05-26 }}</ref>
• <ref>{{cite news |first= |last= |authorlink= |coauthors= |title= |url= |quote= |publisher=[[Time (magazine)]] |date= |accessdate=2024-05-26 }}</ref>
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• <ref>{{cite journal | last = | first = | authorlink = | coauthors = | date = | year = | month = | title = | journal = | volume = | issue = | pages = | publisher = | location = | issn = | pmid = | doi = | bibcode = | oclc = | id = | url = | language = | format = | accessdate = | laysummary = | laysource = | laydate = | quote = }}</ref>
• {{Lifetime|1900|2000|Last, First}}
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• <div class="references-small" style="-moz-column-count:2; column-count:2;">
• <ref name=usgs></ref> <ref name=usgs/>
• {{Maintained|[[User:Richard Arthur Norton (1958- )|'''Richard Arthur Norton (1958- )]]'''}}
• {{Infobox Person | name = | image = | image_size = | caption = | birth_name = | birth_date = {{Birth date|1911|10|20}} | birth_place = | death_date = {{Death date and age|1940|12|23|1911|10|20}} | death_place = | death_cause = | resting_place = | resting_place_coordinates = | residence = | nationality = | other_names = | known_for = | education = | employer = | occupation = | title = | salary = | networth = | height = | weight = | term = | predecessor = | successor = | party = | boards = | religion = | spouse = | partner = | children = | parents = | relatives = | signature = | website = | footnotes = }}