Helmholtz Free Energy vs Gibbs Free Energy in Landau Theory When I was studying thermodynamics potentials for the first time, I learned that keeping the pressure and the temperature constant, the Gibbs Free Energy is a minimum, also, if we maintain the temperature and the volume of a given system constant, the Helmholtz energy will be a minimum. However, when I was studying Landau theory, I was confused if one should minimize the gibbs free energy or the helmholtz free energy. For example, in the Ising Model, one expands the Free Energy and the applies landau theory. However, we need to expand the Gibbs Free Energy in the liquid-gas transition.
In the Landau theory of phase transitions, is the order parameter a thermodynamic variable of state? In the Landau theory of phase transitions, one typically considers a "free energy" $F$ as a function of the temperature $T$ and the "order parameter" $\psi$: $F(T, \psi)$.  For the sake of clarity, let's consider the liquid-gas transition, in which the order parameter is usually taken to be the difference of densities between the ordered phase and the disordered phase.  In this case, it looks that the order parameter $\psi$ is closely related to the the volume $V$, a global variable of state.  What is the correct interpretation of $\psi$ as a thermodynamic variable?   I consider several logical possibilities, but all interpretations look troublesome to me:    $\psi$ is a variable of state replacing the volume. In this case I have a few related concerns:   Is the "free energy" $F$ the Helmholtz or the Gibbs free energy?    If $F$ is the Gibbs free energy, $F$ should depend on the temperature $T$ and the pressure $p$ (the conjugate variable of the volume), so it shouldn't depend on $\psi$.  If it is the Helmholtz free energy, it's ok that it depends on $\psi$, but the experiments are usually carried out at constant pressure, not constant volume.  Usually, state variables are something one can select at will [for instance, think of the phase diagram $(p, T)$]. However, the order parameter $\psi$ is not selected at will, but it is selected by the system as a result of the minimization process according to the Landau theory.  $\psi$ is yet another variable of state in addition to volume and temperature.    Why the pressure is not usually taken into account as a variable of state, and only the temperature is considered? What is the conjugate variable to $\psi$? Can really $\psi$ and $V$ be taken as independent?  $\psi$ is not a variable of state, is something else.   What else can $\psi$ be from the thermodynamic point of view? Same question as above, why is pressure not usually taken into account in Landau theory?
How can I make a tool drop blocks with CanPlaceOn tag In Adventure mode, I'm trying to let a player get blocks that he can place later, is there a way to do it directly or do I have to change them in his inventory?  Situation: The player gets a tool that will allow him to clear a boulder blocking the road. Further down the path, he encounters a broken bridge, and I want him to use the blocks he gathered before to rebuild it.  At the moment, if I build the boulder with blocks having the CanPlaceOn tag, the tag disappear when the player destroys them and they go in its inventory. How do I keep Python print from adding newlines or spaces? In python, if I say  print 'h'   I get the letter h and a newline.  If I say   print 'h',   I get the letter h and no newline.  If I say  print 'h', print 'm',   I get the letter h, a space, and the letter m.  How can I prevent Python from printing the space?  The print statements are different iterations of the same loop so I can't just use the + operator. Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? I just wrote a question before this one and had trouble uploading a screenshot. The drop your image dialog throws no warning or error, but after uploading it won't show.  I also noticed that other questions won't load their images. Could this be some issue with imgur or similar?  Example:     When opening the image in a separate tab, Chrome gives a "too many redirects" error:    This might be linked to How should self-deletes be evaluated in the question ban algorithm? A self-deleted post is often not a low-quality post and may simply be a mistake noticed by the user, unlike moderator- or community-deleted posts which may indicate serious problems that the user may not recognize or may have intentionally posted.  Should they carry less weight in the question ban algorithm? What dried food can I take across the American border from Canada? We will be driving from Banff to Glacier National Park while on our camping trip and I was told to be careful with taking food across the border. We don't eat meat so that's one less issue and I already know that many fresh fruits/veggies can get complicated so I'm only thinking of taking dry stuff across (mostly grains but possibly also dried veggies).  My plan at first was to stock up at BulkBarn in Calgary since they have tons of stuff and you can just take what you need but I'm afraid that since the packaging is generic and unsealed that it might cause more problems. What are the criteria to be able to take dried food across the border? How can I send a custom desktop notification? I have a custom script and I want to send a desktop notification (the one that appears in the top right corner of the screen) with a custom message. How do I do that? Aromatic rings resized with chemfig When I want to reduce the size of a molecule drawn with  I usually put a [scale=0.xx] before the argument of the command. However when I deal with aromatic rings this method doesn't work well, in fact only the carbon skeleton is resized: the 'aromaticity' is not and it's also out of center.  For example:  \chemfig[][scale=0.75]{**6(-------)}\quad   gives me:     How do I achieve a complete resize? In the decimal system: $...-3, -2, -1, 0, 1, 2, 3...$ In the binary system: $...-11, -10, -1, 0, 1, 10, 11...$ In the unary system: $-111, -11, -1, \text{ ??? }, 1, 11, 111...$   I was only able to find a related question but none with an answer so far: I have already seen a proof of this statement here on stackexchange but I would like someone to verify whether "my proof" is correct and also if you could point out issues with the writing style. thanks in advance!  Statement. $\mathbb{Z}_{p}$ is an integral domain $\Leftrightarrow$ $p$ is prime.  Proof.$(\Rightarrow)$ Suppose $\mathbb{Z}_{p}$ is an integral domain. Assume (by way of contradiction) that $n$ is not prime. Then $n=ab$ for some $a,b\in \mathbb{Z}$ where $1&lt;a$ and $b&lt;n$, meaning $ab\equiv 0\pmod n$. This is a contradiction since $\mathbb{Z}_{p}$ is assumed to be an integral domain and therefore does not contain any zero divisors. Therefore $p$ must be prime.  $(\Leftarrow)$ Assume $p$ is prime. Then $a$ does not divide $n$ for any $1&lt;a&lt;n$ where $a\in \mathbb{Z}_{p}$. Thus $\mathbb{Z}_{p}$ does not contain any zero divisors and is therefore an integral domain. I am interested in how uncertainty can be accounted for when considering the risk of extinction of a species. Forgive me for extending a rather tired thought experiment, but at least it's familiar territory and I hope it illustrates what I am trying to understand.  Let's say that Schrödinger was not satisfied with killing and not killing only one cat, so he went out and collected the last 15 remaining Himalayan Snow-Cats. He put each one in a box with a vial of poison, hammer, and a trigger device for releasing the hammer. Each trigger device has a known probability of releasing the hammer within any given hour, and the poison will take 5 hours to kill a cat (things got a bit trippy when he used radioactive decay, so this time he steers clear of quantum mechanics). After one hour, Schrödinger receives a notice from the ethics board telling him he's nuts and ordering him to release the cats immediately. He pushes a button which opens a cat-flap at the back of each box, thus releasing them all back into the wild. By the time the humane society check the boxes, all the cats have bolted. Before anyone can restrain him, Schrödinger detonates the remaining triggers, so nobody knows how many of the cats were poisoned.  The probability that each of the cats is poisoned is as follows:   0.17 0.46 0.62 0.08 0.40 0.76 0.03 0.47 0.53 0.32 0.21 0.85 0.31 0.38 0.69   After 5 hours have elapsed and all poisoned cats have died, what is the probability that there are:   (A) More than 10 Himalayan Snow-Cats still alive  (B) 10 or less still alive (C) 5 or less still alive (D) 2 or less still alive How to get a list of the filenames of the figures embedded in a document I have a TeX document that is embedding a huge amount of figures from a fig folder that is full of another humongous amount of other figures.  I would like to move the document along with its figures, but I don't want to take the whole fig folder, just the figures in there that are embedded by the document.  Is there a package (or any other way) that allows me to, for example, print out the filenames of all the figures embedded by the document? Something like:  fig/figurename1.eps fig/figurename2.eps ... etc   I could then just copy-paste that list and add some cp's in a plain text file and automate the copy of those files to my desired location by running the file in my shell:  $ cp fig/figurename1.eps wherever/fig/ $ cp fig/figurename2.eps wherever/fig/ ... etc   Of course this is just an idea, if somebody has an alternative to the "file listing and copy-pasting in a shell file" I would be happy to listen =) Number of permutations of $n$ elements where no number $i$ is in position $i$ I am trying to figure out how many permutations exist in a set where none of the numbers equal their own position in the set; for example, $3,1,5,2,4$ is an acceptable permutation where $3,1,2,4,5$ is not because 5 is in position 5.  I know that the number of total permutations is $n!$. Is there a formula for how many are acceptable given the case that no position holds its own number? Right now I'm using echo &quot;Hello World&quot; &gt;&gt; file.txt  to append some text to a file but I also need to add text below a certain string let's say [option], is it possible with sed? EG: Input file Some text Random [option] Some stuff  Output file Some text Random [option] *inserted text* Some stuff Textbook approach to modeling non-proportional hazards in the Cox model In Cox models with time varying coefficients, the effect of covariates on the hazard is allowed to change through time. In cases where a coefficient has a linear relationship with time, I am aware of at least two text books ( and an Appendix to ) that advocate for the following model, first in R code:  coxph(Surv(time, death) ~ disease + disease:time)   And now in mathematical notation:  $$ h(t|X) = h_0(t) \exp(\beta_1X + \beta_2X_T) $$ Where $h(t|X)$ is the instantaneous hazard rate at time $t$ given covariates $X$; $h_0(t)$ is the baseline hazard rate; and $T$ is the censoring or event times. $\beta_1$ represents the log hazard ratio for individuals with $T=0$ (which is sort of nonsensical since you can't have $T=0$ in the data), while $\beta_2$ is harder to interpret... $\beta_2$ shows how the log hazard ratio depends on censoring/event times, but that is circular because event times are also the response variable (this point is made in Terry Therneau’s vignette “”).  According to Therneau's vignette, to correctly specify a time varying coefficient, you need to use the following model, first in R code:  coxph(Surv(time, death) ~ disease + tt(disease), tt=function(x,t,...){x*t})   And now in mathematical notation: $$ h(t|X) = h_0(t) \exp(\beta_1X + \beta_2X_T) $$  The difference is that the time interaction is with $t$ in the model rather than with the event times $T$ in the data. The resulting parameters are easy to interpret: they are the intercept and slope of the linear relationship between the log hazard ratio of $X$ and time. This approach seems like the proper way to model a time varying effect.  Can anyone justify or advocate for the first approach to modeling non-proportional hazards in the Cox model? If it is really incorrect (as I suspect it is), what are the practical consequences of using this approach? Turn off paragraph indention for a single line Is it possible to turn off the paragraph indention for a single line? I have the following use case: I open a new section which describes the task of an exercise. Since I use the font size \large for the section I don't want to use the section text for the task description.  Therefore I added the task description after the section. The disadvantage is that an indention is created when starting to write text after the task description.  What are possible solutions to this problem? Feature request: Add pronouns to usercards? The FAQ mentions bounty, but sometimes a question just doesn't really have a right answer:   No solution exists The question as written is unanswerable and unsalvageable There may be a technically correct answer, but the asker is unable to utilize it for some reason.   In these cases, a bounty, and more answers, won't really help. What is the appropriate etiquette in these cases? Should the question be closed? Left hanging? Interscholastic Mathematics League Senior B Division #2 Can a function be made to accept a variable amount of inputs?