Fermi Method Research Articles

Solutions for Fermi Questions, October 2018: Question 1: Automobile air conditioning; Question 2: Falling leaves

Solutions for Fermi Questions, October 2018: Question 1: Automobile air conditioning; Question 2: Falling leaves

Revisiting the Thomas–Fermi equation: Accelerating rational Chebyshev series through coordinate transformations

We revisit the spectral solution of the Thomas–Fermi problem for neutral atoms, urr−(1/r)u3/2=0 on r∈[0,∞] with u(0)=1 and u(∞)=0 to illustrate some themes in solving differential equations when there are complications, and also to make improvements in our earlier treatment. By “complications” we mean features of the problem that either destroy the exponential accuracy of a standard Chebyshev series, or render the classic Chebyshev approach inapplicable. The Thomas–Fermi problem has four complications: (i) a semi-infinite domain r∈[0,∞] (ii) a square root singularity in u(r) at the origin (iii) a fractional power nonlinearity and (iv) asymptotic decay as r→∞ that includes negative powers of r with fractional exponents. Our earlier treatment determined the slope at the origin to twenty-five decimal places, but no fewer than 600 basis functions were required to approximate a univariate solution that is everywhere monotonic, and all of the earlier tricks failed to recover an exponential rate of convergence in the truncation of the spectral series N, but only a high order convergence in negative powers of N. Here, using the coordinate z≡r to neutralize the square root singularity as before, we show that accuracy and rate of convergence are significantly improved by solving for the original unknown u(r) instead of the modified unknown v(r)=u(r) used previously. Without a further change of coordinate, a rational Chebyshev basis TLn(z;L) yields twelve decimal place accuracy for the slope at the origin, ur(0), with 70 basis functions and twenty-four decimal places with a truncation N=100.True exponential accuracy can be restored by using an appropriate change of coordinate, z=G(Z), where G is some species of exponential. However, the various Chebyshev and Fourier series for the Thomas–Fermi function have “plural asymptotics”, that is, an∼aintermediate(n) for 1≪n≪nI but an∼afar(n) for n≫nI for some positive constant nI. The “far-asymptotics” as n→∞ are often of no practical significance. For this problem, the TL-with-sinh method, theoretically the best for huge n, is bedeviled with numerical ill-conditioning and its asymptotic superiority is realized only when the goal is at least forty decimal places of accuracy.This is absurd for engineering, but useful perhaps for benchmarking. To sixty decimal places, ur(0)=−1.588071022611375312718684509423950109452746621674825616765677. To nineteen digits after the decimal point, the constant in the Coulson–March asymptotic series is improved to F=13.2709738480269351535.

Solutions for Fermi Questions, May 2018: Question 1: Throwing out energy; Question 2: Blood pressure

Solutions for Fermi Questions, May 2018: Question 1: Throwing out energy; Question 2: Blood pressure

Solutions for Fermi Questions, April 2018: Question 1: Automobile air use; Question 2: Personal air use

Solutions for Fermi Questions, April 2018: Question 1: Automobile air use; Question 2: Personal air use

Solutions for Fermi Questions, March 2018: Question 1: Air pressure on waves; Question 2: Weight of toner

Solutions for Fermi Questions, March 2018: Question 1: Air pressure on waves; Question 2: Weight of toner

Fermi problem in disordered systems

We revisit the Fermi two-atoms problem in the framework of disordered systems. In our model we consider a two-qubits system linearly coupled with a quantum massless scalar field. We analyze the energy transfer between the qubits under different experimental perspectives. In addition, we assume that the coefficients of the Klein-Gordon equation are random functions of the spatial coordinates. The disordered medium is modeled by a centered, stationary and Gaussian process. We demonstrate that the classical notion of causality emerges only in the wave zone in the presence of random fluctuations of the light cone. Possible repercussions are discussed.

The Plus or Minus Game – Teaching Estimation, Precision, and Accuracy

A quick survey of physics textbooks shows that many (Knight, Young, and Serway for example) cover estimation, significant digits, precision versus accuracy, and uncertainty in the first chapter. Estimation “Fermi” questions are so useful that there has been a column dedicated to them in TPT (Larry Weinstein's “Fermi Questions.”) For several years the authors (a college physics professor, a retired algebra teacher, and a fifth-grade teacher) have been playing a game, primarily at home to challenge each other for fun, but also in the classroom as an educational tool. We call the game “The Plus or Minus Game.” The game combines estimation with the principle of precision and uncertainty in a competitive and fun way.

Fermi Problem and Superluminal Signals in Quantum Electrodynamics

Fermi Problem and Superluminal Signals in Quantum Electrodynamics

Aprendiendo a Enseñar Matemáticas a Partir de la Propia Experiencia

An experience of educational innovation in teaching and learning mathematics with primary student teachers is presented in this report. The obtained data point out the power of emulating Fermi problem solving activities as an alternative to lectures. Our aim is to promote reflection on the characteristics of primary students’ mathematical learning and classroom management.

Analysis beyond the Thomas-Fermi approximation of the density profiles of a miscible two-component Bose-Einstein condensate

We investigate a harmonically trapped two-component Bose--Einstein condensate within the miscible regime, close to its boundaries, for different ratios of effective intra- and inter-species interactions. We derive analytically a universal equation for the density around the different boundaries in one, two and three dimensions, for both the coexisting and spatially separated regimes. We also present a general procedure to solve the Thomas--Fermi approximation in all three spatial dimensionalities, reducing the complexity of the Thomas--Fermi problem for the spatially separated case in one and three dimensions to a single numerical inversion. Finally, we analytically determine the frontier between the two different regimes of the system.

A brief guide to modelling in secondary school: estimating big numbers

Fermi problems are problems which, due to their difficulty, can be satisfactorily solved by being broken down into smaller pieces that are solved separately. In this article, we present different sequences of activities involving Fermi problems that can be carried out in Secondary School classes. The aim of these activities is to discuss problem-solving strategies and to introduce modelling processes in compulsory education.

Question 1: “Up” — A Fermi question about buoyancy; Question 2: Traffic jams

Question 1: “Up” — A Fermi question about buoyancy; Question 2: Traffic jams

The fermi paradox is neither Fermi's nor a paradox.

The so-called Fermi paradox claims that if technological life existed anywhere else, we would see evidence of its visits to Earth--and since we do not, such life does not exist, or some special explanation is needed. Enrico Fermi, however, never published anything on this topic. On the one occasion he is known to have mentioned it, he asked "Where is everybody?"--apparently suggesting that we do not see extraterrestrials on Earth because interstellar travel may not be feasible, but not suggesting that intelligent extraterrestrial life does not exist or suggesting its absence is paradoxical. The claim "they are not here; therefore they do not exist" was first published by Michael Hart, claiming that interstellar travel and colonization of the Galaxy would be inevitable if intelligent extraterrestrial life existed, and taking its absence here as proof that it does not exist anywhere. The Fermi paradox appears to originate in Hart's argument, not Fermi's question. Clarifying the origin of these ideas is important, because the Fermi paradox is seen by some as an authoritative objection to searching for evidence of extraterrestrial intelligence--cited in the U.S. Congress as a reason for killing NASA's SETI program on one occasion. But evidence indicates that it misrepresents Fermi's views, misappropriates his authority, deprives the actual authors of credit, and is not a valid paradox.

Explicit Tracking of Uncertainty Increases the Power of Quantitative Rule-of-Thumb Reasoning in Cell Biology

Back-of-the-envelope or rule-of-thumb calculations involving rough estimates of quantities play a central scientific role in developing intuition about the structure and behavior of physical systems, for example in so-called Fermi problems in the physical sciences. Such calculations can be used to powerfully and quantitatively reason about biological systems, particularly at the interface between physics and biology. However, substantial uncertainties are often associated with values in cell biology, and performing calculations without taking this uncertainty into account may limit the extent to which results can be interpreted for a given problem. We present a means to facilitate such calculations where uncertainties are explicitly tracked through the line of reasoning, and introduce a probabilistic calculator called CALADIS, a free web tool, designed to perform this tracking. This approach allows users to perform more statistically robust calculations in cell biology despite having uncertain values, and to identify which quantities need to be measured more precisely to make confident statements, facilitating efficient experimental design. We illustrate the use of our tool for tracking uncertainty in several example biological calculations, showing that the results yield powerful and interpretable statistics on the quantities of interest. We also demonstrate that the outcomes of calculations may differ from point estimates when uncertainty is accurately tracked. An integral link between CALADIS and the BioNumbers repository of biological quantities further facilitates the straightforward location, selection, and use of a wealth of experimental data in cell biological calculations.

Quantum signaling in cavity QED

We consider quantum signaling between two-level quantum systems in a cavity in the perturbative regime of the earliest possible arrival times of the signal. We present two main results: First, we find that, perhaps surprisingly, the analog of amplitude modulated signaling (Alice using her energy eigenstates $\left|g\right\ensuremath{\rangle},\left|e\right\ensuremath{\rangle}$, as in the Fermi problem) is generally suboptimal for communication, namely, e.g., phase-modulated signaling (Alice using, e.g., ${\left|+\right\ensuremath{\rangle},\left|\ensuremath{-}\right\ensuremath{\rangle}}$ states) overcomes the quantum noise already at a lower order in perturbation theory. Second, we study the effect of mode truncations that are commonly used in cavity QED on the modeling of the communication between two-level atoms. We show that, on general grounds, namely for causality to be preserved, the UV cutoff must scale at least polynomially with the desired accuracy of the predictions.

Devising a plan to solve Fermi problems involving large numbers

Fermi problems are open, non-standard problems that require the students to make assumptions about the problem situation and to estimate relevant quantities before engaging in, often, simple calculations (Arleback The Montana Mathematics Enthusiast, 6(3):331–364, 2009). Our study focuses on the plans—schemes aimed at solving—devised by students aged 12–16 years to solve Fermi problems involving big numbers. The strategies which appear in these schemes are characterized herein and analyzed to find out to what extent they would be suitable to solve the problems. Some of the identified strategies are related to magnitude estimation (use of a reference point), whereas others correspond to modeling processes (grid distribution, use of a concentration measurement, reductions of a problem, and use of a proportion). We conclude that Fermi problems involving large numbers may constitute a good opportunity to discuss heuristics and problem solving strategies in compulsory secondary education classes.

Problemas de estimación de magnitudes no alcanzables: una propuesta de aula a partir de los modelos generados por los alumnos

In this paper we present a proposal to introducing modeling in Secondary School classrooms. The proposal is based on the use of a set of Fermi problems related to estimating large quantities. It uses the results of a previous study which describes the strategies proposed by the students to solve this kind of problems.

Rational Chebyshev series for the Thomas–Fermi function: Endpoint singularities and spectral methods

We solve the Thomas–Fermi problem for neutral atoms, uyy−(1/y)u3/2=0 on y∈[0,∞] with u(0)=1 and u(∞)=0, using rational Chebyshev functions TLn(y;L) to illustrate some themes in solving differential equations on a semi-infinite interval. L is a user-choosable numerical parameter. The Thomas–Fermi equation is singular at the origin, giving a TL convergence rate of only fourth order, but this can be removed by the change of variables, z=y with v(z)=u(y(z)). The function v(z) decays as z→∞ with a term in z−3, which is consistent with a geometric rate of convergence. However, the asymptotic series has additional terms with irrational fractional powers beginning with z−4.544. In spite of the faster spatial decay, the irrational powers degrade the convergence rate to slightly larger than tenth order. This vividly illustrates the subtle connection between the spatial decay of u(x) and the decay-with-degree of its rational Chebyshev series. The TL coefficients an(L) are hostages to a tug-of-war between a singularity on the negative real axis, which gives a geometric rate of convergence that slows with increasingL, and the slow inverse power decay for large z, which gives quasi-tenth order convergence with a proportionality constant that decreasesinversely as a power of L. For L=2, we can approximate uy(0) (=vzz(0)) to 1 part in a million with a truncation N of only 20. L=64 and N=600 gives uy(0)=−1.5880710226113753127186845, correct to 25 decimal places.

Dynamics of atom–atom correlations in the Fermi problem

We present a detailed perturbative study of the dynamics of several types of atom–atom correlations in the famous Fermi problem. This is an archetypal model to study micro-causality in the quantum domain, where two atoms, one initially excited and the other prepared in its ground state, interact with the vacuum electromagnetic field. The excitation can be transferred to the second atom via a flying photon, and various kinds of quantum correlations between the two are generated during this process. Among these, prominent examples are given by entanglement, quantum discord and non-local correlations. The aim of this paper is to analyze the role of the light cone in the emergence of such correlations.

Fermi Problem with Artificial Atoms in Circuit QED

We propose a feasible experimental test of a 1D version of the Fermi problem using superconducting qubits. We give an explicit nonperturbative proof of strict causality in this model, showing that the probability of excitation of a two-level artificial atom with a dipolar coupling to a quantum field is completely independent of the other qubit until signals from it may arrive. We explain why this is in perfect agreement with the existence of nonlocal correlations and previous results which were used to claim apparent causality problems for Fermi's two-atom system.