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bio website unige.ch/math/folks/velenik
location Geneva, Switzerland
age 44
visits member for 2 years, 7 months
seen 7 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!


1d
comment Why doesn't the 1 dimensional ising model have a transition temperature?
I should also mention that a precise discussion of the relation between the local behavior of finite and infinite two-dimensional Ising models is given in Section 3.8.6. It requires some terminology, but the latter is explained earlier in the chapter.
1d
comment Why doesn't the 1 dimensional ising model have a transition temperature?
You might also have a look at Figure 1.11 in this book in progress. It shows the behavior of the magnetization in the 2d Ising model for increasing system sizes (including the infinite system). You can see that the curve is smooth for finite systems, but nevertheless approximate better and better the non-smooth one you get in the thermodynamic limit. BTW, the one-dimensional Ising model is discussed there in Section 3.3.
1d
comment Why doesn't the 1 dimensional ising model have a transition temperature?
I can guarantee you that it is perfectly sound. Just look at your free energy: at finite $N$, it is an analytic function of the temperature (more generally, the partition function is just a polynomial in $e^{-\beta}$ with nonnegative real coefficients, so its logarithm is real-analytic)! The only way you can generate non-analyticities is in the thermodynamic limit. This is a general fact. Of course, "real" phase transitions (i.e., for infinite systems) are well approximated by the behavior of large finite systems, which is why you can observe (apparent) singular behavior in finite systems too.
2d
comment Why doesn't the 1 dimensional ising model have a transition temperature?
Phase transitions (in the sense of nonanalyticities of thermodynamic potentials) can only exist in the thermodynamic limit. So, you have to fix $T>0$ and consider what happens in the limit $N\to\infty$. Note also that, for any finite $N$, typical samples will be perfectly ordered (i.e., magnetization equal to $1$) with very high probability provided you take the temperature sufficiently low, but this is just a finite-size effect, which disappears in the thermodynamic limit.
Oct
6
answered How to calculate critical temperature of the Ising model?
Sep
30
answered Ising model on lattices with (vertical side length) $\neq$ (horizontal side length)
Sep
26
revised Does the q-states Potts become the XY model in large q state?
added 86 characters in body
Sep
26
revised Does the q-states Potts become the XY model in large q state?
Added a precision
Sep
26
answered Does the q-states Potts become the XY model in large q state?
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
30
answered Axioms behind entropy!
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.