Readit NewsDoes someone has a good explanation ?
In thermodynamics, there often isn't really one "best" choice of two coordinate functions among the many possibilities (pressure, temperature, volume, energy, entropy... these are the must common but you could use arbitrarily many others in principle), and it's natural to switch between these coordinates even within a single problem.
Coming back to the more familiar x, y, r, and θ, you can visualize these 4 coordinate functions by plotting iso-contours for each of them in the plane. Holding one of these coordinate functions constant picks out a curve (its iso-contour) through a given point. Derivatives involving the other coordinates holding that coordinate constant are ratios of changes in the other coordinates along this iso-contour.
For example, you can think of evaluating dr/dx along a curve of constant y or along a curve of constant θ, and these are different.
I first really understood this way of thinking from an unpublished book chapter of Jaynes [1]. Gibbs "Graphical Methods In The Thermodynamics of Fluids" [2] is also a very interesting discussion of different ways of representing thermodynamic processes by diagrams in the plane. His companion paper, "A method of geometrical representation of the thermodynamic properties of substances by means of surfaces" describes an alternative representation as a surface embedded in a larger space, and these two different pictures are complimentary and both very useful.
For those who are like me and don't know the term, "a language server for HTML" is referring to the plugin that evaluates your HTML syntax. That might be a narrow explanation of the tool but that's the basic idea I got from trying it.
Consider the function f(x) = Sum_{n=1}^\infty c^(-xn)
Then differentiate this k times. Each time you pull down a factor of n (as well as a log(c), but that's just a constant). So, the sum you're looking for is related to the kth derivative of this function.
Now, fortunately this function can be evaluated explicitly since it's just a geometric series: it's 1 / (c^x - 1) -- note that the sum starts at 1 and not 0. Then it's just a matter of calculating a bunch of derivatives of this function, keeping track of factors of log(c) etc. and then evaluating it at x = 1 at the very end. Very labor intensive, but (in my opinion) less mysterious than the approach shown here (although, of course the polylogarithm function is precisely this tower of derivatives for negative integer values).
This is effectively what OP does, but it is phrased there in terms of properties of the Li function, which makes it seem a little more exotic than thinking just in terms of differentiating power functions.
Here is a intuitive explanation for it from [1]:
“Temperature stems from the observation that if you bring physical objects (and liquids, gases, etc.) in contact with each other, heat (i.e., molecular kinetic energy) can flow between them. You can order all objects such that:
- If Object A is ordered higher than Object B, heat will flow from A to B.
- If Object A is ordered the same as Object B, they are in thermal equilibrium: No heat flows between them.
Now, the position in such an order can be naturally quantified with a number, i.e., you can assign numbers to objects such that:
- If Object A is ordered higher than Object B, i.e., heat will flow from A to B, then the number assigned to A is higher than the number assigned to B.
- If Object A is ordered the same as Object B, i.e., they are in thermal equilibrium, then they will have the same number.
This number is temperature.”
> Mind that all of this does not impose how we actually scale temperature.
> How we scale temperature comes from practical applications such as thermal expansion being linear with temperature on small scales.
An absolute scale for temperature is determined (up to proportionality) by the maximal efficiency of a heat engine operating between two reservoirs: e = 1 - T2/T1.
This might seem like a practical application, but intellectually, it’s an important abstraction away from the properties of any particular system to a constraint on all possible physical systems. This was an important step on the historical path to a modern conception of entropy and the second law of thermodynamics [2].
“Amazon confirms 14,000 job losses,” is not an example of the passive voice.
“14,000 workers were fired by Amazon,” is an example of the passive voice.
There is not a 1:1 relationship between being vague about agency and using the passive voice.