What is nonequilibrium?

TL;DR

Maes' lecture notes define nonequilibrium statistical mechanics through trajectory-based ensembles governed by local detailed balance and entropy flux, connecting Boltzmann and Onsager perspectives to modern dynamical fluctuation theory. The work foregrounds the frenetic (time-symmetric) contribution as essential for understanding response, stabilization, and selection in driven systems, and develops macroscopic fluctuation theory as a unifying framework for nonequilibrium diffusive processes. Through toy models (Kac ring, modified Sutherland–Einstein relation, run-and-tumble dynamics) and broad phenomenology (thermoelectrics, convection, active matter), the notes offer a practical, path-space–centric toolkit for linking microscopic dynamics to emergent nonequilibrium structures with potential implications from soft matter to biology. The overarching goal is to provide a coherent, operational framework that blends entropy production, tempo-symmetric activity, and dynamical ensembles to understand, predict, and control systems far from equilibrium across physical disciplines.

Abstract

Lecture notes on elements of nonequilibrium statistical mechanics: (1) a characterization of the nonequilibrium condition, largely by contrast to equilibrium; (2) a retelling of some of the great performances of the more distant past, including the perspectives of Boltzmann and Onsager; and (3) more recent methods and concepts, from local detailed balance and the identification of entropy fluxes to dynamical fluctuation theory, and the importance of dynamical activity.

Figures (12)

• Figure 1: The red curve is the distribution for a finite (small) $N$ while the blue curve is the distribution for $N\uparrow +\infty$ corresponding to the equilibrium distribution for some energy function.
• Figure 2: $k(x,y)=a(x,y)e^{-\beta\left[\Delta(x,y)-E(x)\right]}$ with prefactor $a(x,y)=a(y,x)$ that depends on the specific path in the energy landscape. The $\Delta(x,y)=\Delta(y,x)$ measures the height of the barrier between states $x$ and $y$. E.g. $\Delta(x,y) = [E(x) + E(y)]/2$ as in \ref{['arh']}.
• Figure 3: A continuous time random walk on ${\mathbb Z}$.
• Figure 4: Left: three-level ladder. Right: the energy landscape, where $\Delta$ denotes the height of the barrier. A molecular switch can indeed be reproduced experimentally with a colloid moving in an optically simulated and flashing landscape.
• Figure 5: Stationary distributions of the three-level ladder given in Example \ref{['ex1']} for different values of $\alpha, \Delta$ and temperature $T$.

Theorems & Definitions (7)

• Example 3.2.1: Molecular switch
• Example 3.2.2: Ratchet
• Example 3.2.3: Parrondo game
• Example 3.2.4: Rower model
• Example 5.2.1: Peltier versus Seebeck
• Example 5.3.1: Action for Markov jumps
• Example 6.5.1: Blowtorch theorem, Landauer1993Machineryheatb