1,732 reputation
616
bio website unige.ch/math/folks/velenik
location Geneva, Switzerland
age 45
visits member for 3 years
seen 42 mins 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!


May
13
comment Statistical Mechanics vs Statistics
First of all, the terminology statistical mechanics is rather unfortunate, and it would have been much better to call this theory probabilistic mechanics, for example. Indeed, the concepts and techniques of probability theory play an essential role in statistical mechanics (even if they are, quite wrongly, somewhat hidden in many books and courses). Statistics, however, is mostly irrelevant to statistical mechanics.
May
11
comment 1 dimensional Ising model
@Math_overview: Well, the way the Hamiltonian and free energy are defined here, the $\log 2$ term is fine. What makes you think this is incorrect if you agree with the computations?
May
3
comment Percolation and number of phases in the 2D Ising model
@Math_overview: Thanks for your positive feedback on the book. I am happy you find it useful. There will be a chapter on the variational approach to (translation invariant) Gibbs states at some point, but for the moment, we're working on one chapter on reflection positivity and one on the Pirogov-Sinai theory.
May
3
comment Percolation and number of phases in the 2D Ising model
@Math_overview: yes, this is correct ($\nu$-almost surely, of course). See Chapter 6 of our book, in particular Section 6.3.1.
Apr
15
comment Legendre transformation: non-convex/non-convave functions
One deals with such situations by using the more general Legendre-Fenchel transformation (see the article "Convex conjugate" on wikipedia). Morally, this amounts to taking the Legendre tranformation of the convex envelope of your function (so you lose information in this way). This turns out to be the relevant conept in particular in statistical physics. In the latter, nonconvex thermodynamic potentials appear naturally in mean-field theories and, when this occurs, equivalence of ensembles is violated.
Mar
26
comment How to understand singularities in physics?
Not sure about your second part. Phase transitions are defined as singularities of thermodynamic potentials, because this is what one seems to observe experimentally. Statistical mechanics then tells you that these singularities are in fact only approximate (they only exist in the ideal limit of infinite systems). The "real" functions remain smooth, although they mimic very closely singular behaviour.
Mar
7
comment Phase Transition at Zero Temperature (Not QPT)
I still don't see what you mean. In the 1d Ising model, the thermal fluctuations restore the symmetry which is broken at $T=0$ (the model is symmetric whenever $T\neq 0$, but is not symmetric when $T=0$). This is the opposite of what you seem to say.
Mar
6
comment Phase Transition at Zero Temperature (Not QPT)
I am not sure I understand your analogy: in the 1d Ising model, $T=0$ is the only temperature at which the symmetry is broken. In any case, this is of course extremely general: the same will be true (at the classical level) for any one-dimensional model with compact spins, and periodic interactions. This, of course, includes models with continuous symmetry (the one-dimensional $O(N)$-models, for example).
Jan
2
comment Why are large scale structures isotropic in the Ising model?
Note that the latter result has been only proved in dimension $2$, and there it results from the fact that surface tension becomes isotropic in this limit, for very much the same reason the 2d random walk converges to Brownian motion.
Jan
2
comment Why are large scale structures isotropic in the Ising model?
What do you mean by "the phase separated 'blob' pattern that forms when the Ising model is quenched"? Do you mean the shape of the droplet of one phase immersed inside the other, when the magnetization is fixed at a value between $m^*$ and $-m^*$? If yes, then this shape is not rotation invariant, since the surface tension is not and the shape is obtained by the Wulff construction. It is true, however, that (properly rescaled) this shape converges to a disk as $\beta\downarrow\beta_c$.
Dec
29
comment What is the central charge of the disordered $q$-state Potts model, for large $q$?
The phase transition of the Potts model is first-order for $q\geq 5$ (in two dimensions), there is no critical point.
Nov
26
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.
Nov
26
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.
Nov
26
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.
Nov
25
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.
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.