Linked Questions

13
votes
3answers
6k 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 ...
8
votes
3answers
1k 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 ...
5
votes
3answers
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 ...
2
votes
5answers
238 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 ...
1
vote
1answer
1k 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}\,\...
7
votes
1answer
461 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 ...
3
votes
1answer
557 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 ...
1
vote
1answer
569 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 ...
9
votes
2answers
392 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 ...
2
votes
1answer
451 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 ...
1
vote
2answers
147 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 ...
2
votes
2answers
137 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 ...
1
vote
1answer
108 views

Boltzmann equation for a collisionless medium?

In the derivation of the Boltzmann equation (link to Wikipedia) for a collisionless gas it is assumed that: $$ f\left( \vec{r} + \frac{ \vec{p} }{m} \Delta t, \ \vec{p} + \vec{F} \Delta t, \ t + \...
2
votes
0answers
67 views

Time symmetry of Boltzmann kinetic equation and H-theorem for calculating time-evolution of entropy

I know the Boltzmann kinetic equation is invariant under T transformation ($t \rightarrow -t$). Also, I could derive the entropy's time evolution from H-theorem as: $$\frac{\partial S}{\partial t} = -...
1
vote
2answers
40 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 - ...

15 30 50 per page