Stirling's Formula Research Articles

Statistical‐Mechanical Theory on the Probability Distribution Function for the Net Charge of an Electrolyte Droplet

AbstractDroplets of electrolyte solutions in an insulating medium are ubiquitous in nature. The net charges of these droplets are normally nonzero, and they fluctuate. However, a theory on the probability distribution function for the net charge of droplets is lacking, so far. Thus, a statistical‐mechanical theory of a charged droplet is developed including the effect of the electrostatic energy of charging as well as the random distribution of ions. Then, the probability distribution function for the net charge of an electrolyte droplet is calculated assuming that it is generated from a macroscopic solution with the different cation and anion concentrations. Using the Gaussian approximation and Stirling's formula, the analytic results for the average and variance of the net charge of a droplet are obtained.

A friendly derivation of Stirling's formula

We demonstrate the Stirling formula, approximating the factorial, utilising accessible and elementary methods in an engaging manner.

Random Walk Regression Problem in One and Two Dimensions

This paper examines the previous discoveries and research on the random walk problem, considers the practical application of random walk. This paper starts from the basic questions and conducts a more detailed study of the basic regression problem with one-dimensional and two-dimensional situations. Markov chains and Markov property are introduced, and then the method and important properties of series convergence are properly explained and demonstrated. Through the introduction of Brownian motion, the key part of the article is introduced, and the explanation and proof of Stirling's formula are carried out, then the proof of one-dimensional cases is entered, and the regression is proved by the above knowledge and deduction, and then the same idea is extended to the two-dimensional case for proof. Through the research and proof of this paper, the basic problem of random walk will be solved and well interpreted, laying the foundation for further research and development.

The Recurrence and Transience of Random Walks

This article reviews the research history and application fields of random walks and abstracts random walks into a time-homogeneous Markov chain to study their recurrent and transient properties. For one-dimensional and two-dimensional random walks, the likelihood of returning in steps and the probability of returning for the first time in steps of each state are first introduced, along with their relationship. Then the Stirling's formula is given, which is utilized to estimate the probability of returning in n steps, and the convergence and divergence of infinite series is used to prove that random walk in one dimension is recurrent. Random walk in two dimension has similar properties, which is a bit more complicated by using the relation between polynomials. Higher dimensional random walks need to consider special states first, and then generalize them to other states. Finally, this paper concluded that one-dimensional and two-dimensional random walks are recurrent, and random walks with dimensions higher than two are transient.

Asymptotic comparison of negative multinomial and multivariate normal experiments

This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean–covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper bound on the Hellinger distance between the jittered negative multinomial distribution and the corresponding multivariate normal distribution. Upper bounds on the Le Cam distance between negative multinomial and multivariate normal experiments ensue.

Plea for the use of the exact Stirling formula in statistical mechanics

In statistical mechanics, the generally called Stirling approximation is actually an approximation of Stirling’s formula. In this article, it is shown that the term that is dropped is in fact the one that takes fluctuations into account. The use of the Stirling’s exact formula forces us to reintroduce them into the already proposed solutions of well-know puzzles such as the extensivity paradox or the Gibbs’ paradox of joining two volumes of identical gas. This amendment clearly results in a gain in consistency and rigor of these solutions.

A Curious Identity Arising From Stirling's Formula and Saddle-Point Method on Two Different Contours

We prove the curious identity in the sense of formal power series:\[\int_{-\infty}^{\infty}[y^m]\exp\left(-\frac{t^2}2+\sum_{j\ge3}\frac{(it)^j}{j!}\, y^{j-2}\right)\mathrm{d} t= \int_{-\infty}^{\infty}[y^m]\exp\left(-\frac{t^2}2+\sum_{j\ge3}\frac{(it)^j}{j}\, y^{j-2}\right)\mathrm{d} t,\]for $m=0,1,\dots$, where $[y^m]f(y)$ denotes the coefficient of $y^m$ in the Taylor expansion of $f$, which arises from applying the saddle-point method to derive Stirling's formula. The generality of the same approach (saddle-point method over two different contours) is also examined, together with some applications to asymptotic enumeration.

Outlier‐robust zonotope set‐membership filter for discrete‐time nonlinear system

AbstractTransient sensor malfunction, improper replacement measures, and large non‐Gaussian noise can cause data outliers in real‐world scenarios. In this article, an outlier‐robust zonotope set‐membership filter is proposed for uncertain nonlinear systems susceptible to outliers. In order to linearize nonlinear systems, a linearization method based on Stirling's interpolation formula is proposed. Then, an outlier detector utilizing the linearization error bounds, the prediction domain, and the output feasible domain is constructed. Based on the linearized system, a tight strip construction method is given to reduce the conservatism of the estimated enclosure. Moreover, a prediction‐correction‐backward reconstruction filter structure is designed to improve estimation accuracy and reduce conservatism. The proposed structure fuses historical data to attenuate the impact of uncertainty on the system. The robustness, guarantee, and convergence of the proposed filter are investigated. Finally, the effectiveness of the proposed approach is evaluated by two simulation examples.

On the Bhargava factorial of polynomial maps

Bhargava introduced a generalization of the factorial function to extend classical results in integers to Dedekind rings and unify them. We study the Bhargava factorial of the images of polynomial maps from an analytic perspective. We first give the -adic closures of the images of polynomial maps, which is the key to compute -adic part of the Bhargava factorial. Then, as a special case, we give the Stirling formula for the image of quadratic polynomials with integer coefficients.

Determine the equilibrium distribution by avoiding utilizing the Stirling formula

The derivations of the most probable distribution are not self-consistent in most statistical physics textbooks since the Stirling’s approximation adopted in the derivations is not valid for systems with very few particles. Any improvement by taking more higher terms of the Stirling series is insufficient due to erroneous divergence for small number. In this paper, we derive the equilibrium distribution by extrapolating the factorial function by the gamma function so that the Stirling’s approximation or the Stirling formula is avoided. For the ideal Bose system in a one-dimensional harmonic trap, we find that the equilibrium distribution given by this paper coincide with the rigorous result perfectly even for very small system size.

Tweaking Ramanujan’s Approximation of n!

About 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirling's approximation of $n!$. He gave a good formula which is asymptotic to $n!$. Since then hundreds of papers have given alternative proofs of his result and improved upon it, including notably by Burside, Gosper, and Mortici. However, Srinivasa Ramanujan gave a remarkably better asymptotic formula. Hirschhorn and Villarino gave nice proof of Ramanujan's result and an error estimate for the approximation. In recent years there have been several improvements of Stirling's formula including by Nemes, Windschitl, and Chen. Here it is shown (i) how all these asymptotic results can be easily verified; (ii) how Hirschhorn and Villarino's argument allows tweaking of Ramanujan's result to give a better approximation; and (iii) that new asymptotic formulae can be obtained by further tweaking of Ramanujan's result. Tables are calculated displaying how good each of these approximations is for $n$ up to one million.

On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators

In this paper, we prove a local limit theorem for the ratio of the Poisson distribution to the Gaussian distribution with the same mean and variance, using only elementary methods (Taylor expansions and Stirling's formula). We then apply the result to derive an upper bound on the Le Cam distance between Poisson and Gaussian experiments, which gives a complete proof of the sketch provided in the unpublished set of lecture notes by Pollard [41], who uses a different approach. We also use the local limit theorem to derive the asymptotics of the variance for Bernstein c.d.f. and density estimators with Poisson weights on the positive half-line (also called Szasz estimators). The propagation of errors in the literature due to the incorrect estimate in Lemma 2 (iv) of Leblanc [32] is addressed in the Appendix.

About the Stirling formula

About the Stirling formula

Design of a robust central difference kalman filter in the presence of uncertainties and unknown measurement errors

In this study, a novel RCDKF11Robust Central Difference Kalman Filter. is proposed for nonlinear plants. The system under consideration is a comprehensive case where the norm bounded uncertainties are supposed in both the state and output equations. Since the filtering methodology has been suggested based on the Stirling formula, it requires no derivative or Jacobian matrix computations. Furthermore, the mean value of uncertainties has been eliminated in the development of the filter structure. Different types of uncertainties are incorporated in the standard formulation of CDKF and it is demonstrated that the upper bound of covariance matrices can be derived according to them. In contrast to the similar works, the stability of the proposed filtering strategy is proved by the Lyapunov theory and the upper bound of state estimation error covariance is found for all admissible uncertainties. To demonstrate the performance of the introduced estimation algorithm, it will be evaluated on the attitude determination system of a three-axis satellite including a star camera and gyro sensor. The simulation results of the proposed filter are compared with the conventional CDKF,22Central Difference Kalman Filter. NRF,33Nonlinear Robust Filter. EKF44Extended Kalman Filter. and PF55Particle Filter. and the better efficiency of the developed method is established.

Comparative study between exponential Boltzmann and Lambert Boltzmann distributions for heat capacity calculation

The Stirling’s estimation to [Formula: see text](N!) is typically introduced to students as a step in the derivation of the statistical expression for the heat capacity. However, naïve application of this estimation leads to wrong conclusions. In this paper, firstly, the heat capacity of some semiconductor compounds was calculated using exponential Boltzmann distribution and compared with experimental data. It has shown a disagreement between experimental results and those calculated. Secondly, by applying the more exact Stirling formula, an analytical formulation of Boltzmann statistics using Lambert W function is shown to be a very good model and proves an excellent agreement between calculated and experimental data for heat capacity over the entire temperature range.

An Elementary Proof of Bevan's Theorem on the Growth of Grid Classes of Permutations

AbstractBevan established that the growth rate of a monotone grid class of permutations is equal to the square of the spectral radius of a related bipartite graph. We give an elementary and self-contained proof of a generalization of this result using only Stirling's formula, the method of Lagrange multipliers, and the singular value decomposition of matrices. Our proof relies on showing that the maximum over the space of n × n matrices with non-negative entries summing to one of a certain function of those entries, parametrized by the entries of another matrix Γ of non-negative real numbers, is equal to the square of the largest singular value of Γ and that the maximizing point can be expressed as a Hadamard product of Γ with the tensor product of singular vectors for its greatest singular value.

Gamma and Factorial in the Monthly

Since its inception in 1894, the Monthly has printed 50 articles on the Γ function or Stirling's asymptotic formula, including the magisterial 1959 paper by Phillip J. Davis, which won the 1963 Chauvenet prize, and the eye-opening 2000 paper by the Fields medalist Manjul Bhargava. In this article, we look back and comment on what has been said, and why, and try to guess what will be said about the Γ function in future Monthly issues.11 It is a safe bet that there will be more proofs of Stirling's formula, for instance. We also identify some gaps, which surprised us: phase plots, Riemann surfaces, and the functional inverse of Γ make their first appearance in the Monthly here. We also give a new elementary treatment of the asymptotics of n! and the first few terms of a new asymptotic formula for invΓ.

One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means

Available proofs of result of the type 'at least one of the odd zeta values $\zeta(5),\zeta(7),\dots,\zeta(s)$ is irrational' make use of the saddle-point method or of linear independence criteria, or both. These two remarkable techniques are however counted as highly non-elementary, therefore leaving the partial irrationality result inaccessible to general mathematics audience in all its glory. Here we modify the original construction of linear forms in odd zeta values to produce, for the first time, an elementary proof of such a result - a proof whose technical ingredients are limited to the prime number theorem and Stirling's approximation formula for the factorial.

Polychronakos statistics and α-deformed Bose condensation of α-bosons

In this paper, we consider the Polychronakos statistics for [Formula: see text]. We use the Stirling formula for the [Formula: see text]-Gamma function to find the distribution function for the [Formula: see text]-bosons. As application, we discuss the [Formula: see text]-deformed Bose condensation for [Formula: see text]-boson gas.

Size effects on the open probability of two-state ion channel system in cell membranes using microcanonical formalism based on gamma function

Ion channel systems are a class of proteins that reside in the membranes of all biological cells and forms conduction pores that regulate the transport of ions into and out of cells. They can be investigated theoretically in the microcanonical formalism since the number of accessible states can be easily evaluated by using the Stirling approximation to deal with factorials. In this work, we have used gamma function (Γ (n)) to solve the two-state or open-close channel model without any approximation. New values are calculated for the open probability (p0) and the relative error between our numerical results and the approximate one using Stirling formula is presented. This error (p0app — p0)/p0 is significant for small channel systems.