# Statistical Mechanics & Dynamical Systems

As a (theoretical) physics student I've taken (advanced) undergrad courses in both statistical mechanics and dynamical systems (which was purely mathematical, treatment of nonlinear differential equations and chaotic maps), and will be headed to grad school in September.

I have a keen interest in phase transitions, non-equilibrium phenomena and bifurcation theory. It seems to me that there exists a strong connection between statistical mechanics and dynamical systems theory.

I wish to strengthen my knowledge of these two. Does anyone have article / text recommendations that would "bridge" or "link" those two fields?

• Concerning the application of the formalism of statistical physics to dynamical systems, classical references are Bowen's "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms" and Ruelle's "Thermodynamic formalism: the mathematical structures of equilibrium statistical mechanics". You can also find many lecture notes if you search for "thermodynamic formalism" on you favorite search engine. Commented Feb 16, 2019 at 10:13
• I second Yvan's recommendations. There is also the book "Ergodic Problems of Classical Mechanics" by Arnold and Avez which, if I remember correctly, has a nice discussion of Markov partitions and the Kolmogorov-Sinai entropy. These are the basic tools for encoding a dynamical system into a stat mech spin system. Commented Mar 3, 2019 at 17:25

A classic is Beck and Schögl's book, Thermodynamics of Chaotic Systems - An Introduction.

For the mathematically inclined, there is the well received Thermodynamics - A Dynamical Systems Approach. MathOverflow offers other suggestions in this vein.

It's always good to take a look at highly-cited papers a search for the appropriate terms in Google Scholar and similar reveal, such as Probability, ergodicity, irreversibility and dynamical systems or Probabilistic and thermodynamic aspects of dynamical systems.).

Entropy is certainly a unifying concept. If you're interested in shorter texts, Appendix A36 - Thermodynamic formalism [of dynamical systems] from the ChaosBook.org briefly covers Rényi entropies and fractals; and some old answers of mine describe topological and KS-entropies and how they can be calculated for a concrete example.

From the same online book, Appendix A39 describes a deep connection between statistical mechanics and dynamical systems:

A spin system with long-range interactions can be converted into a chaotic dynamical system that is differentiable and low-dimensional. The thermodynamic limit quantities of the spin system are then equivalent to longtime averages of the dynamical system. In this way the spin system averages can be recast as the cycle expansions. If the resulting dynamical system is analytic, the convergence to the thermodynamic limit is faster than with the standard transfer matrix techniques.

Here on the site, one can also see that there is at least some analogy between bifurcations in dynamical systems and phase transitions, and, if I'm allowed to advertise yet another answer of mine, a related question is Can Chaos Theory be used to explain the Ising model in paramagnetic phase?.

Renormalization Group (RG) flows are quite important in Statistical Physics and these flows are analyzed as dynamical systems. Many references are suggested in Are there any known models with limit cycles in their RG flow?.

For serious readers the list of book are meantioned below https://www.springer.com/gp/book/9780387971414 https://www.springer.com/gp/book/9780387970684 https://www.springer.com/in/book/9780387001777 https://www.springer.com/in/book/9783642967993 https://www.springer.com/in/book/9781461210412 The above mentioned books are really good for Intermediate level guys.