# 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 ...
1 vote
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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 ...
42 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 ...
21 views

### 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 ...
1 vote
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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 ...
38 views

### 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 ...
96 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 ...
132 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 ...
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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 ...
1 vote
127 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 ...
1 vote
109 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 ...
1 vote
170 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 $$\phi_n(xb^{-1}) = b^{\Delta_n}\phi_x(x)$$ ...
459 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 ...
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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 ...
223 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 ...
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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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### 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, ...
1 vote
58 views

### 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 ...
1 vote
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,\...