13
votes
Accepted
Why do we disorder-average before/after taking the logarithm of the partition function for annealed/quenched disorder?
To fix the idea, let's consider a spin glass Hamiltonian $H(\sigma,J)$, where $\sigma$ are the spins and $J$ is a random variable with distribution $p(J)$ representing the couplings.
An example is ...
- 16.5k
10
votes
Accepted
Why is a symmetric traceless tensor zero when averaged over all directions?
The order parameter is a meso- or macroscopic quantity which is formed by adding together the tensor contributions from a bunch of different molecules: if each molecule $\alpha$ has a tensor $T^{(\...
- 135k
7
votes
Why is a symmetric traceless tensor zero when averaged over all directions?
OK, another proof according to the Remark by Emilio that made clear the interpretation of the question. We have to average the tensor (as a bilinear map) in the space of the rotations. The only way to ...
- 78.1k
7
votes
Scaling theory of Anderson localization
But how is this quantity related to the original problem of Anderson? How is it related to the localization/delocalization of the eigenstates ?
Let say we have a finite disorder system of size $L$ ...
- 2,598
7
votes
What is the intuition behind the statement that non-equilibrium systems with static disorder are self-averaging?
I've seen this claim echoed in other references, e.g. in Jordan Rammer's book Quantum Field Theory of Non-Equilibrium States pg. 455 below equation 12.33, however I haven't found a satisfying ...
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Why is a symmetric traceless tensor zero when averaged over all directions?
Symmetry requirement is not necessary. Let us take an orthonormal basis $e_1,e_2,e_3$ and consider all the unit vectors $n \in S^2$.
$$\int_{S^2} T(n,n) d n = \sum_{i,j=1}^3T(e_i,e_j) \int_{S^2} n^i n^...
- 78.1k
5
votes
What is the intuition behind the statement that non-equilibrium systems with static disorder are self-averaging?
I think the response by user kapaw is largely correct; I will try to simply expand on it in relation to the formalism.
Indeed, the replica trick can be circumvented provided we ensure that the ...
- 164
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votes
Why do we disorder-average before/after taking the logarithm of the partition function for annealed/quenched disorder?
A few additional thoughts prompted by valerio's great answer:
The annealed and quenched disorder cases should be thought of as being described by two completely different Hamiltonians with different ...
- 49.4k
5
votes
What does the second law of thermodynamics really mean?
The second law of thermodynamics states, that the total entropy of an isolated system never decreases with time. There are a couple of details in this sentence, which are worth pointing out:
You can ...
- 9,304
4
votes
Scaling theory of Anderson localization
As you have noticed, the literature on Anderson localization uses several different definitions of localization, including (but not limited to!):
A transition from extended to localized eigenstates. ...
- 7,968
4
votes
Quenched systems - disorder average (SYK model)
If we consider the replica trick:
$$
\overline{\log Z}= \lim_{n \rightarrow 0} \frac{\overline{Z^n}-1}{n}
$$
Then self-averaging is equivalent to equation $\overline{Z^n}=\overline{Z}^n$.
In general,...
- 1,012
4
votes
Are complexity and disorder correlated in entropy?
Here is another take which might be helpful.
Entropy has at least two different manifestations: one in the realm of thermodynamics and another in the realm of information theory (where it is called ...
- 97.8k
3
votes
Accepted
References for prerequisite material for understanding papers on Generalized Global Symmetries
The paper
Symmetries in Quantum Field Theory and Quantum Gravity by Harlow and Ooguri (https://arxiv.org/abs/1810.05338)
is a gold mine of insights about symmetries in general. Prerequisites for ...
3
votes
What is the intuitive meaning of the typical value $e^{\left\langle \log X \right\rangle}$ of a random variable $X$?
Without any knowledge in disordered systems, an intuitive answer is that an average of the type $\exp \langle\log X\rangle$, tries to capture the expected order of magntiude of $X$, which is ...
3
votes
What are disorders in condensed matter parlance?
I'm no expert but this is what I understand. A disorder is something which generically breaks some invariance of the Hamiltonian or lead to some deviation from the lattice periodicity. When some ...
- 27.2k
3
votes
Accepted
Fractal structure in colloidal systems
When diffusion is the primary transport mechanism, fractal patterns may arise as the result of diffusion-limited aggregation or more specifically diffusion limited cluster aggregation (DLCA): where ...
- 12.7k
2
votes
Accepted
Are maximum order and maximum disorder equally easy to describe?
Only if your perspective is coarse-grained in some fashion. A full description of the gas you mention, for example, classically must include the position and momentum of every gas particle, while the ...
- 22.8k
2
votes
Accepted
Do hydroelectric power plants violate the popular scientific notion of entropy = disorder?
So, entropy additively measures our uncertainty about the microscopic state of a system given whatever we can macroscopically measure about it. You can kind of call that "disorder" if you want, but ...
- 39k
2
votes
What is(are) the effect(s) of disorder on electrical conductivity?
Short answer is the resistivity increases until it transitions to insulator.
A band description makes things more clear. Before the transition, the electronic states are pertubatively connected to ...
- 1,444
2
votes
Do the random-bond Ising model correlation functions decay with the disorder strength?
This is not a direct answer, but it may be useful.
Define, for any subset of sites $A$
$$ \sigma_A \equiv \prod_{i\in A} \sigma_i
$$
For systems with only ferromagnetic couplings (ferromagnetic is ...
- 1,058
2
votes
Are complexity and disorder correlated in entropy?
Of the 3 concepts you mention, the one to drop off the discussion is "disorder". While there are cases where it can be useful, it more often than not is simply a shorthand for much clearer ...
- 12.7k
1
vote
Accepted
Find correlation function $\langle R^2(t_1) R^2(t_2)\rangle$ for 2d stochastic dynamics of polymer
I think that you probably already know everything necessary to obtain a solution. The form of the equations for $R_\alpha$ and the fact that $\sigma$ is $\delta$-correlated imply that the random ...
- 6,258
1
vote
Question about molecules and their movement
I'm currently learning about basic thermodynamics and was thinking, if
there is some "average" or median movement pattern or velocity which a
system of molecules reach after some time has ...
- 77.9k
1
vote
Question about molecules and their movement
If you are talking about a gas, the molecules are bouncing around at random, off of each other and the walls of the container (unless “contained” by gravity). They quickly fill any available volume ...
- 12.2k
1
vote
Weak localization, strong localization, and localization without a metal-insulator transition
I can help to explain the physical meaning of weak localization.
First, there are several characteristic lengthes need to be clarified:
The sample size $L$;
The mean free path $\ell_{e}$. It is the ...
- 11
1
vote
Level statistics of many body localization
I'm not too familiar with the exact details, but $\langle r\rangle \gtrsim 0.53$ is completely fine and expected in certain situations. In fact, $\langle r\rangle \simeq 0.60$ for Hamiltonians that ...
- 11
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