Macroscopic Stochastic Thermodynamics

TL;DR

This work extends stochastic thermodynamics to the macroscopic limit by connecting mesoscopic Markov jump dynamics to a deterministic macroscopic drift and an extensive thermodynamic structure. Using large-deviation theory, it builds a macroscopic fluctuation theory that preserves fluctuation theorems, reveals a quasi-potential as a Lyapunov function, and clarifies when Gaussian Langevin reductions fail thermodynamically. It recovers equilibrium and near-equilibrium theories (Einstein–Onsager, Onsager–Machlup) and provides a coherent framework for far-from-equilibrium phenomena via Freidlin–Wentzell theory, including metastability, attractor transitions, and a coarse-grained emergent ST on the space of attractors. Importantly, it demonstrates how dissipation splits into adiabatic and nonadiabatic components, yields bounds on transition rates between attractors, and derives a field-theoretic macroscopic limit (Macroscopic Fluctuation Theory) that clarifies entropy production in continuous space. The framework is illustrated with chemical reaction networks, electronic circuits, and driven Potts models, highlighting the practical relevance for understanding energy dissipation, phase transitions, and robust coarse-grained descriptions of complex nonequilibrium systems.

Abstract

Starting at the mesoscopic level with a general formulation of stochastic thermodynamics in terms of Markov jump processes, we identify the scaling conditions that ensure the emergence of a (typically nonlinear) deterministic dynamics and an extensive thermodynamics at the macroscopic level. We then use large deviations theory to build a macroscopic fluctuation theory around this deterministic behavior, which we show preserves the fluctuation theorem. For many systems (e.g. chemical reaction networks, electronic circuits, Potts models), this theory does not coincide with Langevin-equation approaches (obtained by adding Gaussian white noise to the deterministic dynamics) which, if used, are thermodynamically inconsistent. Einstein-Onsager theory of Gaussian fluctuations and irreversible thermodynamics are recovered at equilibrium and close to it, respectively. Far from equilibirum, the free energy is replaced by the dynamically generated quasi-potential (or self-information) which is a Lyapunov function for the macroscopic dynamics. Remarkably, thermodynamics connects the dissipation along deterministic and escape trajectories to the Freidlin-Wentzell quasi-potential, thus constraining the transition rates between attractors induced by rare fluctuations. A coherent perspective on minimum and maximum entropy production principles is also provided. For systems that admit a continuous-space limit, we derive a nonequilibrium fluctuating field theory with its associated thermodynamics. Finally, we coarse grain the macroscopic stochastic dynamics into a Markov jump process describing transitions among deterministic attractors and formulate the stochastic thermodynamics emerging from it.

Figures (10)

• Figure 1: Sketch of the different levels of description emerging with growing size $V$ and time scale $\tau$. The mesoscopic level (bottom) consists of a Markov jump process in discrete space $n$ that enjoys local detailed balance. For large $V$ the dynamics of the intensive variable $c=n/V$ is dictated by the quasi-potential $I_\text{ss}$ and the nonconservative drift $v_\text{ss}$ (middle). For large $\tau$ the dynamics reduces again to a Markov jump process on the space of the attractors, regarded as emergent states with internal dissipation (top).
• Figure 2: Schematic representation of a system coupled to three reservoirs with fixed temperature $T$, (electro)chemical potential $\mu$, etc. Each of them can exchange energy, (charged) particles, etc. through various transition mechanisms which result in jumps of the system state $n$.
• Figure 3: Schematic representation, for a system without detailed balance, of relaxation and instanton trajectories connecting the local minimum $\mathcal{x}^*_\gamma$ of the rate function $I_\text{ss}$ and a saddle point $\mathcal{x}_\nu$.
• Figure 4: Difference in the barrier height \ref{['delta_barrier']} for the fixed point $\mathcal{x}_2^* > \mathcal{x}_1^*$ as a function of the chemical potential difference $\Delta \mu$ for $k_{-2}=3.5$. Bistability is present above the critical value $\Delta \tilde{\mu} \simeq 3.85$. Inset: The quasi-potential $I_\mathrm{ss}$ given by \ref{['eq:Iss_schlogl']} (solid) and its approximation ${I_\mathrm{NLE}}$ (dashed) for $k_{-2}=3.5$ and $\Delta \mu=5$.
• Figure 5: Instantaneous entropy production rate \ref{['ep']} along a stochastic trajectory (obtained via Gillespie algorithm gil77 with $\Delta \mu =2.84$, $k_{-2}=2.07$ and $V=10^2$) starting in the basin of attraction of $\mathcal{x}^*_1$ and ending in the one of $\mathcal{x}^*_2$. The solid lines are the constant values of the macroscopic mean entropy production in the two fixed points \ref{['ep_metastable_states']} and the mean entropy production rate along the transition trajectory \ref{['ep_transition']}.