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 votes

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 ...
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3 votes
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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 ...
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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 ...
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1 vote
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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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  • 557
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 ...
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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 $...
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1 vote

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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