Probability Distribution Pn Research Articles

A general methodology for population analysis

For a given population with N — current and M — maximum number of entities, modeled by a Birth–Death Process (BDP) with size M+1, we introduce utilization parameter ρ, ratio of the primary birth and death rates in that BDP, which, physically, determines (equilibrium) macrostates of the population, and information parameter ν, which has an interpretation as population information stiffness. The BDP, modeling the population, is in the state n, n=0,1,…,M, if N=n. In presence of these two key metrics, applying continuity law, equilibrium balance equations concerning the probability distribution pn, n=0,1,…,M, of the quantity N, pn=Prob{N=n}, in equilibrium, and conservation law, and relying on the fundamental concepts population information and population entropy, we develop a general methodology for population analysis; thereto, by definition, population entropy is uncertainty, related to the population. In this approach, what is its essential contribution, the population information consists of three basic parts: elastic (Hooke’s) or absorption/emission part, synchronization or inelastic part and null part; the first two parts, which determine uniquely the null part (the null part connects them), are the two basic components of the Information Spectrum of the population. Population entropy, as mean value of population information, follows this division of the information. A given population can function in information elastic, antielastic and inelastic regime. In an information linear population, the synchronization part of the information and entropy is absent. The population size, M+1, is the third key metric in this methodology. Namely, right supposing a population with infinite size, the most of the key quantities and results for populations with finite size, emerged in this methodology, vanish.

Free Will with Afterthoughts: A Quasichemical Model

A model of free will is proposed, appealing to the similarity with simple, two-body chemical reactions where the energy curves for the reagents and for the products cross. The system at the crossing point has a freedom of choice to perform the reaction or not. The Landau-Zener formula, corresponding to the opportunity of meeting twice the crossing point, is interpreted as free will with an afterthought and generalized to the cases when a subject thinks about a choice n times. If the probability distribution pn of afterthoughts is known, the probability of a final yes decision is given. The results are generalized to situations where a preference for or against a change exists or where the freedom is only partial, has to fight with conditioning factors, and possibly decreases with increasing instances of free choice.

Dynamical barriers for the random ferromagnetic Ising model on the Cayley tree: traveling-wave solution of the real space renormalization flow

We consider the stochastic dynamics near zero temperature of the random ferromagnetic Ising model on a Cayley tree of branching ratio K. We apply the boundary real space renormalization procedure introduced in our previous work (Monthus and Garel 2013 J. Stat. Mech. P02037) in order to derive the renormalization rule for dynamical barriers. We obtain that the probability distribution Pn(B) of a dynamical barrier for a sub-tree of n generations converges for large n towards some traveling-wave Pn(B) ≃ P*(B − nv), i.e. the width of the probability distribution remains finite around an average value that grows linearly with the number n of generations. We present numerical results for the branching ratios K = 2 and K = 3. We also compute the weak-disorder expansion of the velocity v for K = 2.

Tuck's incompressibility function: statistics for zeta zeros and eigenvalues

For any function D(x) that is real for real x, positivity of Tuck's function Q(x) ≡ D′2(x)/(D′2(x) − D(x)D″(x)) is a condition for the absence of complex zeros close to the real axis. Study of the probability distribution PN(Q), for D(x) with N zeros corresponding to eigenvalues of the Gaussian unitary ensemble (GUE), supports Tuck's observation that large values of Q are very rare for the Riemann zeros. PN(Q) has singularities at Q = 0, Q = 1 and Q = N. The moments (averages of Qm) are much smaller for the GUE than for uncorrelated random (Poisson-distributed) zeros. For the Poisson case, the large-N limit of PN(Q) can be expressed as an integral with infinitely many poles, whose accumulation, requiring regularization with the Lerch transcendent, generates the singularity at Q = 1, while the large-Q decay is determined by the pole closest to the origin. Determining the large-N limit of PN(Q) for the GUE seems difficult.

Growing random sequences

We consider the random sequence xn = xn−1 + γxq, with γ > 0, where q = 0, 1, ..., n − 1 is chosen randomly from a probability distribution Pn(q). When all q are chosen with equal probability, i.e. Pn(q) = 1/n, we obtain an exact solution for the mean xn and the divergence of the second moment x2n as functions of n and γ. For γ = 1 we examine the divergence of the mean value of xn, as a function of n, for the random sequences generated by power law and exponential Pn(q) and for the non-random sequence Pn(q) = δq,β(n−1).

Exact distribution function for discrete time correlated random walks in one dimension

A discrete time correlated random walk in one dimension is investigated. Combinatorial arguments are used to calculate the exact probability distribution PN(L), the probability that the correlated random walker arrives at a distance L steps to the right of its starting point after exactly N steps. PN(L) is calculated using arbitrary initial conditions which permit the influence of end effects and boundary conditions to be calculated and several special cases are considered in detail. PN(L) with arbitrary initial conditions is calculated both with and without a bias for motion in one direction yielding a useful model for the combined diffusion and drift of charged particles undergoing a correlated random walk in an applied field. The relation of the correlated random walk to the Ising model is also discussed.

A generalization of the Ehrenfest urn model

This article presents a generalization of the Ehrenfest urn model in order to obtain the time evolution of the number of molecules n(t) in any subvolume V of a vessel V0 containing N molecules of a gas. The formalism makes use of the equilibrium (stationary) probability distribution PN(n,V,V0)≡P0(n) characterizing the gas. Two cases of pedagogical value, the ideal gas (binomial distribution) and the lattice gas (hypergeometric distribution) are considered.

Numerical simulations on spin glasses (invited)

We discuss the results of numerical simulations of two spin-glass models, the infinite range Sherrington–Kirkpatrick (SK) model, and a nearest–neighbor model in three dimensions. The probability distribution PN(q) for the spin glass order parameter q has been evaluated for the SK model below Tc with N, the number of spins, up to 192. It shows the same basic features as the Parisi function dx/dq as expected, but with sizeable finite size corrections. Results for the short-range model show evidence for a finite temperature transition but further work is needed to confirm this.

Normal and antinormal ordering of field operators in quantum optics

The relations between the probability distributionp n vt of the number of photonsn vt in a volumeV at timet and the probability distributions PN(W) and PA(W) of the integrated intensityW related to the normal and antinormal ordering of field operators and the relations between corresponding moments are studied for a field containingM modes. As the normally ordered correlations are measured with photodetectors and the antinormally ordered correlations are measured with quantum counters the relations betweenn vt , PN(W) and PA(W) permit the statistical behaviour of light to be determined from measurements with photodetectors and quantum counters. The results obtained here are used for the superposition of coherent and thermal fields. It is also shown that the antinormal correlations depend explicitly on the number of modes and that in the classical limit, when the average photon occupation number per mode becomes large, the distributionsn vt , PN(W) and PA(W) become equal.

Probability Distribution of the Radius of Gyration of a Flexible Polymer

Various aspects of the mathematics of the probability distribution PN(Sx) of one component of the square of the radius of gyration of an ideal Brownian chain with N units are presented. A rigorous expression for PN(Sx) in the form of a contour integral is obtained. The resulting integral is written in terms of Tchebichef polynomials. Several rigorous and approximate results are obtained for both the limiting distribution (N infinite) and for finite N.

Zur Sekund�relektronenemission durch Argonionen und Argonatome im Energiebereich von 1 bis 20 keV

Be-, CuBe-, Cu- and Al-targets are bombarded by single A+ and A0 in the energy range from 1 to 20 keV. The secondary electrons accelerated by 20 keV are counted by a proportional counter backed by a multichannel analyser. The probability distributionPn of emission ofn=0,1, 2, 3, ... secondary electrons can be found from the pulse height distribution. WhenPn is known the yield for A+ and A0 can be calculated. The counting of neutrals in the energy range investigated is discussed.

Untersuchungen zur Mehrfach-Emission von Sekund�relektronen

CuBe- and NaCl-targets are bombarded by single electrons (100–600 eV). The secondary electrons accelerated by 40 kV strike the crystal of a scintillation counter, backed by a multichannel analyser. The probabilityP n of emission ofn=0, 1, 2, 3, ... secondaries can be found from the pulse height distribution. The probability distributionP n =f(n) shows a characteristic deviation from aPoisson's distribution. There was no evidence that there is a preference for even numbers ofn as found byBarrington andAnderson.

Systematic Characterization of mth-Order Energy-Level Spacing Distributions

The mth-order energy-level spacing distributions p(m)(x) for complex spectra are defined in terms of a general joint probability distribution PN(λ1, … λN) for N consecutive eigenvalues. The precise limiting processes involved are explained, and are subsequently used to obtain two formal representations of p(m)(x). Both representations yield p(m)(x) = xm/m! exp (−x) for statistically independent eigenvalues. One of the representations, which is an extension of Dyson's method for m = 0, 1, is applied to the superposition of n independent sequences of levels. General asymptotic results are found for the mth-order distributions for (a) small x, arbitrary n, and (b) arbitrary x with n → ∞.