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bio website unige.ch/math/folks/velenik
location Geneva, Switzerland
age 44
visits member for 2 years, 5 months
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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!


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...
Oct
26
comment Percolation and number of phases in the 2D Ising model
BTW, you might be interested in a book that we are in the process of writing. At the moment, only one chapter has been uploaded, but several are in a rather advanced state and should be made available soon. The drafts of presentable chapters can be found on my homepage (select Book project).
Oct
26
revised Percolation and number of phases in the 2D Ising model
Typo corrected
Oct
26
revised Percolation and number of phases in the 2D Ising model
Better formulation
Oct
26
revised Percolation and number of phases in the 2D Ising model
Better formulation
Oct
26
answered Percolation and number of phases in the 2D Ising model
Oct
21
comment How do you prove the second law of thermodynamics from statistical mechanics?
@BenCrowell: I could not find any. But you might like the following paper (which I've just found): igitur-archive.library.uu.nl/phys/2011-0316-200303/uffink.pdf . I haven't read it, but it seems to discuss Lanford's results at length. Note also that there are many more recent derivations of the Boltzmann equation (under different sets of assumptions), and you should be able to find some on arXiv. One recent result of this kind that I find rather remarkable is this one: arXiv:1305.3397. There are other interesting references therein.
Oct
21
comment How do you prove the second law of thermodynamics from statistical mechanics?
@BenCrowell: I agree with you, of course. However, it is still possible to derive irreversible behavior from reversible dynamics in suitable limits, provided one excludes sets of initial conditions of zero measure (which does break time-reversal symmetry). This is the case of all rigorous derivations of Boltzmann's equation (see, e.g., Lanford's famous work: Time evolution of large classical systems, or its papers in Comm. Math. Phys. 9 1968 176–191 and Comm. Math. Phys. 11 1968/1969 257–292).
Oct
18
comment Emergent symmetries
One should also mention the notion of "enhancement of symmetry" (just google for it). One example: when $q$ is large enough, there is a range of temperatures at which the 2d $q$-state clock model (a discrete spin system invariant under a discrete subgroup of $SO(2)$) has a massless phase, that is, it behaves at large scales like a low-temperature 2d XY model. Everything occurs as if the symmetry group is enhanced to the full $SO(2)$. Another important example is the roughening transition.