Questions tagged [critical-phenomena]
The physics of critical phenomena is the physics of systems close to a critical point, like the critical temperature in a ferromagnetic transition or the critical point of a gas-liquid transition. Examples of critical phenomena include dynamical slowing down, divergence of correlation length and ergodicity breaking.
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First and second order phase transitions
Recently I've been puzzling over the definitions of first and second order phase transitions. The Wikipedia article starts by explaining that Ehrenfest's original definition was that a first-order ...
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Is my interpretation of the fixed points of the renormalization group correct?
I would like to know whether or not I understood the meaning of the fixed points of the RG concerning the phase diagram of a system. This is how I understand it:
Since a RG transformation leaves the ...
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What leads to the existence of critical temperature?
We know that $T_c$ is the temperature above which no amount of pressure could force a gas to liquefy.
But why is this? Somehow I don't buy the point that the gas molecules exert too much pressure back ...
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What is the definition of correlation length for the Ising model?
The correlation length $\xi$ is related to critical temperature $T_c$ as
$$
\xi\sim|T-T_{c}|{}^{-\nu},
$$
where $\nu$ is the critical exponent.
Is this the formal definition of correlation length? ...
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Are the first order phase transitions always associated with a latent heat?
Is the first order ferromagnetic transition below the critical temperature associated with latent heat?
For example, the transition of ferromagnetic configuration with all its spins aligned up to a ...
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Why do spin correlation functions in Ising Models decay exponentially below the critical temperature?
I'm trying to form a better understanding of the 2D Ising Model, in particular the behaviour of the correlation functions between spins of distance $r$.
I've found a number of explanatory texts that ...
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How to measure the spin-spin correlation in a Monte Carlo simulation of the Ising model?
I'm simulating the Ising Model in 2D up to 5D and I want to calculate the spin-spin correlation, correlation length, and critical exponent of the system. What is a good way to go about doing this?
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Critical temperature and lattice size with the Wolff algorithm for 2d Ising model
When I run my implementation of the Wolff algorithm on the square Ising model at the theoretical critical temperature I get subcritical behaviour. The lattice primarily just oscillates between mostly ...
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Scale invariance at phase transitions
The Wikipedia entry for scale invariance states
In statistical mechanics, scale invariance is a feature of phase
transitions. The key observation is that near a phase transition or
critical ...
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What are conditions for the existence of a critical value (for a phase transition)?
Can there only be a critical temperature if there is some natural unit for an observable in the model, i.e. if there is a natural scale for something? Otherwise I don't see how for a system there ...
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Van der Waals model for liquid gas phase transition : Understanding Maxwell construction
I have a question on the context of Maxwell construction, spinodal lines.
In this pdf https://www.uam.es/personal_pdi/ciencias/evelasco/master/tema_III.pdf they first compute the Van der Waals model ...
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What is the physical meaning of "correlation length"?
I am studying phase transitions right now and trying to understand the physical meaning of the concept correlation length. I saw the equations but I still couldn't quite wrap my head around the ...
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Examples of important known universality classes besides Ising
I am working with RG and have a pretty good idea of how it works. However I have noticed that even though the idea of universality class is very general and makes it possible to classify critical ...
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If boiling of water involves change in internal energy, then why does the temperature remain constant?
According to the first law of thermodynamics,
$$\Delta Q=\Delta W+\Delta U$$
Considering boiling of water to be an isothermal process, $\Delta U$ should be zero, but then my textbook says: "we ...
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Non-uniqueness of the Order Parameter and its Critical Exponent
In the theory of phase transitions, an order parameter is usually defined as some quantity which distinguishes the two phases of the system by being zero in one phase, and non-zero in the other (see e....
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Why are there large fluctuations at the critical point and why does Landau theory work despite such large fluctuations?
The question is about the critical point in a second-order phase transition: Why do fluctuations become so large at the critical point?
As I understand, Landau’s theory of phase transition is some ...
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Why correlation length diverges at critical point?
I want to ask about the behavior near critical point.
Let me take an example of ferromagnet.
At $T < T_c$, all spins are aligned to the same direction thus it is in the ordered state, scale ...
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Is Wilson-Fisher fixed point unique?
It is well-known, that in $\phi^4$ theory in 3d there is interaction fixed point:
$$
S_{\Lambda} = \int d^dx \left[\frac{1}{2}(\partial_i \phi)^2 + \frac{1}{2} \mu_0^2 \phi^2 + \Lambda^{d-4} \tilde{g}...
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Why are the eigenvalues of a linearized RG transformation real?
The RG transformation $R_\ell$ maps a set of coupling constants $[K]$ of a model Hamiltonian to a new set of coupling constants $[K']=R_\ell[K]$ of a coarse-grained model where the length scale is ...
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How close to the critical point is sufficient close for measuring critical exponents?
I am learning Monte Carlo and just manage to simulate a phase transition by computing the heat capacity or the susceptibility. I wish I can also compute critical exponents.To this purpose, I have read ...
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The relation between critical surface and the (renormalization) fixed point
In the book, I read some remarks about the criticality:
Iterations of the renormalization (group) map generate a sequence of points in the space of couplings, which we call a renormalization group ...
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Is the Landau free energy scale-invariant at the critical point?
My question is different but based on the same quote from Wikipedia as here. According to Wikipedia,
In statistical mechanics, scale invariance is a feature of phase transitions. The key ...
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Log-law entanglement with large central charge contradict bounds on the entanglement entropy
I am trying to learn more about entanglement entropy in large but finite-size systems at critical points. I am still relatively new to conformal field theory, so it is not unlikely I have ...
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Divergent Coulomb integrals in superfluid fluctuations
In Chapter 3 of Kardar's statistical physics of fields, in the context of lower critical dimension, he works out an example about superfluids where starting from the probablity of a particular ...
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Assumptions behind Ornstein-Zernike correlation function
Let $S(\mathbf q)$ be come correlation function in Fourier space ($\mathbf q$ = wavevector). In the study of condensed matter systems, I have often encountered the statements that a reasonable form ...
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Phase transition vs. critical phenomena
Just trying to get some clarity in terminology: is phase transitions synonymous with critical phenomena? At the first glance they mean the same thing, but I am not sure whether phase transitions ...
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Why is the upper critical dimension of the Ising model 4?
I have read in various sources, that the critical exponents of the Ising Model are identical to the meanfield ones for dimensions $d \geq 4$. In trying to understand this better I came across the ...
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Behavior in renormalization group flow that reaches critical point
First question. Does correlation length in renormalization group flow has to be infinite when it eventually reaches critical point?
Second question. Why does renormalization group flow keep partition ...
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Is Landau free energy really the free energy?
I recall reading in Negele&Orland's book that Landau free energy function is not really the free energy that one obtains from the partition function $$F = -\frac{1}{\beta}\log Z.$$ Indeed, the ...
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Critical exponents and scaling dimensions from RG theory
In most books (like Cardy's) relations between critical exponents and scaling dimensions are given, for example
$$ \alpha = 2-d/y_t, \;\;\nu = 1/y_t, \;\; \beta = \frac{d-y_h}{y_t}$$
and so on. Here $...
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Some Questions about the Critical Point [closed]
I'm currently trying to understand the physics of phase transitions and I'm having a hard time doing that. First of all, the discussions on the topic seem to be confusing and there is no methodical ...
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Nonzero spontaneous magnetization in two-dimensional Ising model
The two-dimensional Ising model with the nearest-neighbour interactions enjoys a $\mathbb{Z}_2$ symmetry under $S_i\to -S_i$; it displays sponatebous symmetry breaking at a finite temperature $T_C=2J[...
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Relating scaling and critical exponents in the Ising model
I'm reading the chapter about the renormalization group in Yeoman's book "Statistical mechanics of phase transitions" and I'm puzzled about how the author relates the scaling of the RG with ...
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Negative critical exponent $\alpha$ for Superfluid helium at lambda point
Due to the positive critical exponent of the transition in liquid helium I would expect there to be no peak at the transition $t=0$. Since the $t$ dependent part of the specific heat should go to 0 as ...
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Was Big Bang a phase transition? [closed]
Was Big Bang a phase transition (critical phenomenon)? If "yes", what is the order parameter and what determined the value of the order parameter chosen?
When talking about phase transitions ...
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Various questions on renormalization in lattice systems
Forgive the long, multi questioned-question. The setting of this question is inspired by this answer.
Consider some theory on a lattice, for example the 2D $0$-field Ising model
$$H=-K\sum_{\langle i,...
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Is combustion a phase transition?
Is combustion a phase transition?
Premise
If we take a chemical reaction
$$
A + B \leftrightarrow AB,
$$
we expect all the three chemicals, $A,B, AB$ to be present in the mixture, in the proportions ...
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What is the state of water at exactly 0°C?
Theoretically speaking, what is the state of water at bang on 0°C - not any lower or higher?
Any lower would make it a solid whereas any higher would make it a liquid. But what about bang on 0°C?
...
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How to interpret a null critical exponent?
In the 2D Ising model the value for $\alpha$ is $0$, but I fail to see how we can have this if the specific heat of the system actually has a divergence in the critical temperature. I've seen this ...
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Is there any zero-order phase transition in nature?
Theoretically, a finite jump in the free energy phase diagrams can naturally be called a zeroth-order phase transition according to the Ehrenfest classification. We always hear about the first- and ...
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Examples of phase transition nuclei whose dynamics impede their own growth?
I recently asked a question over on the Earth Science stack exchange about cumulus cloud formation from (roughly) point sources. These points can form around the same time across large areas, such as ...
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Is it sufficient to consider a small part of a system (without potential energy sources which can be released) to determine if it's chaotic?
The world around us abounds with chaotic systems: dripping taps (when a certain dripping rate is reached the dripping becomes irregular, which can be seen in this old but very entertaining video, ...
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How to obtain a nonzero order parameter for a symmetry-breaking quantum phase transition?
If $\hat{m_z}=\frac{1}{N}\sum_i \hat{\sigma^z_i}$ is an order parameter for finite quantum system (transverse Ising model, say), then it will never break the $\mathbb{Z}_2$ symmetry since $\langle\...
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