arXiv subjects

Chris L. Lin

Publications and source records attributed to Chris L. Lin.

15 recordsLinked to original sources

arXiv2025

The Inverse Velocity Force and Automotive Physics

The inverse velocity force $F(v)=C/v$ is central to the analysis of automotive performance, but is often unmentioned in classical mechanics courses. It represents the maximum force that can be delivered from a power-limited engine over a wide range of speeds. We show how such a force arises from a device called a transmission in internal combustion engines, and more simply from circuit controls in electric vehicles. We also examine how the force needs to be modified and supplemented to describe vehicle acceleration throughout the entire range of speeds, along with the ensuing kinematics. Along the way we compare gasoline vehicles with electric ones and derive the shape of their torque-speed curves.

physics.ed-ph↗

arXiv2021

On the Normalization and Density of 1D Scattering States

The normalization of scattering states is more than a rote step necessary to calculate expectation values. This normalization actually contains important information regarding the density of the scattering spectrum (along with useful details on the bound states). For many applications, this information is more useful than the wavefunctions themselves. In this paper we show that this correspondence between scattering state normalization and the density of states is a consequence of the completeness relation, and we present formulas for calculating the density of states which are applicable to certain potentials. We then apply these formulas to the delta function potential and the square well. We then illustrate how the density of states can be used to calculate the partition function for a system of two particles with a point-like (delta potential) interaction.

quant-ph↗

arXiv2021

Simple Model of a Standing Vertical Jump

In this paper we use Newton's 3rd law to deduce the simplest model of an object that can perform a standing vertical jump -- a two-segmented object with an initial constant repulsive force between the segments, followed by an abrupt attractive force. Such an object, when placed on a sturdy ground, will jump, and the motion can be calculated using only the constant acceleration equations, making the example suitable for algebra-based physics. We then proceed to solve for the motion of an n-segmented object, and determine the optimal number of segments for jumping. We then discuss a few similarities and differences of this simple model from jumping robots and jumping humans, and then conclude by arguing the model's pedagogical merits.

physics.pop-ph↗

arXiv2021

Quantum anomaly and thermodynamics of one-dimensional fermions with antisymmetric two-body interactions

A system of two-species, one-dimensional fermions, with an attractive two-body interaction of the derivative-delta type, features a scale anomaly. In contrast to the well-known two-dimensional case with contact interactions, and its one-dimensional cousin with three-body interactions (studied recently by some of us and others), the present case displays dimensional transmutation featuring a power-law rather than a logarithmic behavior. We use both the Schrödinger equation and quantum field theory to study bound and scattering states, showing consistency between both approaches. We show that the expressions for the reflection $(R)$ and the transmission $(T)$ coefficients of the renormalized, anomalous derivative-delta potential are identical to those of the regular delta potential. The second-order virial coefficient is calculated analytically using the Beth-Uhlenbeck formula, and we make comments about the proper $ε_B\rightarrow 0$ (where $ε_B$ is the bound-state energy) limit. We show the impact of the quantum anomaly (which appears as the binding energy of the two-body problem, or equivalently as Tan's contact) on the equation of state and on other universal relations. Our emphasis throughout is on the conceptual and structural aspects of this problem.

cond-mat.quant-gas↗

arXiv2020

Chirality Through Classical Physics

Chirality, or handedness, is a topic that is common in biology and chemistry, yet is rarely discussed in physics courses. We provide a way of introducing the topic in classical physics, and demonstrate the merits of its inclusion - such as a simple way to visually introduce the concept of symmetries in physical law - along with giving some simple proofs using only basic matrix operations, thereby avoiding the full formalism of the three-dimensional point group.

physics.pop-ph↗

arXiv2019

Time Scale for Velocity to Track a Force

In this paper we derive and discuss the time it takes for a force to turn a velocity. More precisely, we derive the formula for the time $\tau$ it takes a constant force that makes an angle $\alpha$ with the initial velocity $\vec{v}(0)$ to have $\vec{v}(\tau)$ get within an angle $\theta<\alpha$ of the force. We then show how the addition of a viscous force decreases $\tau$ logarithmically. The result can be generalized to any vector quantity whose first time derivative is a constant.

physics.class-ph↗

arXiv2019

A Quantum Field-Theoretical Perspective on Scale Anomalies in 1D systems with Three-Body Interactions

We analyze, from a canonical quantum field theory perspective, the problem of one-dimensional particles with three-body attractive interactions, which was recently shown to exhibit a scale anomaly identical to that observed in two-dimensional systems with two-body interactions. We study in detail the properties of the scattering amplitude including both bound and scattering states, using cutoff and dimensional regularization, and clarify the connection between the scale anomaly derived from thermodynamics to the non-vanishing nonrelativistic trace of the energy-momentum tensor.

hep-th↗

arXiv2018

Virial expansion for the Tan contact and Beth-Uhlenbeck formula from 2D SO(2,1) anomalies

The relationship between 2D $SO(2,1)$ conformal anomalies in nonrelativistic systems and the virial expansion is explored using recently developed path-integral methods. In the process, the Beth-Uhlenbeck formula for the shift of the second virial coefficient $δb_2$ is obtained, as well as a virial expansion for the Tan contact. A possible extension of these techniques for higher orders in the virial expansion is discussed.

cond-mat.quant-gas↗

arXiv2018

Quantum anomaly and thermodynamics of one-dimensional fermions with three-body interactions

We show that a system of three species of one-dimensional fermions, with an attractive three-body contact interaction, features a scale anomaly directly related to the anomaly of two-dimensional fermions with two-body forces. We show, furthermore, that those two cases (and their multi species generalizations) are the only non-relativistic systems with contact interactions that display a scale anomaly. While the two-dimensional case is well-known and has been under study both experimentally and theoretically for years, the one-dimensional case presented here has remained unexplored. For the latter, we calculate the impact of the anomaly on the equation of state, which appears through the generalization of Tan's contact for three-body forces, and determine the pressure at finite temperature. In addition, we show that the third-order virial coefficient is proportional to the second-order coefficient of the two-dimensional two-body case.

cond-mat.quant-gas↗

arXiv2017

Dilational Symmetry-Breaking in Thermodynamics

Using thermodynamic relations and dimensional analysis we derive a general formula for the thermodynamical trace $2\mathcal{E}-DP$ for non-relativistic systems and $\mathcal{E-DP}$ for relativistic systems, where $D$ is the number of spatial dimensions, in terms of the microscopic scales of the system within the grand canonical ensemble. We demonstrate the formula for several cases, including anomalous systems which develop scales through dimensional transmutation. Using this relation, we make explicit the connection between dimensional analysis and the virial theorem. This paper is focused mainly on the non-relativistic aspects of this relation.

hep-th↗

arXiv2016

Bose and Fermi Statistics and the Regularization of the Nonrelativistic Jacobian for the Scale Anomaly

We regulate in Euclidean space the Jacobian under scale transformations for two-dimensional nonrelativistic fermions and bosons interacting via contact interactions and compare the resulting scaling anomalies. For fermions, Grassmannian integration inverts the Jacobian: however, this effect is cancelled by the regularization procedure and a result similar to that of bosons is attained. We show the independence of the result with respect to the regulating function, and show the robustness of our methods by comparing the procedure with an effective potential method using both cutoff and $ζ$-function regularization.

hep-th↗

arXiv2016

Relationship between Fujikawa's Method and the Background Field Method for the Scale Anomaly

We show the equivalence between Fujikawa's method for calculating the scale anomaly and the diagrammatic approach to calculating the effective potential via the background field method, for an $O(N)$ symmetric scalar field theory. Fujikawa's method leads to a sum of terms, each one superficially in one-to-one correspondence with a vacuum diagram of the 1-loop expansion. From the viewpoint of the classical action, the anomaly results in a breakdown of the Ward identities due to a scale-dependence of the couplings, whereas in terms of the effective action, the anomaly is the result of the breakdown of Noether's theorem due to explicit symmetry breaking terms of the effective potential.

hep-th↗

arXiv2015

Path-Integral Approach to Scale Anomaly at Finite Temperature

We derive the relativistic thermodynamic scale equation using imaginary-time path integrals, with complex scalar field theory taken as a concrete example. We use Fujikawa's method to derive the scaling anomaly for this system using a matrix regulator. We make a general scaling argument to show how for anomalous systems, the $β$ function of the vacuum theory can be derived from measurement of macroscopic thermodynamic parameters.

hep-th↗

arXiv2015

Virial Theorem for Non-relativistic Quantum Fields in D Spatial Dimensions

The virial theorem for non-relativistic complex fields in $D$ spatial dimensions and with arbitrary many-body potential is derived, using path-integral methods and scaling arguments recently developed to analyze quantum anomalies in low-dimensional systems. The potential appearance of a Jacobian $J$ due to a change of variables in the path-integral expression for the partition function of the system is pointed out, although in order to make contact with the literature most of the analysis deals with the $J=1$ case. The virial theorem is recast into a form that displays the effect of microscopic scales on the thermodynamics of the system. From the point of view of this paper the case usually considered, $J=1$, is not natural, and the generalization to the case $J\neq 1$ is briefly presented.

hep-th↗

arXiv2015

Path-Integral Derivation of the Non-relativistic Scale Anomaly

In this paper we calculate the scale anomaly for a quantum field theoretic 2D-nonrelativistic Bose gas with contact interactions using Fujikawa's method, both in vacuum and in many-body systems. The use of path integrals for these problems is novel and motivated by a recently developed path-integral framework for addressing questions about scaling in these systems. A natural class of regulators is found that produces the correct value of the anomaly traditionally calculated via other methods, e.g., diagrammatically via the beta function.

hep-th↗