Linked Questions

17 votes
3 answers
9k views

Why do some substances undergo sublimation while others do not?

This question is indeed lengthy, but please go through the question at least. From the study of kinetic theory I know that for intuitive answers we can associate the states (liquid, solid,gas) with ...
Freelancer's user avatar
8 votes
3 answers
2k views

Extending the ergodic theorem to non-equilibrium systems

I try to make this as short and concise as possible. For equilibrium systems in statistical mechanics, we have the Liouville's theorem which says that the volume in phase space is conserved when the ...
user929304's user avatar
  • 4,655
6 votes
3 answers
2k views

What does Liouville's Theorem actually mean?

Basically, the mathematical statement of Liouville's theorem is: $$\frac{\partial \rho }{\partial t}= -\sum_{i}\left(\frac{\partial \rho}{\partial q_i}\,\dot{q_i}+\frac{\partial\rho}{\partial p_i}\,\...
user avatar
7 votes
3 answers
1k views

What is the difference between Liouville's theorem and the Boltzmann transport equation?

From what I understand, Liouville's theorem is about the probability density $\rho$ of an ensemble existing in a differential volume in phase space $d\mathbf{r}d\mathbf{p}$. So the statement for ...
megamence's user avatar
  • 697
20 votes
4 answers
728 views

Do unstable equilibria lead to a violation of Liouville's theorem?

Liouville's theorem says that the flow in phase space is like an incompressible fluid. One implication of this is that if two systems start at different points in phase space their phase-space ...
UtilityMaximiser's user avatar
5 votes
3 answers
1k views

How do shock waves form in space?

Today my aerodynamics professor mentioned that the equations we were learning for atmospheric shock waves can also be applied by astrophysicists to study black hole-related shock waves in space. How ...
techSultan's user avatar
2 votes
5 answers
264 views

How does the lack of information increase as temperature increases?

Suppose one knows nothing about the concept of entropy. How can we argue that the lack of information/ignorance about the system typically increases with the increase in the temperature using the ...
Solidification's user avatar
4 votes
2 answers
2k views

Why does entropy jump across a shockwave?

Using the Rankine-Hugoniot relations for a shockwave, one can show that entropy jumps across the shock, so that the entropy difference between upstream and downstream conditions is given by $$s_2 - ...
Tofi's user avatar
  • 2,661
7 votes
1 answer
786 views

How does the fluctuation theorem dissipation function become entropy?

In "The Fluctuation Theorem" by Evans and Searles, they derive the transient fluctuation theorem from Liouville's theorem (pg 1541). Following their notation $ \Gamma = (\vec{q}, \vec{p}) $, they use ...
user avatar
2 votes
2 answers
496 views

How many temperatures has a plasma?

In nonthermal plasma, not all particles move in the same way. The electrons are different from other particles. Both can be described as having a temperature separately. But that would mean, one piece ...
Volker Siegel's user avatar
5 votes
2 answers
974 views

Gibbs entropy, Clausius' entropy and irreversibility

I have a bunch of doubts and confusions on the concept of entropy which have been bothering me for a while now. The most important ones are of a more technical nature, arisen from the reading of this ...
Qwertuy's user avatar
  • 1,262
1 vote
1 answer
2k views

Mach Number after Normal Shock

Is there any way that someone can give me more of a conceptual explanation for the fact that the Mach number downstream of a normal shock must be less than or equal to 1? I understand the ...
Greg Castaldi's user avatar
2 votes
1 answer
624 views

Current density in phase space

$\newcommand{\dd}{{\rm d}}$ I have a question which arises from looking at the impact free Boltzmann equation. Let $(\vec{x},\vec{v})$ be a vector in our phase space $\Gamma^N = \mathbb{R}^{6N}$. The ...
user avatar
3 votes
2 answers
251 views

Why does distribution density seem to change though Liouville says it shouldn't?

I am trying to get a deeper understanding of Liouville's theorem and the distribution function in general. As an aid, I was thinking of the following simple, one-dimensional case: ball bearings are ...
Damon's user avatar
  • 65
0 votes
1 answer
233 views

Derivation of dispersion relation of electromagnetic waves in a magnetized hot plasma

I look to solve analytically the Vlasov-Maxwell equations for a magnetized hot plasma $$\frac{\partial f_{s1}}{\partial t}+\vec{v}\cdot\frac{\partial f_{s1}}{\partial \vec{r}}+\frac{q_s}{m_s}\Big(\vec{...
Gallagher's user avatar
  • 113

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