1,602 reputation
516
bio website unige.ch/math/folks/velenik
location Geneva, Switzerland
age 44
visits member for 2 years, 8 months
seen 18 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. Your feedback would be more than welcome!


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.
Oct
15
comment Does this type of phase transition exist?
I don't think it matters that much for what you're interested in: you can always Legendre-transform to your favorite set of variables... For the proofs I mentioned, however, the choice of ensemble plays an important role.
Oct
15
comment Does this type of phase transition exist?
There are quite general results guaranteeing strict convexity of the pressure in $\beta$ (and any other parameters appearing linearly in the Hamiltonian), see projecteuclid.org/euclid.cmp/1103857626 (and more recent works citing the latter). Of course, there are known counterexamples, when their assumptions are violated, see for example link.springer.com/article/10.1007/BF01877543.
Oct
8
comment Definition of phase transitions in statistical mechanics
@gatsu: I was thinking of first-order phase transitions without symmetry breaking. It is very easy to construct models with such a property. There is actually a whole mathematical apparatus aimed precisely at establishing rigorously the existence of first-order phase transition in the absence of symmetry (by considerably extending the Peierls argument): the Pirogov-Sinai theory.
Oct
8
comment Definition of phase transitions in statistical mechanics
@gatsu: No, there are many phase transitions with no symmetry breaking.
Aug
27
awarded  Nice Question
Jul
27
comment Phase transitions. Conceptual link of my intuitive notions and definition of Georgii's book in terms of probabilities
Unfortunately, I don't know any book for physicists discussing the abstract mathematical framework (Gibbs measures, etc.). On the other hand, there are some online lecture notes for equilibrium statistical mechanics that I find quite good (they are for physics students, but they discuss the mathematical framework and explain its relevance). These notes are by Yoshi Oono, and can be found at yoono.org/Y_OONO_official_site/download.html . The relevant file is SMHyperS12.pdf (see the part on the course Phys 504 S12). I think that this might provide the type of explanations you want.
Jul
25
comment Phase transitions. Conceptual link of my intuitive notions and definition of Georgii's book in terms of probabilities
Unfortunately, I really don't have time to answer. But it might help you to have a look at introductory books on statistical mechanics. Alternatively (moving from the math side), you might benefit from reading more student-friendly introductions to this topic. Two nice ones are front.math.ucdavis.edu/9210.2732 (in particular section 2) and arxiv.org/abs/math/9905031 .
Jul
13
awarded  Good Answer
Jul
1
awarded  Critic
Jun
18
comment The critical point of Bose-Hubbard model
There's no reason why the critical value should be computable, in general. Of course, it's probably possible to get some value through various approximations. Actually, you don't even specify the dimension, and the critical value will clearly depend on that information. If you mean 3 dimensions, then you have to realize that one does not even know how to compute the critical value in the 3d Ising model!
Jun
5
comment Partition function of a gas of $N$ identical classical particles
@fatema: Sorry, for some reason I wasn't notified of your comment/request. In case you're still looking for it: wase.urz.uni-magdeburg.de/~kassner/itp2/spezialseminar/… .
Jun
3
comment 1 dimensional Ising model
For the change of variables, the situation is similar: fixing the leftmost spin to be $+1$ has the advantage that the transformation from the $\sigma$ to the $\eta$ variables is invertible, otherwise we would have to keep the value of the original leftmost spin $\sigma_1$. I chose not to impose anything at the right end of the interval, since otherwise this would induce a constraint on the product of the $\eta$ spins.
Jun
3
comment 1 dimensional Ising model
@sreeram: in each case, I chose boundary conditions leading to simple computations. FOr example, in the case of low temperature expansion, imposing the $++$ boundary condition would add a constraint that the number of "domain walls" has to be even; of course, this can be done, but it makes the argument slightly less immediate. [TO BE CONTINUED]
Jun
2
comment 1 dimensional Ising model
@Vibert: What you said is correct, but only for the low temperature expansion. The high temperature expansion always gives rise to collections of edges, in any dimensions (well, at least as long as we only consider two-body interactions).
Jun
2
comment Is Brownian motion a deterministic system?
Yes, you can derive (in a suitable limit) the stochastic description of Brownian motion from the purely deterministic dynamics of hard spheres, see the following recent and quite remarkable paper: arxiv.org/abs/1305.3397 .