Weakly Non-Symmetric Case in Stochastic Models

• The weakly non-symmetric case describes stochastic systems with slight symmetry-breaking via small boundary perturbations that induce analytical, linear deviations from equilibrium.
• Linear response and the McLennan ensemble methods precisely quantify the first-order non-equilibrium corrections, preserving a locally equilibrated product measure.
• The resulting spatially varying fugacity and linear density profile reveal how minor asymmetries yield weak irreversible dynamics and localized entropy production.

A weakly non-symmetric case arises in diverse mathematical and physical settings where systems that are otherwise symmetric are subject to slight symmetry-breaking perturbations, often resulting in subtle but analytically tractable deviations from their symmetric behavior. The theory of such regimes is particularly well illustrated in stochastics by close-to-equilibrium properties of interacting particle systems under weakly non-symmetric driving, such as the Symmetric Inclusion Process (SIP) with open boundaries and a small difference of chemical potentials imposed at the system's ends (Vafayi et al., 2014). This situation encapsulates a prototypical weakly non-symmetric case wherein irreversibility, entropy production, and spatially varying currents appear, but only to leading order in a small symmetry-breaking parameter.

1. Symmetric Inclusion Process and Equilibrium

The one-dimensional Symmetric Inclusion Process (SIP) is a stochastic lattice gas model exhibiting attractive interactions between particles during hopping events. The generator for the bulk dynamics is given by

$$L_{\mathrm{bulk}}f(\eta) = \sum_{i} \eta_i(m + \eta_{i+1})[f(\eta^{(i,i+1)}) - f(\eta)] + \sum_{i} \eta_{i+1}(m + \eta_i)[f(\eta^{(i+1,i)}) - f(\eta)]$$

with $m > 0$ controlling the diffusion strength. In the absence of boundary asymmetries (i.e., both particle reservoirs at the boundaries have equal chemical potentials), SIP satisfies detailed balance and admits a reversible stationary product measure

$$\nu_{\mathrm{EQ}}(\eta) = \prod_{i=1}^N \gamma(\eta_i)$$

with site marginals

$$\gamma(n) = \frac{\theta^n \Gamma(m+n)}{Z_\theta n! \Gamma(m)},\qquad Z_\theta = (1-\theta)^{-m}$$

where $\theta\in (0,1)$ parameterizes the chemical potential. The thermodynamic potential is associated as $U(x) = -\log \nu_{\mathrm{EQ}}(x)$.

2. Introduction of Weak Non-Equilibrium: Boundary Perturbation

Weak non-symmetry is induced by perturbing injection/removal rates at the boundaries so that the left and right reservoirs are no longer identical but differ by a small parameter $\epsilon \ll 1$: $$b_1 = b + \epsilon b, \quad b_N = b - \epsilon b, \quad d_1 = d_N = d$$ This asymmetry introduces a weak external force, causing the process to violate detailed balance specifically at the boundaries, while leaving bulk dynamics symmetric.

The modified local detailed balance for transition rates becomes

$$\frac{\lambda(x,y)}{\lambda(y,x)} = \exp\big( U(x) - U(y) + F_\epsilon(x,y) \big)$$

where the additional anti-symmetric “external force” term $F_\epsilon(x, y)$ characterizes the symmetry breaking localized at the boundaries: $F_\epsilon(x, x^{(1+)}) \simeq \epsilon$ This approximation holds for $\epsilon \ll 1$ and explicitly quantifies the weak asymmetry.

3. Entropy Production, Particle Currents, and McLennan Ensemble

The irreversible dynamics engender an entropy production rate

$$w_1(x) = \sum_{y \neq x} \lambda_0(x, y) F_1(x, y) = (b - d)(\eta_1 - \eta_N)$$

with $\lambda_0$ the reversible rates. The global current (and so the entropy production) is thus controlled by the occupation difference at the boundaries and is order $\epsilon$.

In weakly non-symmetric systems near equilibrium, the stationary non-equilibrium measure can be captured via the McLennan ensemble: $\rho(x) \simeq Z^{-1} \exp\{-U(x) + W(x)\}$ where the first-order non-equilibrium correction,

$$W(x) = -\epsilon \int_0^\infty \langle w_1(x_t) \rangle_x^{0} dt$$

is determined as the integrated transient entropy production along the equilibrium trajectory started from $x$. This correction can be explicitly written as a linear shift in occupation numbers,

$-L_0^{-1} w_1(x) = \sum_{i=1}^N c_i \eta_i$

with $c_i$ linear in the site index $i$ and determined by discrete Laplacian boundary value problems reflecting the finite lattice and the boundary drive.

4. Local Equilibrium and First-Order Product Structure

Significantly, the stationary state in the weakly non-symmetric regime retains a product form. The effect of the boundary perturbation is simply to replace the uniform product structure by one with spatially varying parameters. In particular, a spatially varying fugacity (local chemical potential) $\theta_i$ and correspondingly local density $\rho_i$ are defined by

$\rho_i = \frac{m \theta_i}{1 - \theta_i}$

and so the stationary measure, up to $O(\epsilon)$, is given by the local equilibrium (LEQ) product measure

$$\nu_{\mathrm{LEQ}}(\eta) = \prod_{i=1}^N \gamma_{\theta_i}(\eta_i)$$

with

$$\gamma_{\theta_i}(n) = \frac{\theta_i^n \Gamma(m+n)}{Z_{\theta_i} n! \Gamma(m)}$$

The spatial density profile $\rho_i$ is found to be linear to first order in $\epsilon$: $$\rho_i = \alpha(\epsilon) + \beta(\epsilon) i + O(\epsilon^2)$$ with $\beta(\epsilon) \propto \epsilon$ and proportional to the stationary particle current. This concrete, explicit description confirms that the non-equilibrium correction is still of “local equilibrium type”, fully reproducible by matching the first-order current and boundary conditions.

5. Interpretation and Physical Implications

Weakly non-symmetric dynamics in stochastic interacting particle systems—such as the SIP with a small boundary driving—produce a stationary state in which:

• There is a weak particle current of order $\epsilon$, arising solely due to the boundary rate asymmetry.
• The non-vanishing entropy production rate remains localized at the boundaries and is linear in the occupation difference between the ends.
• The stationary state remains amenable to complete analytic description: both its non-equilibrium correction and stationary profile are captured via a product measure consistent with local thermodynamic equilibrium, parameterized by a spatially linear density/fugacity profile.
• The correspondence between the McLennan ensemble construction (from entropy production) and the explicit LEQ product structure validates the usefulness of linear response and transient fluctuation approaches in weakly non-symmetric, weakly driven systems.

The demonstration that, to first order in the boundary asymmetry, the stationary measure remains a product measure tailored to the steady-state profile is profound: in weakly driven non-equilibrium settings, the complicated many-body measure is entirely characterized by local information and a single current-carrying linear mode.

6. Summary and Broader Significance

The weakly non-symmetric case, as analyzed for the SIP with open boundaries, exhibits the following features:

• Symmetry-breaking is introduced via boundary perturbations of $O(\epsilon)$, modifying only the injection/removal rates.
• The stationary state is only weakly irreversible, with the entropy production rate and steady current both of order $\epsilon$ and localized at the boundaries.
• Linear response (McLennan) theory allows precise computation of the stationary measure as a product measure shifted linearly in the occupations—a signature of local equilibrium superimposed with a linear mode responsible for current flow.
• Equilibrium statistical structure and analytic tractability are retained to leading order, despite the physical system carrying a net current and violating detailed balance at the boundaries.

These results provide a rigorous framework for understanding how small symmetry-breaking perturbations in otherwise symmetric stochastic systems yield simple, analytically tractable non-equilibrium stationary states characterized by additivity and locality, thus bridging equilibrium and non-equilibrium statistical mechanics at the linear response level.