New answers tagged statistical-mechanics
0
votes
Are the physical laws scale-dependent?
Scale-Invariant
Classical physics laws should be scale-invariant. Scale-invariancy is tested with such mappings :
$$ \tag 1 \begin{align}x&\rightarrow \lambda x~,\\ \varphi &\rightarrow \...
- 14.9k
1
vote
Accepted
Understanding entropy and its connection to probability distributions
The difference comes from the distinction between microstates and macrostates. It is the distribution over the microstates that is uniform. However, macrostates encompass many microstates with varying ...
- 15.1k
0
votes
Complete ionization temperature of an atom
If we don't over interpret OP and take the question very literally, the desired answer is the temperature scale that is comparable to the energy required to remove the innermost electron:
$$
T \sim Z(...
- 438
1
vote
Complete ionization temperature of an atom
I think the question is slightly misposed: For hydrogen the Saha equation gives (Ionization energy $I_e$=13.6eV)
\begin{align}
\frac{n_\text{e} n_\text{p}}{n_\text{H}}=\frac{n^2_\text{Ion}}{n-n_\text{...
- 56
0
votes
Quantum Phase Transitions in the Standard Model
In the Standard Model, the chiral symmetry breaking process of QCD is a nice example of quantum phase transition, which is "quantum-fluctuation-induced" rather than "thermal-fluctuation-...
- 4,780
0
votes
Why do we integrate the momentum states from 0 to infinity to get the partition function?
Normally when we discuss the partition function, we are implicitly looking at the states of a "small" system in thermal contact with a "large" (effectively infinite) reservoir at a ...
- 51.6k
1
vote
Mean-field theory and coarse-grained modelling: Are these the same methodology?
The two are closely related, but refer to different aspects of the issue: Coarse-graining is a procedure by which the continuous field is introduced, whereas mean-field theory is description of a ...
- 65k
4
votes
Imposition of order parameter constraint into free energy functional
The constraint $\sum_i \nabla_i \phi_i = 0$ can be incorporated into the free energy functional using a Lagrange multiplier. Let us introduce a new field $\alpha(r)$ and modify the functional as ...
- 764
4
votes
How can we neglect fluctuations in the Ising model with mean field approximation?
In mean field approximation, "neglecting fluctuations" does not mean that their size is small compared with the average value of the corresponding average observable. It means that the ...
4
votes
How can we neglect fluctuations in the Ising model with mean field approximation?
You cannot ignore the fluctuations. That's why it's an approxmation.
Only when the system is such that every spin interacts equally with the every other spin --- no matter how far away it is --- and ...
- 56.5k
1
vote
What if gas molecules collide inelastically?
I know that inelastic collision (e=0) of gas molecules does not make sense...
They do.
..., but I wonder, if somehow hypothetically gas molecules made inelastic collision with the container in which ...
- 41.2k
2
votes
Taking derivative with respect to quantum canonical ensemble expectation value
IMHO the OP is mislead by a recourse to imaginary time... which is not warranted here. Indeed, let us assume that we know the exact eigenstates of $H$:
$$
H|n\rangle = E_n|n\rangle,
$$
(these coule be ...
- 65k
4
votes
Taking derivative with respect to quantum canonical ensemble expectation value
On a side note, you can avoid the Trotter formula by using instead the standard interaction picture (if you are already familiar with it from previous QM courses). I will write $\partial_{x_i} = \...
- 15.1k
0
votes
Why does the motion of a gas never stop?
The problem is that, with real gases, the internal area of the container is not a closed system, and the container walls themselves are neither fully stationary nor opaque.
What I mean by "not ...
- 2,833
2
votes
What if gas molecules collide inelastically?
In addition to the good answer of @Roger V. :
When you heat your gas to a temperature high enough, the electronic degrees of freedom start to participate in the energy distribution as well.
Electronic ...
- 8,249
3
votes
Taking derivative with respect to quantum canonical ensemble expectation value
$$\partial_{x_b} \exp(-\beta H/N)\approx -\exp(-\beta H/N) (\frac{\beta}{N})(\partial_{x_b} H) $$
My question is, what is the strict justification for the last step? How to prove that the error ...
- 23.3k
3
votes
What is the reason for power laws appearing at phase transitions?
One can get to these power laws through this sequence:
Continuous phase transition $\to$
physical observables, such as magnetic susceptibility, diverge $\to$
system is scale free $\to$
physical ...
- 12.7k
0
votes
Confusion while deriving kinetic-molecular theory of gases
Projections of the velocities of gas molecules follow Maxwell distribution (i.e., Gaussian/normal distribution):
$$
f(v_x)dv_x=\sqrt{\frac{m}{2\pi k_B T}}e^{-\frac{mv_x^2}{2k_BT}}dv_x,
$$
that is ...
- 65k
8
votes
What is the reason for power laws appearing at phase transitions?
My question is: what is the fundamental reason why power laws should occur at phase transition specifically? What assumptions on the system should we make to obtain this result?
IMHO this is answered ...
- 65k
5
votes
Accepted
Interpretation of Distribution Function in Boltzmann Equation
Since the two are not synonymous
Despite they being not synonymous, they're closely related. If we have a system with $N$ total particles and $f(\mathbf r,\mathbf p, t)$ is the density of particles ...
- 3,227
2
votes
Why does the motion of a gas never stop?
As the molecules of the gas collide with the container do they lose some energy (kinetic)? - as no collision is perfectly elastic in nature.
They can, but not because of elasticity of collision. ...
- 2,421
16
votes
What if gas molecules collide inelastically?
I know that inelastic collision(e=0) of gas molecules does not make sense, but I wonder, if somehow hypothetically gas molecules made inelastic collision with the container in which it is filled and ...
- 65k
2
votes
Why does the motion of a gas never stop?
Why does the motion of a gas never stop?
It does stop, eventually. Well almost. Over an infinite period of time, the motion will approach zero. Consider a closed vessel in space containing a gas ...
- 10.1k
2
votes
Why does the motion of a gas never stop?
As the molecules of the gas collide with the container do they lose
some energy (kinetic)? - as no collision is perfectly elastic in
nature.
In a microscopic system collisions are elastic. Energy is ...
1
vote
Does fixing particle number + only evolving to lower energy states make the simulated system isolated?
If a system is completely isolated, then it will neither gain nor lose energy, i.e., its energy is fixed. A system whose energy is exactly specified will take on states drawn from the microcanonical ...
- 51.6k
1
vote
Why does the motion of a gas never stop?
The real answer is that entropy always increases in a system by fundamental laws of thermodynamics. As entropy decreases within the container, it increases in the container itself and/or in the ...
- 21
0
votes
Why does the motion of a gas never stop?
Gas molecules collide not only with the container walls but also frequently with one another. The energy exchange during these collisions depends on the temperature of the walls. If the walls are ...
- 4,126
1
vote
Why does the partition function require an integral?
One way to look at it is the following: The sum is the correct representation for a regularized system with a countable Hilbert space (by e.g. working with periodic boundary conditions).
The ...
- 12.2k
-1
votes
Why does the motion of a gas never stop?
It actually stops. https://en.wikipedia.org/wiki/Heat_death_of_the_universe
But not anywhere close to our lifetimes.
Probability of emitting a photon from collisions at room temperature is so small, ...
- 232
3
votes
Why does the motion of a gas never stop?
Damping can add energy.
The more damping there is the more the kinetic (or potential) energy of, say, a swinging pendulum is brought toward the thermal equilibrium energy. The equilibrium energy is a ...
- 6,653
0
votes
Anti-Hermitian Hamiltonians in Quantum Mechanics and Open Systems
Question: Can the presence of the anti-Hermitian term in Eq. 2 and Eq. 4 be justified or intuitively explained?
It seems that you proposed a mathematical model and is asking for a physical motivation,...
- 3,227
1
vote
How was Boltzmann able to neglect the possibility of an infinite number of microstates for a given macrostate?
Probably Boltzmann knew calculus. I am not sure whether he used special functions or an equivalent limiting procedure, but one could count all the microstates with energy $E$ for an ideal gas as
$$
\...
- 65k
1
vote
Entropy equation in Pathria
By fundamental derivative definition,
$$ \tag 1 \lim_{\Delta x \to 0} \frac {\Delta f(x)}{\Delta x} = \frac {df(x)}{d x} $$
So when $\Delta x \approx 0$, then
$$ \tag 2 \frac {\Delta f(x)}{\Delta x} \...
- 14.9k
16
votes
Why does the motion of a gas never stop?
It might be worth pointing out that at molecular level, perfectly elastic collisions can happen. Take for example two protons. When they are stationary & at distance $r$ apart, their kinetic ...
- 22.1k
2
votes
A Particular Interaction Matrix in a Fully Connected Ising Model
When I first asked this question, I was honestly just hoping someone would swoop in with an example from condensed matter physics or statistical mechanics. Spoiler alert: nobody did. But the question ...
1
vote
Is there a specialized formula for molecular friction?
It is an interesting thought (+1), but no it is not a good approach.
Friction is complicated if you look at the details. However, it often turns out to have a simple approximate law, $f=\mu N$.
This ...
- 45.6k
75
votes
Accepted
Why does the motion of a gas never stop?
The molecules which make up the walls of the container are also undergoing random thermal motion.
If you put a hot gas into a cold container, the gas molecules will on average lose energy during their ...
rob♦
- 94.2k
4
votes
Deriving the Internal Energy of a gas using Work-Energy Theorem
I’m trying to derive the internal energy (IE) of a gas using a method similar to how the electric potential energy is derived for a system of two charges.
I assume that your intention is to treat a ...
- 22.5k
1
vote
Accepted
Can the simulation box's edge length differ from the periodic distance in an Lennard-Jones (LJ) Monte-Carlo (MC) simulation?
Any is pretty broad, ex. surface studies are necessarily non-periodic in one direction and thus, have no periodic distance. If you're simulating amorphous materials, then you have no periodic distance ...
- 177
2
votes
Accepted
How does a Boltzmann distribution curve look like?
What is known as Maxwell-Boltzmann distribution (which may be sometimes abbreviated as Boltzmann distribution) is the distribution of particle speeds in an ideal gas. One could also recast it as a ...
- 65k
0
votes
How does a Boltzmann distribution curve look like?
The Maxwell-Boltzmann distribution is a distribution for the molecular speeds or energies based on assumptions from the kinetic theory of gases. In terms of the energy, it is
$$ f(E) = 2\sqrt{\frac{E}{...
- 3,430
0
votes
What is the partition function of a classical harmonic oscillator?
The $h^{-3}$ is needed to get the correct units, but there is more to it. $h^{-3}$ is the elementary volume of phase space needed for one quantum state. This constant cannot be found by classical ...
- 51
0
votes
Accepted
Equilibrium in a reversible process
Here's maybe a different answer with some equations. I don't think your two statements are contradictory, because they are too (as you say) ``hand-wavey'' to be definitive. I think establishing a bit ...
- 650
2
votes
Accepted
Spin-1 Ising model
Your formula for the magnetization can be put under the form
$$m={\sqrt T\over Tc}\sqrt{T_c-T}$$
In the neighborhood of the critical temperature, you can approximate the prefactor $\sqrt T$ by $\sqrt{...
- 3,708
0
votes
Equilibrium in a reversible process
However, I also found information that says a system with internal
temperature variations is not in equilibrium. Are these two pieces
contradictory, or am I missing something obvious?
They are ...
- 77.9k
0
votes
Equilibrium in a reversible process
The two statements refer to very different situations and descriptions/approximations. A system in equilibrium is characterized by a temperature, pressure, and other parameters which do not vary - the ...
- 65k
0
votes
Does the canonical distribution assign probability to microstates or macrostates?
I wrote what I found convincing. Sorry this is handwritten.
2
votes
Does the canonical distribution assign probability to microstates or macrostates?
Microstate
Factor $e^{-E_i/(k_BT)}$ is the probability of a microstate $i$. The canonical distribution then gives probability of a microstate:
$$
p_i=Z^{-1}e^{-\frac{E_i}{k_BT}}, Z = \sum_ie^{-\frac{...
- 65k
5
votes
Does the canonical distribution assign probability to microstates or macrostates?
It is actually both.
$$\tag1P^{\text{sys}+\text{rsv}}_\text{micro}=\frac1\Omega\qquad\implies\qquad P^\text{sys}_\text{micro}\propto\mathrm e^{-E_i/k_BT}\qquad\implies\qquad P^\text{sys}_\text{macro}\...
- 11.1k
3
votes
Does the canonical distribution assign probability to microstates or macrostates?
You're confused about the definition of microstate and macrostate. The energy of the microstate is the energy of the whole system. What is meant by the term microstate is just that we're specifying ...
- 3,479
Top 50 recent answers are included