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.
145 questions
2
votes
1
answer
59
views
Scaling wavefunction $\psi(x)\to \psi(2x)$ [closed]
denote $\psi_1(x)=\psi_0(2x)$. Then normalization condition $1=\int^{\infty}_{-\infty}|\psi_0(2x)|\,dx$ should indicates the normalization $\frac{1}{\sqrt{2}}\psi_1(x)=\frac{1}{\sqrt{2}}\psi_0(2x)$; ...
- 21
0
votes
0
answers
32
views
Scaling transformation confusion [duplicate]
I'm reading the yellow book (CFT by Di Francesco) and I came across a bit of a confusing statement.
Consider a scaling transformation $x^\mu \rightarrow x'^\mu = \lambda x^\mu$. Then the metric ...
- 263
0
votes
0
answers
52
views
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:
$$
\langle \phi_1(x_1)\phi_2(x_2)\phi_3(x_3)\phi_4(x_4)\rangle=C_{1234}x_{12}^ax_{13}^...
0
votes
0
answers
33
views
Regarding physical phenomena related to operators of the form $\exp(x\partial_x)$ [duplicate]
I am studying exponential operators, for example, of the form $\exp(x \partial_x)$. Do these operators appear or model any physical phenomena? or are they just abstract entities?
- 223
1
vote
0
answers
29
views
Can students effectively model the temperature effects of passic cooling on a scale architectural model? [closed]
I am helping some students made models for a Science fair. They'll make 1:12 scale (one-scale) models of these in sculpey and wood:
Roman court-houses with water features.
A Persian wind catcher.
...
- 487
4
votes
2
answers
219
views
Why do scalars scale?
The diffeomorphism invariance of scalars is often written as:
$$ \phi'(x') = \phi(x).\tag{1}$$
However, while scaling transformation is a type of diffeomorphism, in many places (say Di Francesco, ...
1
vote
1
answer
52
views
Scaled Hamiltonian and sped up evolution
Suppose there is a time-dependent Hamiltonian and the Schrodinger equation is solved.
$$
i\hbar \partial_t U(t) = H(t) U(t)
$$
Now, how easy is it to solve a scaled version of the Hamiltonian (e.g., $...
- 915
1
vote
1
answer
62
views
How to prove that unit operators are the only operators with zero scaling dimension in an unitary CFT?
In David Simmons-Duffin's Phys 229 notes found on author's github here pg. 147 it is said that the free boson field $\phi$ in bosonic CFT has zero dimension but it is not the unit operator. So the ...
- 300
1
vote
0
answers
25
views
Scaling equation for the external field H in an Ising like system [closed]
i want to show that the following relation is true for the external field H, starting from the scaling form of the free energy. It is an Ising like System close to a critical point with $M \geq 0$ and ...
- 43
1
vote
0
answers
58
views
Using the RG equations to find the free energy scaling form of the 2D Ising Model
i am trying to calculate the scaling form of the free energy of the 2D Ising model, starting from it's RG equations:
$$\frac{d u_I}{dl} = 2 u_I + u_t^2$$
$$\frac{d u_t}{dl} = u_t$$
$$\frac{d u_h}{dl} =...
- 43
5
votes
1
answer
149
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 ...
- 3,888
1
vote
1
answer
96
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 ...
- 93
0
votes
1
answer
533
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 ...
- 151
1
vote
1
answer
148
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 ...
- 437
2
votes
0
answers
127
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 ...
- 862
0
votes
1
answer
87
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 ...
- 922
1
vote
0
answers
19
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 ...
- 3,966
0
votes
1
answer
127
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 ...
- 31
2
votes
0
answers
100
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 ...
- 233
0
votes
1
answer
413
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 ...
- 233
1
vote
0
answers
138
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 ...
- 63
1
vote
1
answer
224
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 ...
- 1,607
1
vote
1
answer
352
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
$$\phi_n(xb^{-1}) = b^{\Delta_n}\phi_x(x)$$
...
- 725
9
votes
1
answer
666
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 ...
- 12.3k
0
votes
0
answers
38
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):
$$
P(\lambda) = \delta(\lambda_\text{max} - \lambda) + \frac{b \Theta(\...
user261609
2
votes
0
answers
136
views
Critical exponents and scaling dimension
It is often stated that the scaling exponents, e.g. $\alpha$ and $\beta$, of the critical 2D Ising model are related to the scaling dimensions $\Delta_{\sigma}$ and $\Delta_{\epsilon}$ of the ...
2
votes
1
answer
63
views
How to Prove A Claim made to Construct the Calabrese-Lefevre Distribution?
My question is a mathematical one based on this physics paper. Suppose that $\lambda_i $ is an eigenvalue of a reduced density matrix. Up to a normalization factor, the distribution of eigenvalues is ...
user261609
1
vote
1
answer
85
views
When scaling down a universe, what is scaled down, and how to calculate gravity?
Assume I want to scale down Earth so that it fit into a 500x500 units (pixels) cartesian plane, at a scale of 1:1,000,000.
Earth, with a radius of 6,371,000 meters, now has a radius of 6.371 units. ...
- 13
0
votes
1
answer
92
views
Scaling of resistance in different dimensions
Reading a pedagogical article by Steve Girvin on the quantum Hall effect, I noticed a result that in principle I have known for a long time, but I had never actually noticed.
In $\rm 1D$, resistance ...
- 2,405
0
votes
0
answers
393
views
How can the scaling dimension of a scalar field be $1$ in a $1d$ CFT?
In a one-dimensional scalar field theory, the kinetic term of the action takes the following form:
$$S_\text{kin} = \int_\mathbb{R} dt\, \frac{1}{2} \dot{\varphi}(t) \dot{\varphi}(t)\,, \tag{1}$$
with ...
- 1,743
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 ...
- 702
2
votes
1
answer
505
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. ...
- 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 ...
- 1
1
vote
0
answers
128
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 ...
- 139
9
votes
2
answers
404
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 ...
- 196
0
votes
0
answers
405
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, ...
- 12.1k
3
votes
1
answer
150
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 ...
- 3,063
8
votes
0
answers
279
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 ...
- 518
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].
$$
RG analysis can be performed exactly by ...
- 83
1
vote
0
answers
216
views
Is this quantum mechanical proof of the virial theorem general?
I have seen the following proof for the virial theorem in QM using the variational method. It goes like this:
Suppose an exact eigenstate of the system is $\psi(\vec{r})$ and consider a variational ...
- 2,654
0
votes
3
answers
82
views
Equations homogeneous in $X$ and $y$
Arfken and Weber in their mathematical methods for physicists edition 6 on page 334 talk about equations homogeneous in $X$ and $y$ and tells they are homogeneous if the combined powers of $X$ and $y$ ...
2
votes
0
answers
33
views
How can one compute an effective probability at the critical point of a first order phase transition
In an excerpt from Finite-Size Scaling by John Cardy I found the following development:
At a first-order transition, the correlation length ξ remains finite, and the finite-size scaling properties in ...
- 127
3
votes
0
answers
137
views
Scaling limit, renormalization group and low-energy effective field theories
Given a quantum lattice theory $T_0$, e.g., in one dimension defined on $L$ sites, I know that there is a scaling limit, which introduces a lattice scale $a_0$ and keeps the following two quantities ...
- 823
1
vote
0
answers
79
views
Dynamic scaling laws derivation
In the book Critical Dynamics by Tauber the following scaling hypotheses are made for the static correlation function and for the characteristic frequency in Fourier space
$$
C(\tau, q) = |q|^{-2+\eta}...
- 53
2
votes
1
answer
91
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 ...
- 2,288
5
votes
0
answers
116
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 ...
- 4,776
3
votes
2
answers
330
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 ...
- 4,776
5
votes
1
answer
349
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 ...
- 4,776
0
votes
1
answer
70
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
$$\...
- 1,379
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 $\...
- 1,691