arXiv subjects

Ronald F. Fox

Publications and source records attributed to Ronald F. Fox.

13 recordsLinked to original sources

arXiv2022

Exploring the Jarzynski Equality for the Harmonic Oscillator

Almost 25 years ago, Jarzynski published a paper in which it was asserted: the work done, W, in driving a system from state A to state B, characterized by the Helmholtz free energies FA and FB, satisfies an equality in which an average over an ensemble of measurements for W determines the difference in Free energy. Several features of this result require more detailed description, to be given in the text. The equality is significant and unexpected. So is the statement that the equality is independent of the rate of change from state A to state B. A few years ago, I had presented three papers in which the contraction of the description from full phase space to coordinate space only was made. This was motivated by the large difference in time scales for momenta relaxation and coordinate relaxation. The Jarzynski equality (JE) will be shown here to be correct only in the limit of slow processes. The proposed example is for a macromolecular harmonic oscillator with a Hooke's spring constant, k, that during the time interval linearly changes from k1 to k2. When performed slowly, we obtain JE in a form in which the free energy is related to the ratio of the k's. However, when the transition is performed more rapidly the equality is lost. Jarzynski has suggested to me that Hummer and Szabo have shown that JE follows directly from the Feynman-Kac formula. We show here that F-K does not reproduce the direct calculation executed in this paper and that it has been misapplied in earlier papers.

physics.gen-ph↗

arXiv2022

A Note on Over- and Under-Representation Among Populations with Normally-Distributed Traits

In every finite mixture of different normal distributions, there will always be exactly one of those distributions that not only is over-represented in the right tail of the mixture, but even completely overwhelms all other subpopulations in the rightmost tails. This property, although not unique to normal distributions, is not shared by other common continuous centrally-symmetric unimodal distributions such as Laplace, nor even by other bell-shaped distributions such as Cauchy (Lorentz) distributions.

math.PR↗

arXiv2021

Jarzynski Equality Counterexample?

Almost 25 years ago, Jarzynski published a paper in which it was asserted: the work done, W, in driving a system from state A to state B, characterized by the Helmholtz free energies FA and FB, satisfies an equality in which an average over an ensemble of measurements for W determines the difference in Free energy. Several features of this result require more detailed description, to be given in the text. The equality is significant and unexpected. So is the statement that the equality is independent of the rate of change from state A to state B. A few years ago, I had presented three papers in which the contraction of the description from full phase space to coordinate space only was made. This was motivated by the large difference in time scales for momenta relaxation and coordinate relaxation. The Jarzynski equality (JE) will be shown here to be correct only in the limit of extremely slow processes. The proposed counterexample is for a macromolecular harmonic oscillator with a Hooke's spring constant, k, that during the time interval linearly changes from k1 to k2. When performed very slowly, we obtain JE in a form in which the free energy is related to the ratio of the k's. However, when the transition is performed more rapidly the equality is lost. Jarzynski has suggested to me that Hummer and Szabo have shown that JE follows directly from the Feynman-Kac formula. We show here that F-K does not reproduce the direct calculation executed in this paper and that it has been misapplied in earlier papers.

physics.gen-ph↗

arXiv2020

A Conjecture Regarding the Riemann Hypothesis as Visualized by t strings

The introduction of strings into the study of the Riemann Hypothesis provides a visualization of the genesis of zeros for the Zeta function. The method is heuristic and when originally introduced suggested strong visual evidence for the truth of the Riemann Hypothesis. Some sort of organizing principle for strings with similar t values is observed and points towards a region between (1, 0) and (0, 0) on the abscissa, and within order unity along the ordinate. Progress in understanding these observations has been made by expanding the domain of sigma from the critical strip, [0, 1], to the half-line [0, infinity]. The nature of the organizing principle is explained. A generic structure for strings over the expanded domain is proffered. New perspective is gained regarding the truth of the Riemann Hypothesis.

math.GM↗

arXiv2018

Critique of the Fox-Lu model for Hodgkin-Huxley fluctuations in neuron ion channels

Using a well known result that every FP equation has an antecedent Langevin equation LE Fox and Lu proposed such a description for ion channels in 1994. Their contraction followed the works of van Kampen and of T. Kurtz. The contraction produces a diffusion term with a state dependent diffusion matrix, D, that arises from the coupling matrix, S, in the LE. This S connected the noise terms to the channel subunit variables in the LE. Fox and Lu and many others later on observed that SS = D. Since D was determined by the contraction of the MC equations into the FP equation, this left the problem of determining the square root matrix, S, for every time step of the simulation. Since this is time consuming, Fox and Lu introduced simplified models not requiring the square root of a matrix. Subsequently, numerous studies were published that showed the several shortcomings of these simplified models. In 2011, Goldwyn et al. [6] rediscovered the overlooked original matrix dependent approach in the Fox-Lu 1994 paper. They showed that it produced results in very good agreement with the MC results. In 1991, Fox and Keizer [7] wrote a paper on an unrelated topic that utilized the work of van Kampen and of Kurtz. In that work the connection between D and S is SST = D. ST is the adjoint (transpose) of S. D remains a positive definite symmetric matrix but S need not be. Fox has reproduced the 2012 results of Orio and Soudry for potassium channels and has also found in closed form the solution for the more complicated sodium channels. The square root problem generally must be done numerically, but the Cholesky is always doable in closed form. Thereby the S matrix for sodium is given explicitly for the first time in this paper

q-bio.NC↗

arXiv2018

Irreducible Characters for the Symmetric Groups and Kostka Matrices

In an earlier paper [1] it was shown that the Frobenius compound characters for the symmetric groups are related to the irreducible characters by a linear relation that involves a unitriagular coupling matrix that gives the Frobenius characters in terms of linear combinations of the irreducible characters. It is desirable to invert this relationship since we have formulas for the Frobenius characters and want the values for the irreducible characters. This inversion is straightforward and yields both the irreducible characters but also the coupling matrix that turns out to be the Kostka matrix in the original direction. We show that if the Frobenius monomial identity is applied a modification of it, equation (22), produces a monomial formula that produces the Kostka matrix inverse without involving the characters of either type. However it is a formidable task to execute this procedure for symmetric groups of even modest order. Alternatively the inversion by means of the unitriangular coupling matrix produces the Kostka matrix and the irreducible characters simultaneously and with much less effort than required for the monomial approach. Moreover there is a surprise.

math.RT↗

arXiv2015

Statistical Thermodynamic Foundation for the Origin and Evolution of Life

In this paper we review and extend our earlier recent work on thermostated systems. A description of nano-biological systems by Markov chains in coordinate space in the strongly overdamped limit is presented. Characterization of the most probable path is given and a new formula for the probability of this special path is provided from recursion formulae. The deterministic limit is derived and the significance of Lagrange multipliers introduced when constructing the most probable path is elucidated. The characterization of the generation of path entropy by the most probable path is given an equivalent interpretation relating to the rate of entropy production by the most probable path. The paper concludes with an account of the biological implications. Here we address why the origin of life and its subsequent evolution took place, not the particular chemical details of how it happened.

q-bio.SC↗

arXiv2015

Hubble's Law Implies Benford's Law for Distances to Galaxies

A recent article by Alexopoulos and Leontsinis presented empirical evidence that the first digits of the distances to galaxies are a reasonably good fit to the probabilities predicted by Benford's law, the well known logarithmic statistical distribution of significant digits. The purpose of the present article is to give a theoretical explanation, based on Hubble's law and mathematical properties of Benford's law, why galaxy distances might be expected to follow Benford's law. The new galaxy-distance law derived here, which is robust with respect to change of scale and base, to additive and multiplicative computational or observational errors, and to variability of the Hubble constant in both time and space, predicts that conformity to Benford's law will improve as more data on distances to galaxies becomes available. Conversely, with the logical derivation of this law presented here, the recent empirical observations may be viewed as independent evidence of the validity of Hubble's law.

physics.data-an↗

arXiv2015

Contributions to the Theory of Thermostated Systems II: Least Dissipation of Helmholtz Free Energy in Nano-Biology

In this paper, we develop further the theory of thermostated systems along the lines of our earlier paper. Two results are highlighted: 1) in the Markov limit of the contracted description, a least dissipation of Helmholtz free energy principle is established; and 2) a detailed account of the appropriateness of this principle for nano-biology, including the evolution of life, is presented.

q-bio.SC↗

arXiv2014

Contributions to the Theory of Thermostated Systems

In this paper, a theory for systems in contact with a thermal reservoir is developed. We call such systems "thermostated systems." Interest in these systems has been vigorous for about the last 20 years and is illustrated by the Crooks theorem and the Jarzinski equality, two results for the description of systems in full phase space where all coordinates and momenta may be followed through time. These fundamental results follow from the behavior of Markovian systems in phase space for which path integral expressions can be derived from the Smoluchowski equation valid for Markovian systems.

cond-mat.stat-mech↗

arXiv2010

A Better Definition of the Kilogram

This article reviews several recent proposed redefinitions of the kilogram, and compares them with respect to practical realizations, uncertainties (estimated standard deviations), and educational aspects.

physics.data-an↗

arXiv2007

A Better Definition of the Kilogram

Fixing the value of Avogadro's constant, the number of atoms in 12 grams of carbon-12, at exactly 84446886^3 would imply that one gram is the mass of exactly 18x14074481^3 carbon-12 atoms. This new definition of the gram, and thereby also the kilogram, is precise, elegant and unchanging in time, unlike the current 118-year-old artifact kilogram in Paris and the proposed experimental definitions of the kilogram using man-made silicon spheres or the watt balance apparatus.

physics.gen-ph↗

arXiv2007

A Proposed Exact Integer Value for Avogadro's Number

An exact value for Avodagro's number, namely NA* = (84446888)^3, is proposed. The number 84446888 represents the side length of a cube of atoms whose volume is closest, among all integral side lengths, to the current official NIST value of Avogadro's number. This value NA* is nearly dead center of the estimated range for the value of Avogadro's number, and is within the official standard level of uncertainty. Adoption of this value as the fixed value for NA would eliminate the current time-dependent definition of Avogadro's number, which depends on the definition of kilogram via an unstable physical artifact. It would also eliminate the need for the kilogram artifact altogether, since then, by definition, a kilogram would be exactly 1000/12 the mass of NA* atoms of carbon-12.

physics.gen-ph↗