There is a link between the vortex-unbinding Kosterlitz-Thouless transition in two dimensions and the two-dimensional Coulomb gas problem, discussed in detail by Minnhagen in the following publication:

[1] P Minnhagen, Rev. Mod. Phys. 59, 1001 (1987).

To answer your question, one may by analogy consider the thermodynamics of the Coulomb gas in one dimension, as dealt with exactly (under different assumptions) by  Lenard and Baxter in the following two publications:

[2] A Lenard, J. Math. Phys. 2, 682 (1961).

[3] RJ Baxter, Proc. Camb. Phys. Soc. 59, 779 (1963).

Briefly, at high temperatures the one-dimensional Coulomb gas behaves as an ideal gas of charged particles, and at low temperatures, as an ideal gas of neutral 'molecules'. You will have to consider the way the two phases connect.

Lastly, by considering the Kosterlitz-Thouless transition in two dimensions as a topological phase transition, you will also have to consider the extensively-studied topological phases of one-dimensional systems. The following book contains a good discussion of one-dimensional topological insulators:

[4] M Franz, and L Molenkamp, Topological Insulators (Elsevier, Amsterdam, 2013). 

Amongst others, consider also the following two important publications:

[5] L Fidkowski, and A Kitaev, Phys. Rev. B 83, 075103 (2011).

[6] AM Turner, F Pollmann, and E Berg, Phys. Rev. B 83, 075102 (2011).

Ps. See also the following important publication:

[7] D Ruelle, Commun. math. Phys. 9, 267 (1968).

How can there be a vortex in one dimension?

@ K. Kassner

Of the form considered by van der Zant et al. [1], van Oudenaarden et al. [2], and Grigorenko et al. [3].

[1] HSJ ven der Zant, et al., IEEE Trans. Appl. Superconductivity 3, 2658 (1993).

[2] A van Oudenaarden, et al., Phys. Rev. Lett. 77, 4257 (1996).

[3] A Grigorenko, et al., Nature 414, 728 (2001).

Dear Professors,

I am working on explaining the transition to turbulence under the perspective of statistical mechanics, Kosterlitz-Thouless transition in particular. So far not successful...

This post is based on the experimental observation of transition to turbulence in pipe flow, which is essentially a one-dimensional system.

Despite the linear stability of laminar profile of pipe flow at all values of Reynolds number, transition to turbulence occurs via the spatial proliferation of localized patches, called puffs, embedded in the laminar base flow. Puffs are arrowhead shaped region of turbulence bounded upstream by a sharp laminar-turbulent interface and downstream by diffusive streaky eddies. At increasing Reynolds number, an isolated puff increases in size and split into subsequent two puffs traveling downstream. The puff splitting resembles the vortex unbinding, and consequently, the onset of turbulence may be described by the Kosterlitz-Thouless transition.

Analog to the one-dimensional Coulomb gas as pointed out by Prof Farid, those laminar-turbulent interfaces can be regarded as "charged" sheet of positive and negative vorticity. However, unlike the Coulomb gas where the interaction falls off as 1/r^2, the velocity field induced by vortical sheet falls off as 1/r^3, resembling the magnetic dipole.

In fact, so long as the vorticity (in the form of thin vortex filaments) are localized and clogged the pipe, it resembles a dipole at large distances, independent of the details of the configuration of vortex filaments. Thus, I expect that the 1/r^3 decay of vorticity induced velocity field is genetic.

I have calculated the kinetic energy due to one vortical sheet located at the center of the pipe, which approaches to a constant at diverging system size. This means, those vortical sheets are always unbounded, and consequently, there is no Kosterlitz-Thouless transition in pipe flow. Thus, the underlying mechanism of observed puff splitting in pipe requires another explanation.

Thank you.

Wang Zhe

Dear Zhe,

The Kosterlitz-Thouless transition was published in 1972 showing that it was possible to find a phase transition in 2-D (spacelike dimensions) against the Mermin-Wagner theorem (1966). Thus the references older than these years are not valid.

This transition was modeled with a XY- two dimensional model at the Kosterlitz-Thouless critical temperature there is a transition from vortex-antivortex pairs to unpaired vorteces and antivortices.

But it is well known that there are no phase transitions at one dimension with short-range interactions. The reason is very simple to understand if you represent the entropy in 1-D and you can see that for T positive and large length this magnitude diverges.

Sometimes there are systems where time is used as one dimension more just using Wick trick or others for making Euclidean metrics, but this is only pure theory at least for the moment.

Dear Zhe Wang,

Following your remarks, I did some search, and it turns out that the problem you are working on has been studied in some detail, both experimentally and theoretically. I shall not go into details, but refer you to a very relevant and interesting talk by Professor Nigel Goldenfeld [1] (see in particular the diagram on page 39 of the lecture); many important works are explicitly cited in this text, such as those by Hof et al. (Phys. Rev. Lett. 101, 214501 (2008)), and Avila et al. (Science 333, 192 (2011)). The paper referred to in the title page of the lecture is that in [2].

[1] https://icmt.illinois.edu/Workshops/Urbana-ChicagoFest2016/GOLDENFELD_v4%204_ChicagoFest%20UIUC%20April%202%202016.pdf

[2] http://www.nature.com/nphys/journal/v12/n3/abs/nphys3548.html

Ps. You might also like to consult the following two publications:

[3] M Shimizu, and S Kida, Fluid Dyn. Res. 41, 045501 (2009).

[4] Masaki Shimizu, et al., Fluid Dyn. Res. 46, 061403 (2014).

Dear Prof Farid,

Thank you for referring me the talk by Prof Goldenfeld, which I am not aware of.

I have read other papers above, and in fact it is those experiments and theories motivate me to look for another possible explanation for the transition to turbulence.

Recently, there are accumulating literature reporting that the transition to turbulence in linearly stable flows belong to the DP class. It is however very uneasy for me to accept "the fact" that the transition to turbulence in the deterministic Navier-Stokes equation follows a fully stochastic process. I read almost all the papers that I can find to seek for a contradiction, and in fact there are many. The most prominent one perhaps that experiments also show that the puff splitting is promoted by increasing system size (i.e., pipe length) [1,2], which seems to be omitted in current theories. At that time, I overestimated my ability and changed my thesis topic (from electrical engineering), now a desperate PhD student having no idea if my work is totoally nonsense.

Nevertheless, as pointed out by Prof Baldomir, it is quite evident that there is no Kosterlitz-Thouless transition in one-dimensional pipe, but we can still calculate the probability of finding a second puff in the vicinity of an initial puff. Since in the absence of external perturbations, a new puff is seeded by vortex filaments disconnected from the initial puff [3], we can interpret this probability as the probability of puff splitting. My result shows that the probability of puff splitting increase linearly with the system size and exponentially with Reynolds number.

Outside the scope of pipe flow, the velocity field induced by filiform vortex filaments falls off like 1/r, leading to the logarithmic diverging of kinetic energy on the system size. Thus, I expect, Kosterlitz-Thouless transition may indeed takes place in filiform vortices dominated transitions, i.e, , in plane Couette flow.

PS, in my work I replace temperature by Reynolds number. Since temperature is just a derived quantity bridging energy and entropy and since Reynolds number is the only control parameter in the Navier-Stokes equation, I think this would be reasonable but not very sure. Please, could you point me out.

[1] M Nishi et al. J. Fluid Mech. 614, 425-446 (2008)

[2] D Moxey and D Barkley P. Natl. Acad. Sci. USA 107, 8091–8096 (2010)

[3] Avila et al. Science 333, 192 (2011)

Thank you.

Wang Zhe

Dear Zhe,

If you base your research on this subject just watching that:

the velocity field induced by filiform vortex filaments falls off like 1/r, leading to the logarithmic diverging of kinetic energy on the system size. Thus, I expect, Kosterlitz-Thouless transition may indeed takes place in filiform vortices dominated transitions, i.e, , in plane Couette flow.

This analogy could be quite dangerous; for instance, the electric field falls also as 1/r and the electric charges don't produce a KT phase transition by themself. There are more ingredients needed as a diverging correlation length inducing a quasi-long range order (having a non trivial topology transformation behind it). But your analogy is not so bad if you think in a II kind superconductor where you have Abrikosov-Nielsen vortices interacting (at distances shorter than the Peierls screening length) to prevent the coherence between the Cooper pairs which have a complex scalar field with a continous local symmetry U(1) (notice that you don't have this symmetry to be broken or at least I don't see it). Below the KT temperature you have superconductivity because there are not free vortices, only vortice-antivortice pairs can be present.

Thus your idea could be possible but more things are necessary than just the kinetic energy logaritmic dependence with relative distances. You need to have a correlation length and winding numbers different than zero (non trivial topological solutions).

Dear Wang Zhe,

I am not expert in the subject matter of your investigation, however the following comments may be relevant.

First, Daniel Baldomir's statement that 'there are no phase transitions at one dimension with short-range interactions' is in general false. The extant considerations concern systems in equilibrium, and, amongst others, for continuous systems the underlying particles / objects are assumed to interact through a hard-core potential, which, in addition, should not display 'unnatural' behaviour at long distance (see the next paragraph). The assumption of hard-core potential is vital in the considerations of Lee and Yang, since only for such potentials the partition function corresponding to a finite volume can be expressed as a finite-order polynomial of the fugacity z. This particular aspect is applicable both to classical and quantum systems.

Absence of phase transition in one dimensional equilibrium systems is often referred to as the van Hove theorem [1], dealt with in some detail in Sec. 5.6 of Ref. [2]. Here, the considerations are restricted to a lattice model and a continuous model with hard-core potential whose long range is not 'unnatural'. Note that the validity of the proposition 5.6.3 (the Perron-Frobenius theorem) is dependent on some very strict conditions, which are not valid in general. Remarkably, the KDP model (an ice-type model) has a finite-temperature phase transition in one dimension [3]. For a cursory discussion of the Lee-Yang theorems, and the relevant references to the original papers by Lee and Yang, consult Sec. 9.3 of Ref. [4].

Second, in my search of yesterday (about which I wrote in my previous comment on this page), I encountered a webpage [5] at Warwick University, which contains some very important details and references relevant to your research. I strongly recommend you to visit this webpage.

Ps. You must absolutely not 'replace temperature by Reynolds number'! To begin with, Reynolds number is dimensionless, while temperature is not. You should use the equation of state of your system and deduce the temperature dependence of the Reynolds number from it. Further, the Navier-Stokes equation being non-linear, it can have very surprising solutions.

[1] L van Hove, Physica 16, 137 (1950).

[2] D Ruelle, Statistical Mechanics (WA Benjamin, New York, 1969).

[3] JF Nagle, Amer. J. Phys. 36, 1114 (1968).

NB. KDP refers to Potassium [K] Dihydrogen Phosphate, KH2PO4.

[4] K Huang, Statistical Mechanics, 2nd edition (John Wiley & Sons, New York, 1987).

[5] https://homepages.warwick.ac.uk/~masax/Research/pipe_models/pipe_models.html

it is not necessary to discuss something very well known as:

thus there is no phase transition in one-dimensional systems with short-range interactions at T > 0

This sentence was just copy from:

https://en.wikipedia.org/wiki/Critical_dimension

and you have there three good references on the subject.

@ Daniel Baldomir

The Wikipedia entry that you refer to concerns systems with continuous symmetry! Everything crucially hinges on the broken symmetry of a continuous group and the attendant Goldstone modes that prevent ordering for T > 0 below the lower critical dimension (Mermin-Wagner theorem and its generalisations). Had you consulted Ref. [4] of my above comment, perhaps you would have paused before commenting.

Dear Benham,

I can understand that you believe in your references, but what cannot hope is that I needed to read them previously for giving my opinion (and frankly I'm not going to it in the future too). Everybody knows that KT transition doesn't exist in 1D (spacelike) and what I cannot understand is that even you can discuss people (i.e. Kassner) who told you that even one vortice in 1D is difficult to imagine. The physics of vortices, and the corresponding topological phase transitions, allowed one to avoid the theorems that opposed spontaneous symmetry breaking in 2D.

I know that the KT transition is remarkably different from the usual Ginzburg-Landau transition. It is a transition without an order parameter. There is a line of conformal fixed points, rather than a single temperature at which the theory is critical. This entire low temperature region displays correlation functions that are power laws of the separation between operators. In fact they can be interpreted withing the conformal group of transformations since the anomalous dimensions ( associated to the critical indices) are continuously varying with temperatures.

Finally, nowadays besides illustrating a defect driven topological phase dtransition, the related systems, the XY model and the two-dimensional (2D) Coulomb gas, became proving grounds for ideas related to the Wilsonian renormalization group.

P.S. I think that the references can be very interesting and useful but only it is worth the work to be taken and read seriously them, if the person convince previously us with some kind of argument before to pass such work.

Sorry the name is Behnam and no Benham.

Greetings, Daniel

@ Daniel Baldomir

Earlier I told you in no uncertain terms that I do not wish to undertake any further discussions with you. You broke some rules, which I neither forget nor forgive. The remark that I have made above, while referring to your name, was by no means addressed to you --- using your name was a necessity, in order to direct any potential reader of this page to the relevant comment.

As for one dimension, the vortices under discussion are not exactly one-dimensional, but quasi-one-dimensional -- they are perfectly three dimensional topological objects, but their dynamics is one-dimensional (this is understood by all concerned, notably the authors of the references that I have presented on this page, in response to K Kassner).

If you had paid attention, in my first comment on this page, I refer to the relationship between the Kosterlitz-Thouless transition and the Coulomb gas in 2D, and suggest that Zhe Wang would consider the Coulomb gas problem in 1D, for which I presented several relevant publications. I said no less and no more.

Lastly, please do not address me any longer, nor refer to me.

Dear Prof. Baldomir,

According to section 1.2 of the attached book chapter, it seems that there is a transition in the one-dimensional Ising model with a long-range interaction between the spins falling off with distance like 1/r2. In one dimension, the topological defect takes the form of domain wall, and I believe that this transition belongs to KT transition? Please, could you point me out?

Wang Zhe

Dear Behnam Farid,

You are even more stupid and crazy if you can defend that you can write in one post:

First, Daniel Baldomir's statement that 'there are no phase transitions at one dimension with short-range interactions' is in general false.

without refering to me. What do you think? Do you think that you only have the authority to say I am a wise man that you must obey and no put my name? I'm an adult as I think you are and the reasoning must be the only limit to put.

Do you think that, without never arguing nothing on the subject and only metioning some references are you helping to people? The references are only important going deepler in the arguments but never for substituting one personal opinion here.

As everybody can see, I am quite polite because I don't say what is my explicit opinion of what I think of your scientific opinions and only I have refered to your behaviour respect to me.

Dear Zhe,

You have only the possibility to analyse the behaviour in 1D and better if Anderson suggest it. But this doesn't imply that you are going to have a KT phase transition in 1D. Later on it is shown that this is not possible to exist. In fact what happens in 1D is that the correlation length between spins is:

Correlation length in 1D= 2Ta

being T the temperature and a the lattice constant (or distance between spins in one magnetic array). Thus you can see that this diverges as T tends to infinity. Notice that this correlation needed to be calculated from a common integral to several different dimensions.

On the other hand the spin correlation spacelike in 1D is

< S(r) S(0)> =exp (-kBT r/2Ja)

tell you that there are not magnetism. In 3D or higher you are going to obtain a very different value for the correlations and only in 2D you can get a topological non trivial phase which is associated to the KT transition phase.