1,879 reputation
1617
bio website unige.ch/math/folks/velenik
location Geneva, Switzerland
age 45
visits member for 3 years, 4 months
seen 1 hour 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!


15h
comment Is the MaxEnt “interpretation” of statistical mechanics the current mainstream approach?
Sure, but a interpretation of probabilities in statistical mechanics is the core of any derivation of the latter. MaxEnt provides a subjective interpretation of the latter, the ergodicity/mixing approaches attempt (and fail) to provide a mechanical interpretation. I don't see how large deviations theory can be considered as an "alternative" approach to MaxEnt in this respect, as it has exactly nothing at all to say about this issue.
1d
comment Is the MaxEnt “interpretation” of statistical mechanics the current mainstream approach?
I don't see what you mean by "getting the statistical physics framework based on the theory of large deviations". Large deviations theory, by definition, requires a probabilistic framework, which is precisely what is difficult to derive in an objective way from the underlying mechanical theory. The only thing that large deviations help you with is moving from one description (say, microcanonical) to another one (say, canonical). But you'd still need to derive the microcanonical probability measure in some way, and this is the hard part.
Aug
13
comment Critical temperature and lattice size with the Wolff algorithm for 2d Ising model
@ACuriousMind: Done (but it is a bit sketchy, as I don't have so much time now; in particular, it would be nice to provide a picture of the specific heat for the 2d Ising model on a finite torus, as well as the limit quantity).
Aug
13
comment Critical temperature and lattice size with the Wolff algorithm for 2d Ising model
This is a well-understood finite-size effect, resulting from the fact that the correlation length in your system cannot become larger than the system size. It implies a shift of the (apparent) critical point of order $1/n$, if you're on an $n\times n$ torus. AFAIK, this was first studied by Ferdinand and Fisher in 1969 (Phys. Rev. 185, 832).
Aug
10
comment What are the definitions of microstates and macrostates?
They are in the same macrostate if they share the same values for the relevant set of macroscopic observables (which usually also imply that they share all macroscopic properties). They are in the same microsctate... well, if they share the same microscopic state.
Aug
5
comment Why does Landau theory not fail when dealing with a first order phase transition?
Yes, basically you have to assume the existence of a metastable branch of the free energy to "justify" the procedure. Note though that this assumption is simply wrong for short-range models (it can, for example, be proved rigorously that there is an essential singularity in the Ising model, in any dimension, which prevents analytic continuation of the free energy beyond the transition point).
Jul
27
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.