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bio website unige.ch/math/folks/velenik
location Geneva, Switzerland
age 45
visits member for 3 years
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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
26
comment Fluctuations in energy for macro and micro canonical ensembles
Consider a super simple example: you have $N$ objects each of which may be either in its "ground state" of energy $0$ or in an "excited state" of energy $\epsilon$. There is no interaction between these objects. The partition function is $\sum_{k=0}^N\binom{N}{k}e^{-\beta\epsilon k}=(1+e^{-\beta\epsilon})^N$, and thus the probability that the energy is equal to $E=M\epsilon$ is $\binom{N}{M}e^{-\beta\epsilon M}/(1+e^{-\beta\epsilon})^N$, which you can check develops a sharp peak (of width of order $\sqrt{N}$) around its expected value $N\epsilon/(1+e^{\beta\epsilon})$.
May
25
comment Fluctuations in energy for macro and micro canonical ensembles
The Boltzmann distribution is not an exponential distribution: the Boltzmann weight has certainly an exponential form, but you're forgetting the entropy (that is, how many microstates of a given energy there are). The distribution of the energy density in the canonical ensemble concentrates sharply on its average (with fluctuations of the order of $1/\sqrt{N}$). This is why you get equivalent results for both microcanonical (fixed energy) and canonical (fluctuating energy) ensembles (well, at least in the absence pf phase transition).
May
23
answered Minimum connectivity required for mean field to be a good approximation?
May
23
comment Minimum connectivity required for mean field to be a good approximation?
If you're interested in critical exponents, then you can, in principle, extract (probably bad) bounds from this paper: arXiv:0712.0312.
May
23
comment Minimum connectivity required for mean field to be a good approximation?
Well, a good approximation in what sense? If you want, say, close free energy and magnetization, or the ability to say that the system has a first order phase transition if its mean-field approximation has, then you might be able to extract estimates (although rather pessimistic, I suspect) from this paper: arXiv:math/0207242. There are also two subsequent papers, if I remember correctly.
Apr
28
revised What happens to the free energy of the two-dimensional ising model with vortices?
Trivial typo corrected
Apr
27
revised What happens to the free energy of the two-dimensional ising model with vortices?
trivial typos corrected
Apr
27
revised Fluctuations of an interface with hammock potential
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Apr
27
revised Fluctuations of an interface with hammock potential
added 365 characters in body
Apr
27
answered What happens to the free energy of the two-dimensional ising model with vortices?
Apr
26
comment What happens to the free energy of the two-dimensional ising model with vortices?
Yes, I agree, it will affect the finite-volume free energy density by a term of order $1/N$ (that is, the energy associated to the defect lines, of order $N$, divided by the total volume, of order $N^2$).
Apr
26
awarded  Yearling
Apr
26
comment What happens to the free energy of the two-dimensional ising model with vortices?
I am not sure what you call vortices in the context of the Ising model (you can have them in continuous spin systems of course). What you can do (and maybe what you mean) is to force the presence of interfaces in the system. Forcing the presence of finitely many will not affect the (limiting) free energy density.
Apr
25
comment Ising model. What is large fluctuations of magnetization?
@Elias: if you have specific questions about our book, please do not hesitate to ask us directly (the relevant email address is given on the book page). For us, it is important to know what are the parts that should be clarified.
Mar
28
comment Symmetry Breaking And Phase transition
@Adam: But, stricto sensu, at the critical point, there is no transition from gas to liquid phases (the two are undistinguishable at that point). So, I keep with what I said: there is no symmetry breaking at the liquid/gas phase transition.
Mar
28
comment Symmetry Breaking And Phase transition
@Adam: But the question was "Is every phase transition associated with a symmetry breaking? If yes, what is the symmetry that a gaseous phase have but the liquid phase does not?", and the answer is negative.
Mar
28
comment Symmetry Breaking And Phase transition
@Adam: your argument does not apply to the first order phase transition between the liquid and gas phases, which I am discussing.
Mar
28
comment Symmetry Breaking And Phase transition
Note that it is easy to construct models with no symmetry in which 1st order phase transitions occur (actually this is the generic situation! You have to work harder to construct models with symmetry breaking). There is a mathematical theory devoted to this problem: the Pirogov-Sinai theory.
Mar
28
comment Symmetry Breaking And Phase transition
@Adam: It's just one rough model of the liquid/gas transition, and the symmetry is accidental. You can actually consider infinitely many perturbations of the Ising lattice gas, which we'll be just as good approximations to the real fluids, and in which there is no symmetry. There is no reason to expect that there is any, even hidden, symmetry for the real liquid/gas transition.
Mar
26
comment Symmetry Breaking And Phase transition
@Roopam : yes, of course they are not always related to symmetry breaking.