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This is a soft question. I am currently studying statistical mechanics and I found this one by chance: Algebraic statistical mechanics

And I also found some workshops on interactions between statistical mechanics and geometry, as follows(UPDATE):

Symplectic and Algebraic Geometry in the Statistical Physics of Polymers: October 12-16, 2015

Geometry, Statistical Mechanics, and Integrability

It is well known that algebraic geometry is quickly developing, and it has a lot of applications in various fields in physics, and I also think that it may be a key to understand physical phenomena, because algebra expresses the essence while geometry unveils the structure. I really willing to know:

  1. Is it possible to apply algebraic techniques in statistical physics? If one can apply, then usually what kind of purpose? What kind of information can be gotten algebraically?
  2. More specifically, I am very much interested in nonequilibrium processes (for example, chemical reactions between gas molecules), because it is very central in statistical physics. I would like to know where there has been any attempt to describe these things algebraically/geometrically.

Any suggestions are welcome.

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    $\begingroup$ You may find the works of people like Borodin and Okounkov interesting $\endgroup$
    – Martin Hairer
    Commented Mar 10 at 23:17
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    $\begingroup$ Could you please provide more details of their works? $\endgroup$
    – FFjet
    Commented Mar 11 at 6:19
  • $\begingroup$ Borodin and collaborators: '(quantum) integrable probability', for the 2d classical stat mech models that I mentioned, and connections to symmetric functions $\endgroup$
    – Jules Lamers
    Commented Mar 11 at 9:43
  • $\begingroup$ I'm not sure which works of Okounkov @MartinHairer has in mind, but he's certainly very famous for the geometric representation theory that I mentioned $\endgroup$
    – Jules Lamers
    Commented Mar 11 at 9:45
  • $\begingroup$ .. and then there's also the work of Okounkov--Kenyon on dimer models in the early 2000s $\endgroup$
    – Jules Lamers
    Commented Mar 11 at 9:48

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The answer depends on the precise interpretation of the question. Carlo Beenakker's answer mentions a possible (general, universal) algebraic approach to the field as a whole.

Let me complement this with certain specific examples that are particularly amenable to (specialised) algebraic methods, related to the topic from the slides in the OP. Note that I will focus on algebra, or more precisely: representation theory. If you're after algebraic geometry, I hope someone else can give pointers.

Roughly speaking, a model from physics is quantum integrable if various quantities of physical interest can be computed using representation theory of quantum groups. (As an aside, part of this is closely related to geometric representation theory, which is maybe closer to what you're after, but I am not aware if that has connections to nonequilibrium physics.) To evaluate thermodynamic properties as the number of particles tends to infinity, this has to be supplemented by (often nonrigorous) methods. In the context of statistical mechanics, quantum integrable models come in two broad classes: (i) as quantum mechanical systems in $1+1$ dimension (time + space), and (ii) as classical statistical mechanics in $0+2$ dimensions (space only, usually a lattice).

Quantum integrability can be used in some nonequilibrium settings. I am not an expert, so the following is probably biased and surely incomplete.

  • The prototype is a stochastic transport model called the asymmetric simple exclusion process (ASEP). It is closely related to the Heisenberg XXZ spin chain (first class above) and six-vertex model (second class), and is based on the representation theory of Hecke (or Temperley-Lieb) algebras and quantum enveloping or affine algebras (usually of $\mathfrak{gl}_2$). See e.g. this review.
  • Another topic that comes to mind are so-called 'Lindblad (super)operators'. I didn't find a review.
  • For some other aspects of non-equilibrium integrability, you can try the Les Houches lecture notes 2018-08: Integrability in Atomic and Condensed Matter Physics.
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  • $\begingroup$ I am not sure if this is even vaguely in the direction you're after, but hope it is of some help $\endgroup$
    – Jules Lamers
    Commented Mar 10 at 22:41
  • $\begingroup$ Thanks a lot for your answer! It sounds interesting. $\endgroup$
    – FFjet
    Commented Mar 11 at 6:10
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An algebraic approach is used in physics to develop a rigorous theory of systems with an infinite number of degrees of freedom, as they appear in quantum field theory and in the thermodynamic limit of statistical mechanics. This is textbook material, see for example

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