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Apr
30
comment A conceptual question related to statistical mechanics
I would also recommend that you have a look at the oldish but beautiful little book "Statistical Mechanics and the Foundations of Thermodynamics" by Anders Martin-Löf (Lecture Notes in Physics 101, 1979). The discussion of the pressure starts on page 23.
Apr
26
awarded  Yearling
Apr
20
comment Why don't we observe spontaneous symmetry restoration in nature?
I agree with the above comments. Nevertheless, one can find (rather restrictive) settings in which such a result actually become a theorem. One is that of ferromagnetic (classical) spin systems, for which, under suitable assumptions, one can prove that, if $T<T'$, then the internal symmetry group at temperature $T$ is necessarily a subgroup of the internal symmetry group at temperature $T'$. See Theorem 4 of Chapter 4 in the book "Group Analysis of Classical Lattice Systems", C. Gruber et al., Lecture Notes in Physics 60, 1977. I am not aware of more recent results, but there should be some.
Apr
10
revised Inverting density in favour of fugacity
typo
Apr
10
revised Inverting density in favour of fugacity
Added more details and explanations
Apr
9
answered Inverting density in favour of fugacity
Mar
22
comment 3-state Potts model - probability of finding a site in state 1
Once you have computed the free energy, just observe that differentiating it with respect to h gives you the density of $1$ in the system. (Just differentiate $N^{-1}\log Z_N$ at finite $N$ to see that.)
Mar
21
comment What does the solid phase in a two-dimensional system with Lennard-Jones potential look like?
@AdiRo : Using precise definitions for the concepts one uses is very important. There are very good reasons for defining them using the thermodynamic limit. Then, of course, one should also say what happens in finite ("real") systems. In particular, these "soft-crystal" phases can and must be studied: they are very interesting and more tractable than the 3d solid phases. However, they should not be called solid.
Mar
21
comment What does the solid phase in a two-dimensional system with Lennard-Jones potential look like?
@AdiRo : He is actually taking a pragmatic position: in finite systems, you see something that looks like a solid phase, so let's call that a solid phase! That might seem reasonable, but is actually conceptually a bad idea: with such a point of view, you can have long-range order in the one-dimensional Ising model, for example! Indeed, for finite systems and sufficiently low temperatures (depending on the system's size), you get a magnetized sample.
Mar
21
comment What does the solid phase in a two-dimensional system with Lennard-Jones potential look like?
@AdiRo: This is even discussed explicitly in the paper you refer to: just read Sections 2 and 3. What the author does is to argue that a new definition of a solid phase is needed. But there is no discussion (it is a mathematical fact) that in two dimensions translation invariance cannot be broken.
Mar
21
comment What does the solid phase in a two-dimensional system with Lennard-Jones potential look like?
@AdiRo : of course not. The proofs in the papers I link to apply just as well to interactions of infinite range. In particular, they apply to Lennard-Jones interactions. In two dimensions, you do not see true solid phases, in the sense that there cannot be positional long-range order. What is possible, however, is to have some kind of "soft crystal" phases, which look like solid locally; this is due to the very slow deformations of the crystal over long distances. All this is completely analogous to the massless (Kosterlitz-Thouless) phase in the 2d XY model.
Mar
17
revised What does the solid phase in a two-dimensional system with Lennard-Jones potential look like?
Added links to freely available versions and corrected a claim. Added a reference.
Mar
16
revised What does the solid phase in a two-dimensional system with Lennard-Jones potential look like?
Added a brief discussion and a reference to results on ground states.
Mar
16
comment What does the solid phase in a two-dimensional system with Lennard-Jones potential look like?
@Couchyam : Sure, I agree with you: they are the analogues of the massless phase in $O(N)$ models. However they are not considered to be solid phases (the latter are characterized by the breaking of translation invariance), and I doubt that's what the OP had in mind.
Mar
16
answered What does the solid phase in a two-dimensional system with Lennard-Jones potential look like?
Feb
18
comment Why does Triple point exist?
I don't understand why is this downvoted without even leaving a comment? It is clearly not universally true, but it is at least partially correct. However, I would have added a link to the Gibbs phase rule.
Feb
15
comment Difference between theoretical physics and mathematical physics?
@PeterShor : Sure, I agree with you. And let me stress that this was not meant to be derogatory at all. I only wanted to emphasize that many "string theorists" in mathematics departments are generating conjectures (and developing tools to do so) much more often than they prove theorems, and theirs is obviously also a very valuable activity.
Feb
7
answered Why do we work in thermodynamic limit in statistical physics?
Feb
1
comment Relation between the $N$ particle partition function and probability?
I agree with the answer, but disagree with the characterization of the partition function as the most important quantity in statistical physics (even though it is often presented as such in textbooks). This central role should be reserved to the probability measure itself. Indeed, the latter gives you access to much more information (even about macroscopic quantities) than the partition function, which only gives you access to the statistical properties of the macroscopic quantities appearing in the Boltzmann weight: energy, magnetization, etc.).
Jan
28
awarded  Custodian