Questions tagged [scaling]

Questions involving the laws which are scale invariant, i.e. that apply to different scales equally. Also laws that involve exponential behavior, expressed in terms of certain magnitudes to the power of certain exponents.

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What does the exponentiated generator of scale transformation do when it acts on a function? [duplicate]

We know that $d/dx$ is the generator of translation in the sense that $$e^{ad/dx}f(x)=f(x+a)\tag{1}$$ which can be easily be proved from the Taylor series of $f(x+a)$. Studying the very basics of ...
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How to Understand the First Term in the Calabrese-Lefevre Distribution?

I am currently reading the following paper and I am trying to understand the first term in equation (6) (reproduced below): $$ P(\lambda) = \delta(\lambda_\text{max} - \lambda) + \frac{b \Theta(\...
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Critical exponents and scaling dimension

It is often stated that the scaling exponents, e.g. $\alpha$ and $\beta$, of the critical 2D Ising model are related to the scaling dimensions $\Delta_{\sigma}$ and $\Delta_{\epsilon}$ of the ...
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How to Prove A Claim made to Construct the Calabrese-Lefevre Distribution?

My question is a mathematical one based on this physics paper. Suppose that $\lambda_i $ is an eigenvalue of a reduced density matrix. Up to a normalization factor, the distribution of eigenvalues is ...
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When scaling down a universe, what is scaled down, and how to calculate gravity?

Assume I want to scale down Earth so that it fit into a 500x500 units (pixels) cartesian plane, at a scale of 1:1,000,000. Earth, with a radius of 6,371,000 meters, now has a radius of 6.371 units. ...
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Scaling of resistance in different dimensions

Reading a pedagogical article by Steve Girvin on the quantum Hall effect, I noticed a result that in principle I have known for a long time, but I had never actually noticed. In $\rm 1D$, resistance ...
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Threaded Ring Polymer - Scaling approach to Diffusion

A monodisperse melt of linear chains is mixed with ring polymers, when a linear chain threads through the opening of a ring, that rings movement becomes confined along the backbone of the chain. (A ...
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How can the scaling dimension of a scalar field be $1$ in a $1d$ CFT?

In a one-dimensional scalar field theory, the kinetic term of the action takes the following form: $$S_\text{kin} = \int_\mathbb{R} dt\, \frac{1}{2} \dot{\varphi}(t) \dot{\varphi}(t)\,, \tag{1}$$ with ...
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What kind of matter's energy density scales as the inverse of the scale factor

We know that radiation energy density scales as $a^{-4}$ with EoS parameter ($w=\frac{1}{3}$), matter as $a^{-3}$ with ($w=0$), curvature as $a^{-2}$ with ($w=-\frac{1}{3}$). Then which kind of matter ...
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2 votes
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In terms of compressive stress $F/A$, what is the cross-sectional area of a sphere?

Any physics textbook chapter on stress-strain curves will generally mention that stress is force acting upon an area, and when a shape is three-dimensional, that area is the cross-sectional area. ...
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Test tank physics

In a test tank scaled-down simulation of, for example, a ship stability problem, is it not incorrect to assume that water will behave in a scaled-down fashion with regard to wave period? Don't we need ...
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Mathematical Reasoning In Physics

Does this form of reasoning have a name. I often see it but am a little confused on how to read/understand it and wanted to look more into it but don't know what to call it Ex. Let's call the force ...
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How do water waves scale up as the size of a bathtub scales up?

If I fill a bathtub with water to the point that it is spilling out of the far end of the tub, the waves in the tub caused by the water coming into the basin stabilize at a given height -- roughly two ...
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Are infinitesimal dilatation transformations local?

In quantum field theories, a local transformation of a scalar field $\phi(x)$ is a transformation that involves the field and its derivatives at same point. See for instance Weinberg's QFT textbok, ...
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Scaling analysis of the free energy of a polymer

We can perform a scaling analysis for polymers by looking at it on two scales. A small scale on which the chain follows a random walk and one (larger) where we collect many monomers in blobs that are ...
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Scaling dimension/weight of $\partial^{\mu}$

Under a scale transformation $x^{\mu}\rightarrow x^{'\mu}=\lambda x^{\mu}$. The operator $\partial_{\mu}$ has the conformal weight $1$ as $\partial_{\mu}^{'}=\frac{1}{\lambda}\partial_{\mu}$. I'm ...
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Fisher exponent and fractal structure

In the context of critical phenomena, there are several critical exponents commonly used to characterize the singular behaviour at the point of phase transition. The Fisher exponent $\eta$ is defined ...
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What is the meaning of a non-canonical fixed point in the RG flow?

Consider, as an example, the Gaussian model: $$ S[\phi]=\int\mathrm{d}\vec x\left[\frac{\gamma}{2}\left(\nabla\phi\right)^2+\frac{\mu^2}{2}\phi^2\right]. $$ RG analysis can be performed exactly by ...
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Is this quantum mechanical proof of the virial theorem general?

I have seen the following proof for the virial theorem in QM using the variational method. It goes like this: Suppose an exact eigenstate of the system is $\psi(\vec{r})$ and consider a variational ...
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Equations homogeneous in $X$ and $y$

Arfken and Weber in their mathematical methods for physicists edition 6 on page 334 talk about equations homogeneous in $X$ and $y$ and tells they are homogeneous if the combined powers of $X$ and $y$ ...
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How can one compute an effective probability at the critical point of a first order phase transition

In an excerpt from Finite-Size Scaling by John Cardy I found the following development: At a first-order transition, the correlation length ξ remains finite, and the finite-size scaling properties in ...
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Scaling limit, renormalization group and low-energy effective field theories

Given a quantum lattice theory $T_0$, e.g., in one dimension defined on $L$ sites, I know that there is a scaling limit, which introduces a lattice scale $a_0$ and keeps the following two quantities ...
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Dynamic scaling laws derivation

In the book Critical Dynamics by Tauber the following scaling hypotheses are made for the static correlation function and for the characteristic frequency in Fourier space $$ C(\tau, q) = |q|^{-2+\eta}...
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2 votes
1 answer
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Could gravity be much stronger (or weaker) at the atomic scale?

If gravity is mediated by particles and you are at a scale where those particles are relatively much larger does that perhaps imply that gravity can't work exactly the same way at very small scales as ...
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Analytic change of free energy after renormalization

Suppose we have some model in statistical physics with Hamiltonian $H$ and partition function $$Z=\mathrm{Tr}\left(e^{-H}\right) $$ the free energy per site is defined as $$ f =\frac1N\log Z$$ A ...
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3 votes
2 answers
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Relating scaling and critical exponents in the Ising model

I'm reading the chapter about the renormalization group in Yeoman's book "Statistical mechanics of phase transitions" and I'm puzzled about how the author relates the scaling of the RG with ...
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5 votes
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Identifying the relevant directions in the Ising model renormalization

I'm reading the chapter about the renormalization group in Yeoman's book "Statistical mechanics of phase transitions" and I'm puzzled about how the author relates the scaling of the RG with ...
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1 answer
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Conformal primaries in momentum space

Consider the Fourier transform of a conformal primary $O$ $$\tilde{O}(k) = \int d^dx e^{ik\cdot x} O(x)$$ Now consider the transformation of the momenta $k \to \lambda k$, so that the above reads $$\...
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29 votes
6 answers
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Why do objects with big size break easily?

Why do objects with big size break easily? For example: if I drop a chalk of length $L$ from height $h$ then there is a greater probability that it might break, when compared it to a chalk of length $\...
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5 votes
1 answer
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One-point function in CFT on an infinite strip through scaling analysis

In Philippe Di Francesco's book on Conformal Field Theory in section 11.2.3 on the Infinite Strip, the one point function of a primary operator (with scaling dimension $\Delta$) is calculated by ...
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Does the action remain dimensionless after the renormalization?

After the renormalization procedure, fields will gain an anomalous dimension, $\gamma$, which means that their scaling dimension will be different from what we would guess from the dimensional ...
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4 votes
1 answer
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Picture of poor man's scaling for AFM/FM interaction in Kondo problem

Poor man's scaling in Kondo problem For the Kondo model: $$H=-t\sum_{i,j}c_i^\dagger c_j+JS\cdot \sigma(0)$$ which only including itinerant electrons with the band-width $ W \in[-D,D]$, and $S$ is the ...
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1 vote
2 answers
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Explain how scaling of the inverse square law breaks down at a stars surface

If the radiation pressure at distance $d>R$ from the center of an isotropic black body star is found to be $$P_{rad}=\large{\frac{4\sigma T^4}{3c}}\left[1-\left(1-\frac{R^2}{d^2}\right)^{\frac{3}{2}...
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2 votes
0 answers
209 views

Scale invariance of lagrangians and transformation properties of fields under dilations

Consider a field theory, and a rescaling transformation of the coordinates \begin{equation} T_\epsilon[\phi(x)]=\phi((1+\epsilon)x). \end{equation} From what I understand, one usually requires that, ...
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1 vote
1 answer
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If space expands, why does a liter of water stay a liter? [duplicate]

From observing the universe, we know that space expands. I see what that means on cosmic scales. But what does it mean on smaller scales? If I have a graduated beaker of 2 liter, and 1 liter of ...
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3 votes
1 answer
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Why traditional turbulence theory concerns so much about statistics such as correlations?

I have been wondering why the traditional turbulence theory, e.g., Kolmogorov's 1941 theory, concerns so much about things like two-point correlations, structure functions, their scalings, and so ...
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1 vote
1 answer
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Scaling transformations, definitions and all that's not mentioned

If we transform the massless scalar field Lagrangian $$\mathcal{L}=\frac{1}{2}(\partial_\mu\varphi)^2-\frac{\alpha}{4!}\varphi^4$$ with the simultaneous transformations $$x\mapsto x^\prime= \lambda x,\...
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2 votes
1 answer
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Anomalous dimension for 1D quantum Ising model

I am reading Chapter 10.2, Quantum Phase Transition--Subir Sachdev(P144), it said All previous scaling dimensions of the d = 1 Ising model coincided with their so-called engineering dimension; the ...
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4 votes
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Why/When would one study Renormalization Group flow of a system?

It is not that I am looking for a cheap way out of reading a book about RG flow, but I would like to know few key insights that RG flow study provides, backed with some specific examples. I know a ...
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1 vote
1 answer
347 views

Scale diagram confusion

I’m familiar with scale diagrams where three forces act upon an object and trigonometry is used to find the resultant force, but I’ve come across questions that use units of velocity instead of force. ...
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2 votes
1 answer
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Why does the universe manifest scale?

I'll try outline my question in clear terms, articulating specific aspects that are its primary motivators. I'm just beginning in my exploration of physics as a student, but a persistent question that ...
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1 vote
1 answer
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Scaling of nuclear and electromagnetic force

The Wikipedia article on alpha decay stated: The strength of the attractive nuclear force keeping a nucleus together is thus proportional to the number of nucleons, but the total disruptive ...
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1 vote
0 answers
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Apparent similarity paradox

Consider a (p)rototype consisting in an incompressible and newtonian fluid flowing in a pipe of diameter $D_P$, studied by similarity in a (m)odel of the same fluid in another pipe of diameter $D_m<...
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12 votes
1 answer
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Why is the ratio of two extensive quantities always intensive?

Is this something that we observe that always happens or is there some fundamental reason for two extensive quantities to give an intensive when divided?
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1 vote
4 answers
118 views

Do smaller aircraft have lower take-off speeds?

Assuming there are two aircraft, each of the same density and each the same shape, am I correct in understanding that the smaller aircraft would have a lower take-off speed? I have explained how I ...
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9 votes
4 answers
900 views

Rigorous Definitions of Intensive and Extensive Quantities in Classical Thermodynamics

Most undergraduate books on Thermodynamics offer intuitive definitions for intensive and extensive thermodynamic variables. Authors assert, for example, that the former is independent of the system's ...
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What is the magnitude of a super-nova on the Richter scale?

What is the richter scale of a super-nova? If one could measure compare it to standard earthquakes measured in logarithmic richter scale, what would be the value for a super-nova?
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663 views

What is finite size scaling theory?

The question is in the title. I read some articles about it but I could not understand it completely. What does it mean when we say scaling of any system? Can you please give a brief introduction ...
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1 vote
1 answer
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UV Preconditioning test [closed]

I wanna build a solar panel with a new material and I wanna to test the endurance of the material against UV light. I've found in internet a Standard for UV tests. It says it has to be irradiated with ...
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0 votes
3 answers
120 views

Scaling of force between cubes? [closed]

I found an interesting problem online which has been confusing me for quite a while. Basically, two solid cubes of side length $a$ touch each other along one of their faces, and I am to find how many ...
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