1,829 reputation
1617
bio website unige.ch/math/folks/velenik
location Geneva, Switzerland
age 45
visits member for 3 years, 3 months
seen 29 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!


2d
comment Simple mean-field “lattice gas” model
You can have a look at the discussion in Section 4.9, in chapter 4 here.
Jul
2
comment Random walk recurrence term and the self-energy
@DanielSank : if you want more detail on such computations, this is spelled out very explicitly in Chapter 7 here.
Jun
24
comment Phase transition without the Peierls' counter argument
@MathOverview: Peierls' argument is much more robust, as it only uses the spin-flip symmetry of the model (and even that can be removed, which leads to the powerful Pirogov-Sinai theory). Of course, the computation of the free energy provides a lot of additional information.
Jun
24
comment Phase transition without the Peierls' counter argument
@MathOverview: no Onsager's proof (or the many other approaches to the computation of the free energy) does not depend on the Peierls argument. It is an algebraic approach that does rely on very specific features of the Ising model. – Yvan Velenik just now
Jun
24
comment Phase transition without the Peierls' counter argument
Onsager's computation implies the existence of a phase transition in dimension $2$ (for the nearest-neighbor model). Combined with correlation inequalities, this implies existence of a phase transition in any dimension $d\geq 2$, and interactions of any range (provided that nearest-neighbors also interact). So, yes, this is an alternative proof.
Jun
4
comment Local and global detailed balance
Well, I am not sure about the physicists's terminology, but I guess that they refer to local detailed balance for the condition you wrote (it should be $k(x,y)\rho(x)=k(y,x)\rho(y)$ (for all $x$ and $y$), by the way), and to global detailed balance for the weaker condition $\sum_x \rho(x) k(x,y) = \rho(y)$ (for all $y$). In Markov chain theory, the latter characterizes the stationary distribution, while the former implies in addition that the Markov chain is reversible w.r.t. this distribution.
May
31
comment Is 'Boltzon' an accepted name for particles following Maxwell-Boltzmann (MB) statistics?
Some people also use Boltzmannons. There is no final consensus on this terminology.
May
28
comment What is normal fluctuation?
He probably means that the fluctuations of the relevant quantity (here the energy) follow a normal distribution. Actually, here he seems happy with the weaker claim that the scale of these fluctuations are of the right order (the variance is of order $N$).
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.