8
votes
Accepted
Currently self-studying QFT and The Standard Model by Schwartz and I'm stuck at equation 1.5 in Part 1 regarding black-body radiation
It just a clever use of the geometrical series:
$$\frac{1}{1-q} = \sum_{j=0}^\infty q^j$$
which is valid for any real number $q<1$.
Here $q = e^{-E_n\beta} \equiv e^{-\hbar\omega_n\beta}$. As long ...
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4
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How to derive Shannon Entropy from Clausius Theorem?
These are not the same.
Shannon entropy (Information entropy), $H_\alpha=-\sum_i p_i\log_\alpha p_i$ applies to any system with specified probabilities $p_i$.
Boltzmann entropy, defined via the famous ...
- 37.4k
3
votes
Accepted
How to derive Shannon Entropy from Clausius Theorem?
You can't. It is not possible to derive a more general formula from a less general one. Of course, one can find hints for the generalization, but the validity of the generalization has to be proved ...
- 23.3k
2
votes
Master equation with a coherent bath
You get a Master equation with an additional drive term that explicitly remains in the Master equation (see e.g. https://link.springer.com/book/10.1007/978-3-319-62843-1). This term is semi-classical ...
- 10.8k
1
vote
Accepted
Regarding calculation the moments of a random variable whose probability distribution obeys the Fokker Planck equation
The procedure that you propose is right! I am actually not aware of any other strategy.
Try the integration by parts bearing in mind that you know the distribution $P(y,t_0|y_0,t_0)=\delta(y-y_0)$.
If ...
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1
vote
Metropolis-Hastings and underlying Markov process
I will answer as I found the mistake, my apologies for having been careless in posting I can assure you I spent much time doing hand calculations but sometimes things just evade your attention.
The ...
- 427
1
vote
Boltzmann vs Gibbs definition of entropy
Provided that the average of a quantity is defined by
$$
\langle A\rangle = \sum_i P_i A_i,
$$
the entropy by definition can be viewed as the average of the negative logarithm of the probability of a ...
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1
vote
Are photons with different frequency distinguishable?
Let's consider photons in a cubical box (a cavity) with side length $L$. The modes of the electromagnetic field are labeled by:
The integers $\{n_x, n_y, n_z\}$, which are related to the wavenumbers $...
- 33.3k
1
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How to derive Shannon Entropy from Clausius Theorem?
A better approach would be to use the Shannon Entropy to derive Gibbs entropy: $S=−k\cdot∑p_n \cdot \ln(p_n)$. The two equations are very similar and therefore it is much easier understand. From ...
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