New answers tagged statistical-mechanics
1
vote
Why should we use time independent states to derive the ideal gas law?
The basis being chosen is the basis of eigenstates of the Hamiltonian operator. One reason this basis is useful in statistical mechanics because often a crucial question in statistical mechanics is to ...
- 52.3k
1
vote
Algebraic formulation of statistical mechanics
It's a bit more complicated. Pure states are represented by vectors in a Hilbert space representing the algebra of observables by linear maps.
A pure state is a linear form, mapping each element of ...
- 21
2
votes
Does Poincare recurrence show that Gibbs entropy is not strictly increasing?
Intuitively, one starts with a macro state and a corresponding micro state, eventually that micro state is arbitrarily close the original one, so the entropy should be get closer to the original value,...
- 61.9k
3
votes
Can diffusion create a vacuum?
I have actually done this very experiment, starting with a sealed, permeable vessel (Teflon) filled with helium at 1 atm surrounded by air at 1 atm. What happens is over several days the pressure ...
- 10.2k
0
votes
Can diffusion create a vacuum?
So after enough years, I’ll have a vacuum that I could use to do work.
Yes, if the container is permeable only to helium, then after a long time, the state will be very low pressure helium inside, ...
- 39.7k
16
votes
Accepted
Does Poincare recurrence show that Gibbs entropy is not strictly increasing?
Your perplexity and observations about Huang's statement are well-founded. Indeed, Huang acritically repeats Zermelo's and Poincaré's arguments against Boltzmann's ideas.
One flaw in using the ...
0
votes
Why do the entropy approach and force approach lead to the same result for the Boltzmann distribution?
The expression for the force balance that you wrote down, is strong enough to get to Boltzmann distribution. I mean, why should it have that form?
If you really think harder about the problem, you ...
- 10.2k
1
vote
Why do conserved quantities vanish when integrated against the collision integral?
I'm not completely sure I got your concern.
There might be two questions:
Why is this integral equal to 0?
It is 0 by definition of a collisional invariant. If we define:
$$\langle A\rangle(r, t)\...
- 1,508
4
votes
Accepted
Radial distribution of ideal gas in a cylinder
Let's make it simpler. You have particle on a 2d plane with an Hamiltonian that depends on the distance to some center $r$ and also to the angle with respect to this center $\theta$. You can imagine ...
- 1,508
0
votes
Does entropy outside of thermodynamics also increase?
There exist several quantities called entropy, which are related, but not quite the same (e.g., Jaynes distinguishes at least 6 different entropies: see Is information entropy the same as ...
- 61.9k
1
vote
Does entropy outside of thermodynamics also increase?
You're trying to answer, the rate of change in entropy. Therefore, it is convenient to consider Stochastic processes, with time-dependent probability densities.
Now, for Markov processes, which have ...
- 3,111
8
votes
Accepted
Spin-Spin Correlation Function
Since
$$
M^2
=
\biggl( \sum_{i} s_i \biggr)^2
=
\biggl( \sum_{i} s_i \biggr)\biggl( \sum_{j} s_j \biggr)
=
\sum_{i,j} s_i s_j,
$$
and
$$
\langle M\rangle^2
=
\biggl( \sum_{i} \langle s_i\rangle \biggr)...
- 10.6k
2
votes
Why do systems with greater energy fluctuation have better heat dissipation ability?
Your interpretation is on the right track. Larger fluctuations contribute to better dissipation for several reasons:
Why do larger fluctuations in equilibrium systems contribute to better dissipation ...
- 654
1
vote
Why is entropy's definition useful?
One thing entropy is NOT is a measure of energy dispersal. This erroneous idea is dangerously attractive because it is easy to understand and is sometimes true, but it leads to mistakes and should ...
- 3,168
0
votes
How can events on the quantum-level be random but not on the macro-level?
How can events on the quantum-level be random but not on the
macro-level?
One aspect to the answer to this question is emergent properties. For example we cannot assign a temperature to a single atom,...
- 7,606
0
votes
How can events on the quantum-level be random but not on the macro-level?
I know that even if an atom has a very high chance of undergoing decay (say, more than 0.99999999), when taken to the power of the number of atoms that are in the chair, the probability of the chair's ...
- 9,278
1
vote
How can events on the quantum-level be random but not on the macro-level?
Central Limit Theorem and the thermodynamic limit provide that if a variable $X$ has a certain distribution, as you take the mean of $N$ instances of the variable (energy, momentum, etc), as $N$ gets ...
- 10.2k
4
votes
How can events on the quantum-level be random but not on the macro-level?
All models are wrong; some are useful
"How science is explained" is really a topic for the math & science educators stack exchange. But if you want to look at what scientists really ...
- 50.8k
0
votes
How can events on the quantum-level be random but not on the macro-level?
A useful way to think about stuff like this is as follows:
As we "zoom out" from the quantum world to the macro world we inhabit, the individual screwy histories of quantum particles get ...
- 96.7k
8
votes
Accepted
Pressure of a gas on the inside walls of a cylinder canonical ensemble
That's a good question. Which is similar to the (more common) question: "What's the pressure of a gas in a gravitational field" (for which you will be able to find more information). The ...
- 1,508
5
votes
Accepted
Beta function of a marginal operator
No, $H'$ shouldn't grow for all values of $u$. This can be seen just from the solution of the RG equation with the given $\beta$-function. Let us find the solution of the RG equation. The RG equation ...
- 1,113
-1
votes
How hot can one heat a single atom?
I have to begin by being pedantic. Thermodynamics (the subfield of physics where temperatures, entropy, and pressures are explained) does not make statements about the state of a single atom. Like ...
- 7,560
0
votes
How hot can one heat a single atom?
I'll start with question #1:
For the radiation to have any effect on the hydrogen atom, it must have a frequency (and therefore a well-defined energy content) that the atom is capable of responding to....
- 96.7k
5
votes
How hot can one heat a single atom?
Heat is a thermodynamic concept, applicable to systems with large number of particles (taking Avogadro number $N_A\propto 10^{23}$ as a typical number of particle sin the system). In fact, ...
- 61.9k
0
votes
How do you Find the Variance of Temperature in a Maxwell Boltzmann Distribution?
The exponential distribution of kinetic energy has a parameter, called temperature. Its the limit of many subsystems in contact with an infinite bath of given temperature. Anything interesting is ...
- 1
2
votes
Maxwell–Boltzmann distribution derivation using only thermodynamic equations
Your approach is doomed to fail.
The classical ideal gas obeying Maxwell-Boltzmann distribution rather famously has the quantum-defying Sackur-Tetrode entropy. It is unbounded below as $T\to0$, and so ...
- 10.2k
2
votes
Accepted
Maxwell–Boltzmann distribution derivation using only thermodynamic equations
Probably not legal, for a few reasons. The thermal energy $U$ tends to depend on $T$, so that side $dU/T$ needs to be integrated properly. Additionally, the probability should be proportional to $\...
- 5,317
0
votes
Landau & Lifshitz and the law of increase of entropy
I think it is better to differentiate $d\Gamma$ and $\Delta \Gamma$; the former is a function of $E_a$ and the latter $\overline{E}_a$. L & L tries to find a description for unknown function $d\...
- 168
0
votes
Reversible vs Quasistatic
I've always had your same wondering.
My own explanation is below.
Quasistatic is clear to everybody: it means that the process dynamics is so slow that the system undergoing the process is always very ...
1
vote
Accepted
Does the gradient of the free energy give the direction in which the system most likely moves?
Just to go on with T.P. Ho answer. The relaxation dynamics of a system coupled to an equilibrium bath is indeed complicated and not universal. Here are some thought.
The relaxation dynamics of a ...
- 1,508
1
vote
Does the gradient of the free energy give the direction in which the system most likely moves?
Non-equilibrium dynamics isn't quite encoded in free energy alone. The coupling with a heat bath allows relaxation, but just looking at $F(L,N,M)$ we have no slightest idea how the relaxation occurs. ...
- 442
5
votes
Canonical ensemble, ergodicity and Liouville’s equation
The canonical ensemble is best understood as a small subsystem of a larger microcanonical ensemble. Specifically, the canonical ensemble is assumed to have a fixed volume and particle number, but it ...
- 20.5k
2
votes
Canonical ensemble, ergodicity and Liouville’s equation
Short version
To be precise, the idea of an ensemble is that to consider the same system to exist in multiple configurations (microstates), with each microstate associated with a probability. Further, ...
- 3,111
3
votes
Accepted
Find speed with which a steel bar sinks through ice (Reif)
Important background: In the Clausius–Clapeyron equation, $\frac{\Delta P}{\Delta T}$ refers to the slope of the equilibrium phase-change coexistence curve, not necessarily to any actual pressure and ...
- 27.1k
1
vote
Microcanonical ensemble through Maximum Entropy method
As an earlier related question correctly pointed out:
Jayne's introduced the effectiveness of Bayesian Inference in deriving different statistical ensembles. His idea was that the distribution that ...
- 61.9k
3
votes
Interpretation of the ideal gas single particle partition function
Here comes the part that troubles me. How can the """number of particles that can fit on average in the real space volume"""" be related to the number of states ?
...
- 61.9k
7
votes
Accepted
Interpretation of the ideal gas single particle partition function
In fact, it is not clear why one can assume that $V/\Lambda^3$ is equal to the average number of particles that can fit into a volume $V$. What is the distribution of the number of particles from ...
- 6,040
0
votes
Showing how RG generates new interactions in Ising
Strictly speaking, even in 1D, you will generate infinitely many coupling terms, all the way to infinite range, after one single step of blocking transformation.
The blocking procedure is supposed to ...
- 442
7
votes
Accepted
Self-Contained Explanation of the Fluctuation-Dissipation Theorem?
By no mean, this will be a holistic answer. Altough I hope it might clarify some things. In any case, any question you might have would be answered in the papers
Fluctuation-Dissipation: Response ...
- 1,508
2
votes
Showing how RG generates new interactions in Ising
A pedagogical paper by Maris and Kadanoff provides a convincing argument to explain why a coarse-graining procedure may generate new couplings in more than 1D (you can look for it with Google, and ...
0
votes
Thermodynamics and the state postulate: should it be a Fourth Law?
To your first question,
I think the state postulate is a derived consequence of the equilibrium-state postulate of thermodynamics. The latter includes the definition of thermodynamic equilibrium ...
- 11
0
votes
Non-interacting gas in homogeneous gravitational field
Attempting to build on Leo Webb's answer, let the height of the lower layer be $h$ so that the height of the upper layer is $h+dh$. Also denote the number density at each height to be respectively $n(...
- 51
0
votes
Constructing a gapped family of Hamiltonians in the trivial paramagnet
Let us label the states using the eigenstates of the paul-X operator $|\pm\rangle$. The ground states of $H_0$ and $H_1$ are, respectively $|-++++\ldots\rangle$ and $|+-+++\ldots\rangle$. The state ...
- 181
1
vote
Does the gradient of the probability density tell me in which direction the system is most likely moving?
Now that I am home, here is the general answer.
Your claim is, as far as I understood: Probability densities relax to a constant. Like Fick's law.
This means $\partial_t \rho \propto -\nabla \rho$.
My ...
- 387
1
vote
Accepted
Does the gradient of the probability density tell me in which direction the system is most likely moving?
The hypothesis is false. In some sense, in the case of a stationary probability density, things may go the other way around.
Indeed, let's assume that we have a probability density $\rho(x)$ and that ...
2
votes
Statistical mechanics of a deterministic system
Entropy, unlike energy or volume, is not a quantity that is unique to a physical system. It is bound to its description.
I like this example of stochastic thermodynamic where you have the following ...
- 1,508
2
votes
Statistical mechanics of a deterministic system
The physical system will do what it will do regardless of how we describing it. You knowing the microstate, will not affect how the system behaves.
You have discussed two different frameworks you can ...
- 52.3k
0
votes
How Subjective is Entropy Really?
I'm a computer scientist doing some research that touches on basic concepts in statistical mechanics: macrostate, microstate and entropy.
Since you are a computer scientist and researcher, I think my ...
- 358
4
votes
Feynman tricks to reproduce Onsager's solution of the 2D Ising model
This question and its answer are intertwined with several beautiful ideas and have rich historical significance. Yet, it remained overlooked for some reason—neither upvoted nor answered. Here, I’d ...
1
vote
Notion of Entropy as a functional - is it neccessary?
Methods 1 and 2 are not fundamentally different. Quite often, max entropy methods are introduced with reference to discrete probability spaces to leave aside some technical complications connected to ...
Top 50 recent answers are included