Yvan Velenik

1,241 reputation

314

bio	website	unige.ch/math/folks/velenik
	location	Geneva, Switzerland
	age	43
visits	member for	1 year
	seen	10 hours ago
stats	profile views	94

Professor at the Mathematics department of the University of Geneva. Mostly working at the intersection of probability theory and statistical physics.

	1,241 reputation	bio	website	unige.ch/math/folks/velenik	visits	member for	1 year
314 badges		location	Geneva, Switzerland		seen	10 hours ago

summary answers questions tags badges favorites bounties reputation activity

136 Actions

suggestions reviews revisions comments badges posts accepts all

Mar 12	comment	Measure of Lee-Yang zeros @genneth: I'm not sure I see which of my comments you are adressing... If you mean to say that we can be sure that the free energy is real analytic in finite systems, then yes, of course I agree. But if you don't make it quantitative, this does not tell you anything of interest about the limiting behaviour. Concerning your other comment (to WNY), the LY approach can be used to derive consequences of analyticity of the free energy to other quantities (say, exponential decay of correlations, for example); many interesting results were obtained this way a few decades ago.
Mar 3	comment	Measure of Lee-Yang zeros @WNY: I guess that it depends what information you want to extract (obviously, if the coupling constants are uniformly bounded and decay fast enough with the distance between the corresponding spins, then there won't be a phase transition in the model). Now, independently of what you want to extract, I'd guess that the determination of the asymptotic locations of zeros is certainly difficult (and usually impossible), in general, when the interaction is not finite-range and periodic...
Mar 2	comment	Measure of Lee-Yang zeros @genneth: you're right. Note, however, that, as far as I know, one can say essentially nothing about the limiting density (in dimensions 2 and higher, at low temperature). It is, e.g., not known whether the distribution of zeros in the thermodynamic limit allows analytic continuation from {Re(h)>0} to {Re(h)<0} (it is only known that such an analytic continuation cannot be done through h=0, since there is an essential singularity there).
Feb 13	comment	Thermodynamic limit “vs” the method of steepest descent @user6818 : this is standard textbook material. You should read, e.g., chapter VIII in this nice book.
Dec 12	comment	Mermin-Wagner theorem in the presence of hard-core interactions @RonMaimon: Any news? If you have thought about it and concluded that it is indeed less trivial than you expected, then I'd also be interested to know it (although, of course, I'd prefer a solution ;) ).
Oct 12	comment	Mermin-Wagner theorem in the presence of hard-core interactions @RonMaimon: Great, I'm looking forward to reading your argument.
Oct 12	comment	Mermin-Wagner theorem in the presence of hard-core interactions @RonMaimon: But would such an argument: (i) be rigorous (or might be made so by working hard enough), as real-space renormalization behaves very badly from a mathematical point of view; (ii) give us the invariance of all the infinite-volume Gibbs states (in the usual sense of mathematical statistical physics, i.e., solutions of the DLR equations)? Thanks for all the time you spend on that :) . I'd really like to understand your approach better. Do you have a ref. for a renormalization group approach to MW (even non rigorous, of course)?
Oct 11	comment	Mermin-Wagner theorem in the presence of hard-core interactions [continuation] The difficulty when hard-core is present is that most configurations become forbidden (have infinite energy) after deformation by the spin wave. One can then try to resort to a configuration dependent deformation (this is what is done by Richthammer in his treatment of the hard discs model), but this is technically quite difficult, unfortunately.
Oct 11	comment	Mermin-Wagner theorem in the presence of hard-core interactions @RonMaimon: I think that you're somewhat unfair (or misinformed) when you say "your methods are not in the spirit of Mermin-Wagner". The methods we used (which were introduced in the late 1970s by Dobrushin&Shlosman and by Pfister) seem to me very much in the spirit of MW: roughly speaking, what they do is compare the (finite-volume) Gibbs measure before and after deforming the configurations by a global spin wave, and show that the effect of this deformation becomes negligible when the system size gets big (and thus the wave-length large). [to be continued]
Oct 11	comment	Mermin-Wagner theorem in the presence of hard-core interactions I am looking forward to seeing your argument. Note that in the rigorous statistical mechanics community, the extension beyond smooth interaction potentials (as we did) remained open for some 20 years (read the mathscinet review if you don't believe that). So, if a simple approach can solve this can kind of problems and more, that would be great.
Oct 11	comment	Mermin-Wagner theorem in the presence of hard-core interactions great, I'm looking forward to seeing that :) .
Oct 10	comment	Mermin-Wagner theorem in the presence of hard-core interactions Still I think that you misunderstand my very question: I don't care about the precise counter-example we gave in the paper (and which was just a remark in the paper). What I care about is a (not necessarily rigorous) mathematical argument that derives absence of continuous symmetry breaking in situations where you have a hard-core interactions. You have not given me that, unfortunately. As I said above, even a nice heuristic (but mathematical!) argument in the case of non-differentiable interactions would be interesting (even though we treated those in the paper).
Oct 10	comment	Mermin-Wagner theorem in the presence of hard-core interactions I understand the Mermin-Wagner theorem very well, thank you. You seem to be misunderstanding the very definition of an infinite-volume Gibbs state, that's the single reason we disagree, I think... Note that the Gibbs state we obtain has fluctuations (non zero variance of difference of angles between distinct spins). (Or at least, this should follow from a more careful analysis along the lines of our proof...)
Oct 10	comment	What are some critiques of Jaynes' approach to statistical mechanics? [part 3] As I said above, this can't work without some further assumptions, because all provable statements will require taking initial conditions outside a set of measure zero w.r.t., say, Liouville measure. But there is no reason a priori to define typicality in terms of the Liouville measure (and this can't follow from the Hamiltonian evolution, as the latter has nothing to say about initial conditions).
Oct 10	comment	What are some critiques of Jaynes' approach to statistical mechanics? [part 2] In particular, the interpretation of probabilities in Jaynes' approach is as a description of the state of knowledge of the observer. One might, in principle, hope to give a more mechanical meaning to these probabilities (that's exactly one the other approaches, based on mixing properties of the dynamics) are trying to achieve. [To be continued]
Oct 10	comment	What are some critiques of Jaynes' approach to statistical mechanics? @JoséFigueroa-O'Farrill : The only information one retains from the micrscopic theory in Jaynes' approach is the Hamiltonian, basically. In his approach, the precise form of the microscopic dynamics (say, Hamiltonian evolution) does not play any role. In this sense, he does not derive Statistical mechanics from the apriori more fundamental microscopic theory. [to be continued]
Oct 9	answered	What are some critiques of Jaynes' approach to statistical mechanics?
Oct 9	comment	What are some critiques of Jaynes' approach to statistical mechanics? @genneth: Yes, but as a satisfactory theoretical framework for "non-equilibrium statistical mechanics" does not exist at the moment, discussing its foundational issues seems less important ;) . (I am not talking about linear response theory and such, obviously.)
Oct 9	comment	What are some critiques of Jaynes' approach to statistical mechanics? @genneth: I think that the reference you give has been published in Maximum Entropy Formalism, R. D. Levine and M. T. (Eds), The MIT Press (1978).
Oct 9	comment	What are some critiques of Jaynes' approach to statistical mechanics? @Matt Reece : But apparently, Section IIB is only a criticism of Jaynes' elementary approach to the second principle of thermodynamics. It does not seem to give any argument against MaxEnt as a logical foundation for equilibrium statistical mechanics.