1,542 reputation
516
bio website unige.ch/math/folks/velenik
location Geneva, Switzerland
age 44
visits member for 2 years, 4 months
seen 9 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!


Sep
16
comment MIcrocanonical and Canonical - The thermodynamic limit
You need to fix the total energy in your microcanonical computation.
Aug
30
comment Axioms behind entropy!
No. The approach is extremely general (much more so than previous ones). Concerning phase transitions, read the few lines just after the statement of the theorem you quote.
Aug
29
comment Can an Ergodic dynamical system approach equilibrium?
I am not sure I understand your concern. I assume that your dynamical system is measure-preserving (as is the relevant setting for application to stat. mech.). In that case, it is clear that the system does not "spend its future time in a smaller region of the phase space", at least not as measured with $\mu$. Maybe you'd enjoy reading this very nice paper, which although not discussing specifically the issue you have, might well clarify things for you.
Aug
2
comment Why is there a 'loophole' in Mermin Wagner for rotations?
Yes, that would be one way of stating it. It's pretty clear that one can have "almost crystalline" structures in 2d, even experimentally (graphene!). The reason a strict breaking of translation invariance does not occur is the presence of large distance fluctuations (the analogue of spinwaves in this context). But they generate deformations that are much too weak to restore rotation invariance. I am aware of the paper you mention, which is quite nice, but, in my opinion, it is still too ad hoc to really improve our understanding of the situation.
Aug
1
comment Precise statement of Mermin–Wagner theorem
@leongz: as I stumbled again on this question, I realized that I misunderstood your question about isotropy. In the $2d$ n.n. XY model, for example, the interaction between two spins at neighboring vertices $i$ and $j$ takes the form $-J \vec S_i\cdot \vec S_j$ (the spins are unit-vector in $\mathbb{R}^2$) and isotropy is essential in the sense that if we replace the above interaction by $-J_1 S_x(1) S_y(1) - J_2 S_x(2) S_y(2)$, with different constants $J_1$ and $J_2$, then there would be spontaneous magnetization at low temperatures. This is automatic in the settings of the above answer.
Jul
31
comment Why is there a 'loophole' in Mermin Wagner for rotations?
[...] Rigorous proofs that translation invariance is not broken have been obtained by Fröhlich and Pfister, Commun. Math. Phys. 8, 277 (1981) (in the absence of hard-core) and Richthammer, Commun. Math. Phys. 274, 81 (2007) (including hard-core).
Jul
31
comment Why is there a 'loophole' in Mermin Wagner for rotations?
From a technical point of view, it is the fact that under such a rotation, far away atoms "move with arbitrarily large speed". There are no rigorous proof that there is a breaking of rotation invariance in two-dimensional particle systems (one expects existence of "soft crystals", which would break rotation invariance); there are heuristic arguments (I think you can find a discussion in Nelson and Halperin's paper in Phys.Rev.B 19, 2457 (1979). [...]
Jul
7
comment Numerical Ising Model - Wolff algorithm and correlations
@Learningisamess: Let me reiterate what I said as a comment to the other answer. You can completely avoid the issue of determining the number of steps required to thermalize: just use a perfect simulation algorithm; when the latter stops, you are guaranteed that the system has thermalized! One such (cluster) algorithm is described in this paper: M. Huber, a bounding chain for Swendsen-Wang, Random Structures & Algorithms, 22(1) :43–59, 2003. It is trickier to implement than the standard algorithm, but it is very efficient and you get all the benefits of perfect simulation.
Jul
6
comment Numerical Ising Model: Swendsen–Wang algorithm, Percolation theory?
If you want to understand the relation between the Swendsen-Wang algorithm and percolation, relevant keywords are Fortuin-Kasteleyn's random cluster representation and Edwards-Sokal coupling. There exist several mathematical analyses of relaxation properties of this dynamics. See for example Mario Ullrich's PhD thesis and references therein; the latter can be found on his homepage.
Jul
5
comment Numerical Ising Model - Wolff algorithm and correlations
Why not implement a perfect simulation algorithm? Then you'll be guaranteed that each sample is drawn without any statistical error from the Gibbs distribution. It is possible to do that also with cluster algorithm. The one I implemented (years ago, to make illustrations for a course I was giving) was based on this paper: M. Huber. A bounding chain for Swendsen-Wang. Random Structures Algorithms, 22(1) :43–59, 2003. It's a bit tricky to implement properly, but very efficient, and you get all the benefits of perfect simulation.
Jul
2
comment Critical temperature difference between Ising and XY model
No it's just that for the Ising model, the loop and spin models coincides. This is really due to the fact that the spins take values $\pm 1$, since this implies that $e^{\beta \sigma_i\sigma_j} = \cosh(\beta)(1+\tanh(\beta)\sigma_i\sigma_j)$. (Of course the prefactor $\cosh(\beta)$ in the right-hand side has no impact on the probability distribution.)
Jul
1
comment Critical temperature difference between Ising and XY model
I have no idea what the value of the coupling constant of the spin O(2) model on the triangular lattice is. There are very efficient cluster methods for this model (as far as I know, I am not a specialist in the numerical aspects) which should provide precise estimates.
Jul
1
comment Critical temperature difference between Ising and XY model
The value of the critical points of the spin and loop models should not be related at all. However, the critical behaviors are expected to be the same (they should belong to the same universality class).
Jul
1
comment Critical temperature difference between Ising and XY model
No, the loop model is an approximation of the O(n) spin model. You can look, for example, at the explanations in Nienhuis' Les Houches lectures on Loop models, which can be downloaded from his homepage.
Jun
28
comment Critical temperature difference between Ising and XY model
Your intuition is correct. It is not difficult to prove that the 2-point function of the Ising model is always an upper bound on the corresponding quantity for the XY model. In particular, if $\beta_c^{XY}$ denotes the inverse temperature at which the Kosterlitz-Thouless phase transition occurs, then $\beta_c^{XY}\geq 2\beta_c^{I}$, where $\beta_c^I$ is the inverse critical temperature for the Ising model. You can find the proof here. Note that the formula you give above is not for the $O(n)$ spin model, but for the $O(n)$ loop model.
Jun
24
comment Is there a reasonable lower bound for free energy per site of the 2D Ising model in the presence of an external field?
I am not sure that I understand your question. There are many possible explicit lower bounds on the free energy. What additional properties do you want? For example, a well-known lower bound on the partition function (and thus on $\psi$) is given by the mean-field partition function.
Jun
5
comment Why is the application of probability in QM fundamentally different from application of probability in other areas?
@NikosM.: he says "there are no known examples of classical probabilities that don't have a quantum mechanical origin", and wants to reduce it to QM. He's not limiting his discussion to application of PT to QM, but speaking of PT in general. From this point of view, this is plain wrong.
Jun
4
comment Why is the application of probability in QM fundamentally different from application of probability in other areas?
@CountIblis: You're missing my point. There is no need to have any randomness to apply probability theory. So there is absolutely no need of quantum mechanics to derive or justify classical probability theory. To say that it requires quantum mechanics because the universe is fundamentally quantum, would be the same as saying that, say, literature is reducible to quantum mechanics because the brains of writers are quantum objects... Maybe true, but utterly useless.
Jun
4
comment Why is the application of probability in QM fundamentally different from application of probability in other areas?
@CountIblis: Even if in a fully deterministic universe, probability theory would be indispensable. Including in its application to physics. You (and the authors of this paper) have an extremely narrow view on this field.
Jun
3
comment Why is the application of probability in QM fundamentally different from application of probability in other areas?
Real randomness (inasmuch as such a thing exist) might indeed have quantum origin, but one does not need anything truly random to be able to apply probability theory successfully, just situations of sufficient complexity that renouncing a precise description is the only way to proceed.