All Questions
Tagged with classical-mechanics statistical-mechanics
189
questions
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
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
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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
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
12
votes
1
answer
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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
12
votes
3
answers
9k
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In what limit do we *really* get Maxwell-Boltzmann statistics from Bose-Einstein and Fermi-Dirac?
Fermi-Dirac and Bose-Einstein energy occupation number $n(\epsilon)$ in natural units ($[T]=[\epsilon]$) read
$$n(\epsilon) = \frac{D(\epsilon)}{e^{(\epsilon-\mu)/T}\pm 1},$$
where $D(\epsilon)$ is ...
- 20.4k
10
votes
2
answers
853
views
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
10
votes
1
answer
1k
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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
10
votes
1
answer
3k
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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
10
votes
3
answers
1k
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Is $k_B \rightarrow 0$ the classical limit of stat. mech., as $\hbar \rightarrow 0$ is in QM?
I hear very often among my peers and seniors that just as how $\hbar\rightarrow0$ takes me to classical mechanics from quantum mechanics, $k_B\rightarrow0$ will take me to classical thermodynamics ...
- 6,381
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
9
votes
2
answers
2k
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Intuition behind classical virial theorem
I am continuing to brush up my statistical physics. I just want to gain a better understanding. I have gone through the derivation of the classical virial theorem once more. I have thought about it, ...
- 1,258
9
votes
2
answers
1k
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Does quantum mechanics halve the dimension of phase space?
In classical mechanics, a particle confined to move along only the $x$-direction can be fully described by a 2-tuple $(x_1,p_1)$ in phase space. In this case, the phase-space is clearly 2-dimensional. ...
- 1,581
9
votes
7
answers
2k
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How does such strange microscopic behavior at the atomic level (quantum mechanics) lead to the macroscopic behavior at our level?
So, I'm only a high school student researching quantum physics, and I find it very interesting. However, there's one question that keeps nagging at me in the back of my head. How exactly do odd ...
- 193
9
votes
1
answer
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Adiabatic invariant and Liouville's theorem
It appears that many people have tried to show adiabatic theorem from Liouville's theorem, e.g., Li's note, or at least tried to find some relations, e.g., Rugh, Adib and Tong's lecture notes Sec. 4.6....
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