An Irrational hybrid in an Analogue-Digital world?

Nature's four great irrationals ẟ, ϕ, 𝝿 and e together seem to be encoded in much of what we describe as physical reality. An instructive (if arbitrary) way to characterise them could be to demarcate them according to their discrete or continuous ("analytic") roots.

A fifth irrational, the lesser-known Euler-Mascheroni constant,  ɣ traverses the analogue functions of the analytic (mathematical) modeller and the apparent discrete reality of the physical world. It appears in their regularisation programs, that coarsen an otherwise too finely-tuned view of the world. That is in the development of a non-divergent, non infinity-riddled quantum field theory of electrodynamics. Does ɣ deserve its place in Nature's hall of fame?

• The "Digitals" of sequencing, non-linear growth:

• ẟ-Feigenbaum's number 4.669201609..., that universal constant of chaos theory being the limiting value characterising the velocity of period-doubling, the ratio of the intervals between bifurcation points of dynamical system approaching chaotic behaviour (planetmath.org).

• ϕ- phi-golden ratio (√5-1)/2=1.618033988...that irrational limit to which ratios of successive numbers in the iterative Fibonacci sequence 1,1,2,3,5,8,13,.. tends. The ratio is characterised by the log spiral and the non-Markovian (see below) nature of the sequence growth.

• The "Analogues" of analyticity of Calculus and trigonometric linearity:

• 𝝿 - pi, 3.14159265359.. that ratio of circumference over diameter that reflects the flatness of the space in which a locus of points a fixed distance from a central point scribe. Alternatively the area that that scribed circle (x² +y²=r²) encapsulates if its radius, r is one unit.

•  e- Euler's number, 2.71828182845.. that number beyond x=1 (on the real number line) at which the definite integral (area) under the hyperbola y=1/x is one unit.That is when the domain of function is (1,a=e) as in the graph below,

To these four irrationals should we add to Nature's true natural numbers the "hybrid"(as yet to be confirmed definitively irrational):

•  γ≈0.5772156649…, Euler-Mascheroni constant?

Logarithmically growing Harmonic Series

Whereas  ln(n) is the limit of the area of the analytic hyperbola 1/x as we extend "a" in the graph above to infinity, the "digital" harmonic series:

cumulatively grows"logarithmically' slowly.

The Euler-Mascheroni constant,  ɣ  is the finite difference (the light grey below) between this "logarithmically slowly" cumulatively growing divergent Harmonic series and the divergent hyperbola: 

It is the grey area in the graph below, the difference between the two series 

being summarised as follows:

setting its range between 0.4851 and the limit of the convergent alternating harmonic series:

Rather than exploring the pinning down of the number in terms by using the Psi (digamma) and Gamma functions which is done ably here (mae.ufl.edu) we merely nod at its acquaintances both mathematical and physical.

Alternating Harmonic Series limit of ln(2)

The alternating harmonic series, unlike its corpulent brother is a convergent series,

converging to ln(2), which itself is equivalently the area under the hyperbola defined between the domain interval of (1, 2):

This ln(2)limiting value is a number familiar to those acquainted with exponential radioactive decay. The deterministic model for a decaying (aggregate) set of discrete elements, N(t), governed by quantum selection rules, says that the mean lifetime an element remains in the set relates to the decay rate, λ (characteristic) of the sample:

The time required for the decaying quantity to fall to one half of its initial value is the ratio of this limit to decay rate:

When this expression is inserted in the exponential equation above we move from base e to base 2:

The statistics of this exponential decay can be derived using a stochastic Markov chain process, that satisfies the ("memory-less") property that predictions for the future of the process can be based solely on its present state, ignoring the process's full history. Unlike the memory-full(er) growth determined by golden ratio ϕ, ɣ rather characterises that decay which retains no memory of the history that delivered the aggregate to its present state.  Rather than associate Euler's constant, e with decay, perhaps it is more appropriate to consider such spontaneous decay as governed by ɣ.

We may summarise the irrationals then according to the following:

• 𝝿 and e embody the reductionist principle of describing the scattering of light (electromagnetic waves) with homogenous solids by sets of interacting linear trigonometric functions;
• ϕ, ẟ encode the complex non-linear dynamics inherent to interacting systems involving non-homogenous condensed matter such as gases and fluids as modelled for example by Navier-Stokes equations;
• ɣ traverses the regime between these memory-less and memory-full processes. Somewhere between the analogue functions of the mathematical analytic model and the discrete reality of the physical world.  

In theories of Quantised fields, renormalization is the process of subtracting counter terms at each order of the "perturbation" (about the linearised) theory. Regularization is the modification of the theory at (shorter) distance (“cutoff") scales so that the theory becomes well-defined (UMD physics). A "regulator" is a process which renders finite a momentum integral which is superficially divergent.  The Euler-Mascheroni constant γ appears in the finite part of the integral having no apparent physical significance being merely an "artefact" of a coarse-graining subtraction scheme to counter divergences arising from deploying otherwise analogue idealised functions.