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.
37 questions with no upvoted or accepted answers
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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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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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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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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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0
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137
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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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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 ...
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0
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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 ...
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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 ...
2
votes
0
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50
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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 ...
- 83
2
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0
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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 ...
- 127
2
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0
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513
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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, ...
- 325
2
votes
0
answers
42
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In what sense can a "manned model" boat handle identically to a much larger boat?
A YouTube video(relevant part is 1:24 - 1:46) suggests that if you take an oil tanker, scale it down 25 times, and give it an engine with 0.4 horsepower, "they behave exactly like a [larger] ship".
...
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2
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Scale-covariant decomposition of capacitance
I'm wondering if there is any good insight of how to evaluate a given capacitive geometry in such a way that it would be expressed as a function that depends only on two components:
as a geometric ...
- 14.7k
2
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Scaling in Vlasov equations
This question is in reference to the paper,
http://arxiv.org/abs/1301.7182
What exactly is the argument being made on page 6 and 7?
One deduces that the function $\Delta$ has to be such that, $\...
- 4,709
2
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1
answer
562
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How to scale variables in a classical Hamiltonian?
So I looked at some research articles where one has a classical Hamiltonian $H(p,q,t) = p^{2}/2 + V(q,t)$. If one introduces the scaling transformation
$$t \mapsto t/\sqrt{s}, \quad H \mapsto Hs, \...
- 331
1
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0
answers
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Using the RG equations to find the free energy scaling form of the 2D Ising Model
i am trying to calculate the scaling form of the free energy of the 2D Ising model, starting from it's RG equations:
$$\frac{d u_I}{dl} = 2 u_I + u_t^2$$
$$\frac{d u_t}{dl} = u_t$$
$$\frac{d u_h}{dl} =...
- 43
1
vote
0
answers
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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 ...
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1
vote
0
answers
138
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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 ...
- 63
1
vote
1
answer
224
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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 ...
- 1,607
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vote
0
answers
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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 ...
- 139
1
vote
0
answers
216
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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 ...
- 2,654
1
vote
0
answers
79
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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}...
- 53
1
vote
0
answers
47
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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<...
- 392
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vote
0
answers
138
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Restoring Scale Symmetry
To comprehend more about Scale Symmetry.. I need to know what it would take to restore Scale Symmetry that would make mass and length vanish. For example.. to restore Electroweak symmetry breaking.. ...
- 455
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vote
0
answers
316
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irrational conformal dimension
I know examples of Conformal Field Theories in which the scaling dimension of certain operators is an integer number or a fractional number.
However I do not know any example in which the scaling ...
- 2,241
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vote
0
answers
76
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Scaling dimension in Kerr/CFT
Let's consider the scattering of a scalar field around a Kerr-AdS Black Hole. In terms of the Kerr/CFT correspondence how can I get the relation between the scaling dimension and the mass of the field ...
- 153
1
vote
0
answers
124
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scaling laws for density and temperature of high-energy explosions
I'm wondering if there are heuristic ways to derive how the peak density and temperature of nuclear explosions scale with the amount of fissile/fusible material.
Does it matter what the explosion ...
- 14.7k
0
votes
0
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Four-point function in CFT, two constraints are missing
I am deriving the four-point functions, using translation and Lorentz invariance I start with the following form:
$$
\langle \phi_1(x_1)\phi_2(x_2)\phi_3(x_3)\phi_4(x_4)\rangle=C_{1234}x_{12}^ax_{13}^...
0
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1
answer
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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 ...
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0
answers
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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):
$$
P(\lambda) = \delta(\lambda_\text{max} - \lambda) + \frac{b \Theta(\...
user261609
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How can the scaling dimension of a scalar field be $1$ in a $1d$ CFT?
In a one-dimensional scalar field theory, the kinetic term of the action takes the following form:
$$S_\text{kin} = \int_\mathbb{R} dt\, \frac{1}{2} \dot{\varphi}(t) \dot{\varphi}(t)\,, \tag{1}$$
with ...
- 1,743
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1
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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 ...
- 1
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votes
0
answers
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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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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 ...
- 1,437
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1
answer
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Criticality and the number of paths on a lattice
In the review "Scaling, universality, and renormalization: Three pillars of modern critical phenomena" by Stanley, he makes the following claim towards the end of the paper, which is neither derived ...
- 292
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Effect on Q factor and resonant frequency when scaling down a damped oscillator to micro / nano scales
I've always just accepted that as you scale down a mechanical system the frequency and Q factor both increase. But how exactly do they scale? Linearly? With the square of reduction in size? Or maybe ...
- 311
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votes
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Ignore terms in the equation of motion
While working on a problem relative to particles motion, I ended up with the following second order nonlinear differential equation:
$$\tag{1} \ddot{y}(t)+\frac{y(t)}{b^2}\left(a-\frac{\dot{y}(t)^2}{2}...
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