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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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 ...
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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 ...
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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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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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The scaling form of the free energy density
I came across ''the finite size scaling'' while reading on the scaling hypothesis in statistical mechanics. I was wondering how can one derive a scaling function for the free energy which is dependent ...
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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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Renormalisation flow and conformal symmetry
In mathematics, a meta-principle is that when a group of symmetries $G$ acts on Euclidean space $\mathbf{R}^n$ and fixes the origin $0$, then if $Y\to \mathbf{R}^n$ is any structure sitting over ...
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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 ...
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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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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 ...
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Finding the parameter regime over which a phase transition is observable
Suppose, two variables $P$ and $Q$ follow a relation like,
$P=AQ^n$, where $A$ is a constant.
If this relation describes the phase diagram of a system obtained numerically, how can I determine the ...
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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 ...
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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 ...
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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
$$\phi_n(xb^{-1}) = b^{\Delta_n}\phi_x(x)$$
...
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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):
$$
P(\lambda) = \delta(\lambda_\text{max} - \lambda) + \frac{b \Theta(\...
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Critical exponents and scaling dimension
It is often stated that the scaling exponents, e.g. $\alpha$ and $\beta$, of the critical 2D Ising model are related to the scaling dimensions $\Delta_{\sigma}$ and $\Delta_{\epsilon}$ of the ...
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How to Prove A Claim made to Construct the Calabrese-Lefevre Distribution?
My question is a mathematical one based on this physics paper. Suppose that $\lambda_i $ is an eigenvalue of a reduced density matrix. Up to a normalization factor, the distribution of eigenvalues is ...
user261609
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When scaling down a universe, what is scaled down, and how to calculate gravity?
Assume I want to scale down Earth so that it fit into a 500x500 units (pixels) cartesian plane, at a scale of 1:1,000,000.
Earth, with a radius of 6,371,000 meters, now has a radius of 6.371 units. ...
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Scaling of resistance in different dimensions
Reading a pedagogical article by Steve Girvin on the quantum Hall effect, I noticed a result that in principle I have known for a long time, but I had never actually noticed.
In $\rm 1D$, resistance ...
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How can the scaling dimension of a scalar field be $1$ in a $1d$ CFT?
In a one-dimensional scalar field theory, the kinetic term of the action takes the following form:
$$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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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 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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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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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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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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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
$$\...
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Why do objects with big size break easily?
Why do objects with big size break easily? For example: if I drop a chalk of length $L$ from height $h$ then there is a greater probability that it might break, when compared it to a chalk of length $\...
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One-point function in CFT on an infinite strip through scaling analysis
In Philippe Di Francesco's book on Conformal Field Theory in section 11.2.3 on the Infinite Strip, the one point function of a primary operator (with scaling dimension $\Delta$) is calculated by ...
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Does the action remain dimensionless after the renormalization?
After the renormalization procedure, fields will gain an anomalous dimension, $\gamma$, which means that their scaling dimension will be different from what we would guess from the dimensional ...
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Picture of poor man's scaling for AFM/FM interaction in Kondo problem
Poor man's scaling in Kondo problem
For the Kondo model:
$$H=-t\sum_{i,j}c_i^\dagger c_j+JS\cdot \sigma(0)$$
which only including itinerant electrons with the band-width $ W \in[-D,D]$, and $S$ is the ...
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Explain how scaling of the inverse square law breaks down at a stars surface
If the radiation pressure at distance $d>R$ from the center of an isotropic black body star is found to be $$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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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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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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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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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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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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