Skip to main content

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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
23 views

Scaling equation for the external field H in an Ising like system [closed]

i want to show that the following relation is true for the external field H, starting from the scaling form of the free energy. It is an Ising like System close to a critical point with $M \geq 0$ and ...
Dorek's user avatar
  • 43
0 votes
0 answers
22 views

Scaling Magnetic Fields Properties

Do magnetic fields of magnets of different mass differ in scale rationally or do the fields change in size and shape exponentially. In other words, would a microscopic magnet magnetic field look the ...
Justintimeforfun's user avatar
1 vote
0 answers
55 views

Using the RG equations to find the free energy scaling form of the 2D Ising Model

i am trying to calculate the scaling form of the free energy of the 2D Ising model, starting from it's RG equations: $$\frac{d u_I}{dl} = 2 u_I + u_t^2$$ $$\frac{d u_t}{dl} = u_t$$ $$\frac{d u_h}{dl} =...
Dorek's user avatar
  • 43
5 votes
1 answer
124 views

Why do we rescale momenta after integrating out high momenta in Wilsonian renormalization?

In Section 12.1 of Peskin & Schroeder they motivate Wilson's approach to renormalization by asking how a quantum field theory changes after changing the momentum scale. To answer this they start ...
CBBAM's user avatar
  • 3,350
0 votes
0 answers
15 views

Change of scaling due to a perturbation

I am looking for known examples of models where the introduction of a perturbation changes the scaling law of one or more observables. I would appreciate suggestions relevant to any branch of Physics, ...
AndreaPaco's user avatar
  • 1,232
1 vote
1 answer
80 views

Is there a reason why $\det(g)$ is a scalar density of weight 2 and not 1, 3 or 4?

$\det(g)$ is not coordinate-independent - it is a scalar density of weight +2 (or −2 depending on convention) which generally changes across spacetime. In Minkowski space equipped with spherical polar ...
MartyMcFly's user avatar
0 votes
0 answers
17 views

Connectivity of random geometric graph with open boundary conditions

I have a question regarding the existence of a closed-form solution of the connectivity in terms of the radius of vertices (disks) in a two-dimensional ($d=2$) random geometric graph (RGG) with open ...
Johannes Nauta's user avatar
0 votes
1 answer
302 views

Could a human-sized flea really jump over the Eiffel Tower?

I am puzzled why so many people think such a large flea could jump as high as the Eiffel Tower. We know that a 3mm long flea can jump 150mm, which is 50 times its own length. So I suppose it's ...
Brian F's user avatar
  • 151
1 vote
1 answer
101 views

Landau/Lifshitz "Mechanics" Mechanical Similarity

In Landau/Lifshitz "Mechanics", 3e, subsection 10, a question asks to "find the ratio of the times in the same path for particles having the same mass but potential energies differing ...
CW279's user avatar
  • 349
2 votes
0 answers
109 views

Is there a relatively quick way to obtain the scale factor on the metric as a response to special conformal transformation on spacetime?

For a special conformal transformation on spacetime positions $$x'^\mu=\frac{x^\mu-x^2b^\mu}{1-2b\cdot x+b^2x^2}$$ I was able to derive the scale factor $$\eta'_{\mu\nu}=\Omega(x)\eta_{\mu\nu}$$by ...
Rescy_'s user avatar
  • 838
0 votes
1 answer
68 views

How does safe fall distance scale with the size of an animal?

An animal has a maximum safe fall distance, the distance it can fall without getting hurt. If you take an animal and a similar animal twice as large, what would we expect to hold about the maximum ...
causative's user avatar
  • 910
1 vote
0 answers
15 views

Time scale of earthquake simulations on scaled building models

The effect of earthquakes on a building can be investigated by creating a scaled down model of the building, and simulate an earthquake on a shake table. A recording of a natural earthquake can be ...
Volker Siegel's user avatar
0 votes
1 answer
98 views

A Problem in Scaling

Question: - If the size of the nucleus (in the range of $10^{-15} m$ to $10^{-11} m $) is scaled up to the tip of a sharp pin, what roughly is the size of an atom? (Assume the tip of the pin to be in ...
AshCAD's user avatar
  • 31
2 votes
0 answers
97 views

Dependence of phase transition on scale factors

In a numerical experiment, I have obtained a phase diagram of the system under study. The phase diagram is obtained between two scaled quantities say, $P^{\prime}$ and $Q^{\prime}$ of the system. I ...
bubucodex's user avatar
  • 233
0 votes
1 answer
319 views

Normalized units versus dimensionless units

In a molecular dynamics code, suppose, the distances are expressed in units of a characteristic length of the simulated system, $R_0$. In some papers it is written as, " distances are normalized ...
bubucodex's user avatar
  • 233
1 vote
0 answers
134 views

Does this DEFINE a CFT?

I have a discrete correlated system defined on a squared grid in $d=2$, all euclidean. I have a random field at each point given by a local function of Grassmann variables (I wouldn't say fermions ...
almostsurely's user avatar
1 vote
1 answer
171 views

Scaling dimension and inversions

Defining an inversion transformation in coordinates as $$ x^\mu\rightarrow \mathcal{I}x^\mu = \frac{x^\mu}{x^2}, \tag1 $$ if we want to study these transformations on tensor operators $\mathcal{O}$ we ...
Vicky's user avatar
  • 1,597
1 vote
1 answer
278 views

Intuitive interpretation of the scaling dimension of an operator?

I am reading Field Theories of Condensed Matter Physics by Fradkin and in equation (4.10) it shows that an operator transforms irreducibly under scalings as $$\phi_n(xb^{-1}) = b^{\Delta_n}\phi_x(x)$$ ...
physics_fan_123's user avatar
9 votes
1 answer
596 views

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 ...
Solidification's user avatar
0 votes
0 answers
37 views

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(\...
user avatar
2 votes
0 answers
126 views

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 ...
2 votes
1 answer
61 views

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 ...
user avatar
1 vote
1 answer
79 views

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. ...
Aden's user avatar
  • 13
0 votes
1 answer
83 views

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 ...
Clara Díaz Sanchez's user avatar
0 votes
0 answers
366 views

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 ...
Pxx's user avatar
  • 1,723
4 votes
0 answers
97 views

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 ...
Faber Bosch's user avatar
2 votes
1 answer
444 views

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. ...
Andy's user avatar
  • 23
0 votes
1 answer
66 views

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 ...
sqeeezy's user avatar
1 vote
0 answers
126 views

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 ...
CatsOnAir's user avatar
  • 139
9 votes
2 answers
365 views

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 ...
SRM's user avatar
  • 196
0 votes
0 answers
365 views

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, ...
Adam's user avatar
  • 11.9k
3 votes
1 answer
142 views

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 ...
aitfel's user avatar
  • 3,043
8 votes
0 answers
268 views

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 ...
SaMaSo's user avatar
  • 498
2 votes
0 answers
50 views

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 ...
Iris Allevi's user avatar
1 vote
0 answers
191 views

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 ...
Alex Gower's user avatar
  • 2,604
0 votes
3 answers
79 views

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 $y$ ...
Faheem Aalijah's user avatar
2 votes
0 answers
33 views

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 ...
Xelote's user avatar
  • 127
3 votes
0 answers
134 views

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 ...
Yuan Yao's user avatar
  • 813
1 vote
0 answers
73 views

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}...
Jacopo.R's user avatar
2 votes
1 answer
86 views

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 ...
releseabe's user avatar
  • 2,238
5 votes
0 answers
109 views

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 ...
user2723984's user avatar
  • 4,736
3 votes
2 answers
290 views

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 ...
user2723984's user avatar
  • 4,736
5 votes
1 answer
327 views

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 ...
user2723984's user avatar
  • 4,736
0 votes
1 answer
68 views

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 $$\...
Dr. user44690's user avatar
29 votes
6 answers
6k views

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 $\...
Bhavay's user avatar
  • 1,691
6 votes
1 answer
229 views

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 ...
M111's user avatar
  • 372
3 votes
1 answer
212 views

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 ...
user253491's user avatar
4 votes
1 answer
261 views

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 ...
Merlin Zhang's user avatar
  • 1,602
1 vote
2 answers
130 views

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}...
User1997's user avatar
2 votes
0 answers
488 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, ...
Tanatofobico's user avatar