1,597 reputation
516
bio website unige.ch/math/folks/velenik
location Geneva, Switzerland
age 44
visits member for 2 years, 6 months
seen 21 mins ago

Professor at the Mathematics department of the University of Geneva. Mostly working at the intersection of probability theory and statistical physics.

With a colleague, we are in the process of writing an introductory book on the (classical) equilibrium statistical physics of lattice spin systems, from a mathematically rigorous point of view. A more detailed description of the project, as well as early drafts of several chapters can be found here. Your feedback would be more than welcome!


Jun
1
answered 1 dimensional Ising model
Jun
1
comment 1 dimensional Ising model
@sreeram: I don't know any, but I'll sketch an answer later if nobody does.
Jun
1
comment 1 dimensional Ising model
No, these expansions do apply in any dimensions. In dimension 1, they allow a simple computation of the free energy of the model, provided there is no magnetic field (otherwise the computations become more complicated than using the transfer matrix). This is also true of the change of variables method (which I assume refers to the change of variables $\eta_i = \sigma_i\sigma_{i+1}$ mapping the model to independent spins), which is only useful in dimension $1$.
May
23
revised Precise statement of Mermin–Wagner theorem
Added some information
May
23
revised Precise statement of Mermin–Wagner theorem
Added some references
May
23
comment Precise statement of Mermin–Wagner theorem
Not only is it not necessary to have isotropic coupling constants, but results even apply to a very large class of graphs (which can be extremely irregular). I'll add a couple of references in my post.
May
22
revised Precise statement of Mermin–Wagner theorem
One typo, and some clarifications
May
22
answered Precise statement of Mermin–Wagner theorem
May
20
awarded  Citizen Patrol
May
18
comment Question about the Boltzmann distribution
@yarnamc: Note that if the system $A$ is macroscopic, then the fluctuations of the energy density are extremely small. This is why the corresponding quantity in thermodynamics is considered deterministic.
May
15
comment What is the minimum non-integer dimension for which the XY model shows a phase transition? (if well-defined)
Of course, this universal behavior does not extend to $O(1)$: this is just the Ising model, which has a phase transition in dimension $2$ (and should have one for any dimension $d>1$, if you set up the model correctly). The difference is due to the fact that the symmetry of the latter is discrete, while it is continuous for $O(N)$ with $N\geq 2$.
May
14
comment What is the minimum non-integer dimension for which the XY model shows a phase transition? (if well-defined)
Depending how you set up your model in fractional dimensions, there will be long-range order as soon as $d>2$. The existence of long-range order is closely related to the transience of the random walk on the underlying graph. See this paper for one direction and this one for the other direction. Note that it is not clear how to make the latter paper rigorous, but it should be convincing for a theoretical physicist.
May
14
comment Is there anything physically infinite?
@user12345: Again, your statement "there is no physical law that we know of that would prevent a photon from travelling infinitely far" is about our current theories (it is a statement about a property of the set of physical laws we currently use to model physical reality), it is not about Nature. The only scientifically meaningful statements are those that can (at least in principle) be tested in experiments. Yours cannot.
May
13
comment Is there anything physically infinite?
That's obviously a correct statement. I don't understand how somebody can downvote this answer. Probably somebody clicking before thinking... not a very scientific attitude.
May
13
comment Is there anything physically infinite?
@user12345: I am only saying that such a statement is meaningless from an operational point of view (in that sense it's not science, but philosophy). Again, how would you test this hypothesis?
May
13
comment What is the interface tension between ordered and disordered phases of the Potts model?
Actually, I have no idea about that. Certainly, this must be completely open from a mathematical point of view. But I'd be surprised to hear that it's still a controversial issue for theoretical physicists (although what is known might depend on the geometry of the lattice)... I am sorry, but I really can't provide any information; my knowledge is limited to the mathematical aspects.
May
12
comment What is the interface tension between ordered and disordered phases of the Potts model?
@hlew: from the point of view of a mathematical physicist, there are plenty of open problems concerning the Potts model (including such basic ones as determining the order of the phase transition for the various values of the number of states and of the dimension). About what problem would still be considered open for a theoretical physicist, I unfortunately have no idea.
May
11
comment What is the interface tension between ordered and disordered phases of the Potts model?
@hlew: You're right. I should have checked ;) .
May
11
revised What is the interface tension between ordered and disordered phases of the Potts model?
Still more typos...
May
11
revised What is the interface tension between ordered and disordered phases of the Potts model?
Corrected a typo