Excess dissipation shapes symmetry breaking in non-equilibrium currents

TL;DR

The paper presents a universal geometric framework for non-equilibrium currents in Langevin systems by using the inverse diffusion matrix as a metric to define a current velocity and a non-equilibrium availability. It decomposes this velocity into housekeeping and excess components, deriving exact relations such as $oldsymbol{S}_{ m tot}=oldsymbol{S}_{ m hk}-oldsymbol{S}_{ m ex}$ and a variational principle governing the NESS through excess dissipation, tying symmetry breaking in trajectory space to dissipative structure. The authors classify NESS geometries (locally closed, locally balanced, local equilibrium), extend the analysis to weak and strong noise as well as multiplicative fluctuations, and demonstrate the framework on paradigmatic systems like driven circular potentials and Brownian gyrators, and on many-body coupled oscillators to illustrate energy transduction and synchronization. This work provides a unifying, gauge-like view of how dissipation, geometry, and symmetry breaking shape emergent organization in non-equilibrium systems, with potential applications to molecular machines, active matter, and complex fluids.

Abstract

Most natural thermodynamic systems operate far from equilibrium, developing persistent currents and organizing into non-equilibrium stationary states (NESSs). Yet, the principles by which such systems self-organize, breaking equilibrium symmetries under external and internal constraints, remain unclear. Here, we establish a general connection between symmetry breaking and dissipation in mesoscopic stochastic systems described by Langevin dynamics. Using a geometric framework based on the inverse diffusion matrix, we decompose the velocity field into excess (gradient) and housekeeping (residual) components. This provides a natural entropy production split: the excess part captures internal reorganization under non-equilibrium conditions, while the housekeeping part quantifies detailed-balance violation due to external forces. We derive an exact equality linking the two, along with an inequality identifying accessible thermodynamics. A weak-noise expansion of the stationary solution reveals the general geometry of the NESS velocity field, enabling a unified classification of steady states. We apply this framework to systems ranging from molecular machines to coupled oscillators, showing how symmetry breaking in trajectory space constrains NESS organization. We further extend our approach to systems with multiplicative noise, deriving how additional symmetry breaking relates to curved (space-dependent) metrics. Finally, we show that both the NESS velocity field and stationary distribution can be derived through variational functionals based on excess dissipation. This work sheds light on the intimate connection between geometric features, dissipative properties, and symmetry breaking, uncovering a classification of NESSs that reflects how emergent organization reflects physical non-equilibrium conditions.

Figures (6)

• Figure 1: Parity symmetry breaking for a driven particle in a circular potential. (a) A sketch of the model. (b) In the NESS, the stationary availability (red dashed curve) coincides with the equilibrium potential (gray curve). Hence, the total stationary entropy production coincides with the housekeeping one, $\dot{S}^{\mathrm{st}}_{\rm tot} = \dot{S}^{\mathrm{st}}_{\rm hk}$. The inset show the pdf projected in the $(x,y)$ plane (in color-scale) with the velocity vector field on top (a bigger arrow corresponds to a stronger field). It clearly shows the tendency to rotate in a preferential direction, therefore breaking parity symmetry.
• Figure 2: Parity symmetry breaking in a Brownian gyrator. (a) A sketch of the model. (b) Steady-state distribution (in color-code) with the adiabatic velocity field on top (larger brighter arrows indicate a stronger field). (c) Availability (red dashed curve) and potential (gray curve) along the line $x=y$ show that excited states can be explored solely due to non-equilibrium fluctuations. In the inset, excess and house-keeping velocity at steady-state are presented in color-code and as a stream plot. (d) As a function of time, we show that the system tends to minimize $\dot{S}_{\rm tot}$ until reaching a non-zero value $\dot{S}_\mathrm{a}$ signaling a parity symmetry breaking. At the same time, the housekeeping flux $\langle \bm{v} * \bm{v}_\mathrm{hk}\rangle$ is the only surviving term at stationarity, with the excess availability flux going to zero (not shown). (e) The excess dissipation functional $\mathcal{G}_\mathrm{ex}$ is plotted, showing that it converges to $(1/2)\dot{S}_{\mathrm{ex}}$ at the steady state without being necessarily minimized along the dynamics.
• Figure 3: Beyond weak noise for a Brownian particle in bistable potential and shear flow. (a) Contour plot of the steady-state distribution in the weak-noise limit with the adiabatic velocity as a vector field in color-code (brighter arrow corresponds to stronger field). The minima of the equilibrium potential are shown: they do not coincide with the peaks of the availability, leading to a non-zero excess availability. We report both the corresponding excess velocity as a stream plot in color-code and the housekeeping component that coincides with the shear flow. (b) Same as in panel (a), but for the strong-noise scenario. Notice that the availability minima do not coincide with the deterministic solution, and the excess velocity shows the presence of repellors at $(x_1, x_2) \approx \pm (0.5, 1.8)$. However, the housekeeping component is unchanged, since it only depends on external non-equilibrium conditions. Parameters are: $a = 1, b = 1.5, \gamma = 1$, and $T = 0.1$ (weak noise) or $1$ (strong noise).
• Figure 4: Geometry of velocities for underdamped dynamics with thermal gradient. (a) The stationary solution of the underdamped Fokker-Planck equation up to the first order in $\tau$ is shown (Eq. \ref{['eq:P_under']}), for a linear thermal gradient $T(x) = T_0(1 + \alpha x)$ and a quadratic potential $U = u(x - \mu)^2$. Arrows represent the stationary velocity in color-code (brighter arrows indicate stronger field). (b) Comparison between availability and equilibrium distribution for ${\rm v} = 0$. Notice that the availability along this line coincides with $\phi^{(0)}_{\rm st}(x)$ (Eq. \ref{['eq:P_under']}). The availability minimum does not coincide with the equilibrium one. (c) Streamplot in color-code of housekeeping (reversible) and excess (irreversible) velocities. In this figure, the spatial domain is $x \in [-1/\alpha, +\inf]$ and we set all parameters so that the probability distribution and its derivative vanish at the boundaries. The model parameters are: $m = 1$, $\gamma = 0.01$, $\mu = u = 2$, $T_0 = 1$, $\alpha = 0.8$.
• Figure 5: The geometry of current velocities for coupled molecular oscillators. (a) Pictorial sketch of two coupled molecular biochemical oscillators. Each oscillator is represented by a phase $\phi_i\in [0,2\pi]$ (in shaded blue color) that grows with a constant velocity $ke_g$ (solid black arrows) fueled by ATP consumption. Furthermore, oscillators can interact with a phase exchange potential $E_{12}(\varphi_{12})$ depending on their phase difference $\varphi_{12}=\phi_1-\phi_2$ (in shaded red), eventually leading to anti-correlated phase changes (small red arrows). (b) Stream plot of $v^{\phi}_{\mathrm{a}}$ for a typical oscillator with phase $\phi_1$ as a function of the collective phase $\psi$ with the heat map of $P_{\mathrm{st}}(\theta_1)$, $\theta_1=\phi_1-\psi$ in the mean-field limit in the synchronized phase. Here, $v^{\phi}_{\mathrm{a}}$ drives the synchronized oscillations (yellow arrows and light-blue regions, high probability density) but also indicates that interactions tend to destabilize them (dark blue regions of extremely low probability). (c) Stream plot of $v^{\varphi}_{\mathrm{a}}$ for two typical oscillators with the heat map of $P_{\mathrm{st}}(\theta_{1},\theta_{2})$, where , with $\psi$ the mean-field collective phase. $v^{\varphi}_{\mathrm{st}}$ tends to reduce the oscillators phase differences $\varphi$, stabilizing the dynamics and promoting synchronization. (d) Plot of the synchronization order parameter as a function of the excess entropy for different values of $E_0$. Model parameters are $k=0.5, e_g=4\pi, E_0=8, \Omega=1$.