All Questions
Tagged with classical-mechanics statistical-mechanics
30
questions
67
votes
5
answers
8k
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Is there a Lagrangian formulation of statistical mechanics?
In statistical mechanics, we usually think in terms of the Hamiltonian formalism. At a particular time $t$, the system is in a particular state, where "state" means the generalised coordinates and ...
- 34.3k
40
votes
8
answers
6k
views
How is Liouville's theorem compatible with the Second Law of Thermodynamics?
The second law says that entropy can only increase, and entropy is proportional to phase space volume. But Liouville's theorem says that phase space volume is constant.
Taken naively, this seems to ...
- 103k
10
votes
1
answer
1k
views
Are there necessary and sufficient conditions for ergodicity?
What are the necessary and sufficient conditions (if any) for ergodicity (or non-ergodicity)?
I see for instance that some integrable systems are not ergodic. For instance a linear chain of harmonic ...
- 17.8k
3
votes
1
answer
678
views
Collision Term in the Classical Boltzmann Transport Equation
I cannot get over the feeling that in the classical derivation of the collision term of Boltzmann's transport equation molecules that are already knocked out of a $(\textbf r, \textbf v)$ space volume ...
- 31
10
votes
2
answers
853
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Partition function containing QM?
I am wondering about the partition function of the classical microcanonical ensemble. It contains Planck's constant and also an indistinguishability argument about the particles I am looking at and I ...
- 1,880
8
votes
3
answers
2k
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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 ...
- 4,685
7
votes
3
answers
1k
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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 ...
- 697
13
votes
2
answers
2k
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Is Liouville's equation an axiom of classical statistical mechanics?
Suppose we have a classical statistical problem with canonical coordinates $\vec{q} = (q_1, q_2, \dots, q_n)$ and $\vec{p} = (p_1, p_2, \dots, p_n)$ such that they fulfill the usual Poisson brackets:
\...
- 3,119
9
votes
5
answers
3k
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Are the physical laws scale-dependent?
If you read the article "More Is Different", by P.W. Anderson (Science, 4 August 1972), you will find a deep question: are the physical laws dependent of the size of the system under study?
As an ...
- 560
7
votes
1
answer
612
views
necessary and sufficient conditions for an isolated dynamical system which can approach thermal equilibrium automatically
Given an isolated $N$-particle system with only two body interaction, that is
$$H=\sum_{i=1}^N\frac{\mathbf{p}_i^2}{2m}+\sum_{i<j}V(\mathbf{r}_i-\mathbf{r}_j)$$
In the thermodynamic limit, that ...
- 5,971
2
votes
2
answers
244
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Connection Helmholtz free energy and $H,M,B$ fields
Consider a magnetic system subject to a magnetic field. Here we work with the fields $H,M,B$.
Now, how does a change in the Helmholz free energy depend on $H,M,B$? I have three sources that seem to ...
- 1,440
71
votes
5
answers
12k
views
Why does a system try to minimize its total energy?
Why does a system like to minimize its total energy? For example, the total energy of a $H_2$ molecule is smaller than the that of two two isolated hydrogen atoms and that is why two $H$ atoms try to ...
- 26.8k
12
votes
1
answer
10k
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Derivation of differential scattering cross-section
I'm trying to follow the derivation of the Boltzmann equation in my Theory of Heat script, but have a little trouble understanding the following:
The cross-section $d\sigma$ is defined as: The amount ...
user17574
10
votes
1
answer
3k
views
Quantum and Classical Liouville operators
In the Heisenberg picture of Quantum Mechanics, for an observable $\hat{A}$, we have the famous Heisenberg equation giving the time evolution of the operator: ($\hat{H}$ is the Hamiltonian operator)
$$...
- 1,611
7
votes
1
answer
378
views
Is the principle of indifference enough to derive the microcanonical ensemble?
The microcanonical ensemble is usual motivated solely by the principle of indifference. Textbooks usually say something along the lines of "If the only thing we know about a system is its total ...
- 48.4k