1,517 reputation
516
bio website unige.ch/math/folks/velenik
location Geneva, Switzerland
age 44
visits member for 2 years, 2 months
seen 6 hours 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.


Jan
10
comment Divergent Series
A good place to start: terrytao.wordpress.com/2010/04/10/…
Jan
6
comment Probability of finding n particles in a volume v
I don't understand what you're trying to do in the second approach. How do you encode configurations? Just keeping track of how many of the $N$ particles are in $v$? If yes, then these are not equiprobable: just think of what happens when $v\ll V$; in that case, having all $N$ particles in $v$ has certainly not the same probability of having all of them in $V\setminus v$.
Dec
4
comment Physical intuition for independence of components of velocity in derivation of Maxwell–Boltzmann distribution
This assumption was actually the weakest point in Maxwell's derivations, as he fully realized himself. Indeed, in his subsequent paper "On the dynamical theory of heat" he found an alternative derivation avoiding this assumption. Later, Boltzmann further improved on these aspects in his paper "Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten".
Nov
28
comment Math needed for undergrad Statistical Mechanics/Thermal Physics
Well, it depends how far you want to go, but there is no way you can achieve a deep understanding of stat. mech. without having at least a reasonable understanding of probability theory...
Nov
20
comment Why isn't the 3 pts function vanishing in the Ising model by Z_2 symmetry?
If your computation is correct, yes, that seems to be the conclusion.
Nov
20
comment Why isn't the 3 pts function vanishing in the Ising model by Z_2 symmetry?
At zero external field, $n$-point functions with $n$ odd always vanish (when there is a unique infinite-volume Gibbs state, which is the case at the critical point). That's actually trivial if you start from the measure with free (or periodic) boundary conditions in a finite box, since then $\langle \sigma_i \sigma_j \sigma_k\rangle = -\langle (-\sigma_i) (-\sigma_j) (-\sigma_k)\rangle = -\langle \sigma_i \sigma_j \sigma_k\rangle$, by symmetry.
Nov
16
comment 2D Ising model simulations: Wolff algorithm acceptance probability with an external magnetic field
If the field is very close to zero, this should be ok. If not, then I don't see the point of using a cluster algorithm, just do spin flips.
Nov
10
comment Paramagnetism and large N
Well, if you assume that there is no interaction between the spins and that there is no external magnetic field, this follows immediately from the law of large numbers (for a collection of N independent Bernoulli random variables of parameter $1/2$).
Nov
8
comment Can Lee-Yang zeros theorem account for triple point phase transition?
Note also that if you have three phases, you should consider two external fields.
Nov
8
comment Can Lee-Yang zeros theorem account for triple point phase transition?
If I remember correctly, the paper by Biskup et al, General Theory of Lee-Yang Zeros in Models with First-Order Phase Transitions, arXiv:math-ph/0004003, Phys. Rev. Lett. 84, 4794–4797 (2000), discusses, among others, the Blume-Capel model (which has a triple point).
Nov
4
comment What is the origin of the kinetic theory of temperature?
Well, the early history of the kinetic origin of heat is discussed in en.wikipedia.org/wiki/History_of_heat .
Nov
2
comment What is known about the statistical mechanics of systems with normally distributed energies?
@wvoq: another place you can consult is Mezard+Montanari's book *Information, Physics and Computation", Chapter 5.
Nov
2
comment What is known about the statistical mechanics of systems with normally distributed energies?
@Trimok: This makes sense for a disordered model, such as the Edwards-Anderson model, the Sherrington-Kirkpatrick model, or the random energy (REM) model. In that case, the partition function is itself a random variable (depending on the realization of the disorder). What he asks for seems actually similar to the REM.
Nov
2
comment What is known about the statistical mechanics of systems with normally distributed energies?
This looks pretty much like the random energy model. See, for example, Chapter 9 in Anton Bovier's book Statistical Mechanics of Disordered Systems: A Mathematical Perspective.
Nov
2
comment Difference between theoretical physics and mathematical physics?
@DIMension10: The particular status of String Theory can also be seen in the fact that Witten has received the Fields medal, that string theorists regularly give conferences at the International Congress of Mathematicians, and that many math. departments have people working in this field. This shows that one can contribute (in important ways) to mathematics without necessarily doing rigorous work. Of course, not all mathematicians are happy with this kind of semi-rigorous maths. I don't have any problem with that myself, but I come from physics, after all...
Nov
2
comment Difference between theoretical physics and mathematical physics?
@DIMension10: Sure, there are various degrees of rigor, depending on the subfield. String theory has a quite distinct status, as it is often not fully rigorous, but yields many interesting conjectures. This entire field lies somewhere between Math (because of the advanced math. framework required, and the many fascinating predictions, etc.) and Physics (due to lack of direct experimental verification). But the situation is very different for the other fields. In most (all?) other fields, CMP publishes fully rigorous works. This is the case, for example, in my field, statistical physics.
Oct
31
comment Percolation and number of phases in the 2D Ising model
I am happy that this helped you. Concerning the rest of the paper, I should have a look. I must confess that the last time I read the paper must have been when it came out in 2000 or so. Can't say I remember the details... It is a quite beautiful proof, but the fact that I have now an argument that I find physically much more compelling (though probably less elegant) does not encourage me to read it again ;) . That being said, it is certainly true that if you understand this proof, then the rest should be much easier to read, since the same type of arguments are used again and again.
Oct
31
comment Percolation and number of phases in the 2D Ising model
Yes. And you use the fact that $\Gamma$ is specified from outside, so that $\mathcal{A}_{\Lambda,\Gamma}$ is $\mathcal{F}_{\Gamma^c}$-measurable.
Oct
30
comment Fluctuations of an interface with hammock potential
Well, I wouldn't be surprised if this has been done long ago (maybe for a variant of this model). There is actually no doubt that the result is as claimed above. The question is really about good heuristic arguments about why this should be true (this might in turn lead to a rigorous proof of this fact, which is what I'm after). Unfortunately, numerical simulations do not lead to any sort of real understanding... By the way, some colleagues have made substantial progress about this problem (in a slightly different setting), which I'll describe once their preprint is made available.
Oct
29
comment Percolation and number of phases in the 2D Ising model
I have noticed that you (again) asked the same question on Math.StackExchange. I think that in such a situation, you should always link both questions, so that readers from one site can access answers given on the other...