A new preprint shows that nonequilibrium steady states (NESS) on large dense networks satisfy analogues of the Boltzmann distribution.

Model §

Rate matrix parameterization §

The setting is a random $N \times N$ transition rate matrix $\boldsymbol{W} = (W_{ij})_{i,j}$, which is always irreducible for sufficiently large $N$ under the assumptions. Following the physics convention, a generic off-diagonal entry $W_{ij}$ is the transition rate from $j$ to $i$. The diagonal entries of $\boldsymbol{W}$ are the negated exit rates $-W_{jj} = w_j = \sum_{i(\neq j)} W_{ij}$. When $\boldsymbol{W}$ is irreducible, it has a unique positive stationary distribution $\boldsymbol{\pi}$ (a column vector by this convention), defined by the matrix equation $\boldsymbol{W} \boldsymbol{\pi} = \boldsymbol{0}$.

We redundantly parameterize the transition rates as $W_{ij} = e^{E_j} X_{ij}$, where $E_j$ is a real-valued vertex parameter, and $X_{ij}$ is a nonnegative edge weight. One reason for doing so is that, if $\boldsymbol{X}$ is symmetric, then the stationary distribution $\boldsymbol{\pi}$ of $\boldsymbol{W}$ is equal to $\boldsymbol{\pi}^{\mathrm{eq}}$ where $\pi_i^{\mathrm{eq}} \propto e^{-E_i}$. In other words, the stationary distributions of such $\boldsymbol{W}$ have the form of the Boltzmann distribution, with the vertex parameters playing the role of dimensionless energies.

Arrhenius interpretation. An important special case of this model is when the edge weights are parameterized as $X_{ij} = e^{-B_{ij} + F_{ij}/2}$ in terms of symmetric “barriers” $B_{ij} = B_{ji}$ and anti-symmetric “forces” $F_{ij} = -F_{ji}$, all real numbers. In this case, the transition rates have the form $W_{ij} = e^{E_j - B_{ij} + F_{ij}/2}$, which is a common model of stochastic thermodynamics. As Owen et al. (PRX, 2020) explain, these transition rates are reminiscent of the Arrhenius formula for reaction kinetics on an energy landscape with wells of depth $E_j$, separated by barriers of height $B_{ij}$, and driven by non-conservative forces $F_{ij}$. In accordance with this analogy, in the absence of forces, the $X_{ij}$ are symmetric and $\boldsymbol{\pi} = \boldsymbol{\pi}^{\mathrm{eq}}$.

Model assumptions §

The starting point of this new paper is to model $\boldsymbol{X}$ as a random matrix. In order to cover the special case of Arrhenius-like transition rates, we must allow the pairs $(X_{ij},X_{ji})$ of off-diagonal entries to be dependent. We define a class of random transition rate matrices by taking the off-diagonal pairs $\{(X_{ij}, X_{ji}) \}_{i < j}$ to be independent copies of a representative pair $(X,X')$. We assume that this pair is exchangeable, i.e., $(X,X')$ has the same distribution as $(X',X)$, so that there is no favored orientation of each edge. We further assume that a representative entry $X$ has positive, finite variance. Heavier-tailed edge weights could effectively sparsen the underlying transition graph, in a sense.

Why dense? Under these assumptions, the state space has the connectivity of a dense random digraph. In the special case when $X$ and $X'$ are independent, it is an Erdős–Rényi random digraph on $N$ vertices with each edge independently present with probability $p = \mathbb{P}(X>0)>0$. Because $p$ depends only on the distribution of $X$ and not $N$, this graph is well within the dense regime.

Two extremes. An interesting aspect of this model is that it is flexible enough to accommodate two “extremes.” One extreme is local detailed balance, where $X_{ij} > 0$ only if $X_{ji} > 0$, as in the Arrhenius-like transition rates. Local detailed balance is a typical assumption of papers in this area of the physics literature, as it ensures the consistency of conclusions with stochastic thermodynamics.¹ The other is total irreversibility, where $X_{ij} > 0$ only if $X_{ji} = 0$. This flexibility is especially desirable considering the recent suggestion by some experts to develop theory without the assumption of local detailed balance.²

Main results §

Boltzmann-like occupation §

Statement. Fix an arbitrary sequence of vertex parameters $(E_i)_i$ and assume the edge weight matrix $\boldsymbol{X}$ arises from a pair $(X,X')$ satisfying the assumptions. Denote by $\delta_i = \pi_i / \pi_i^{\mathrm{eq}} - 1$ the state-wise relative distance between $\boldsymbol{\pi}$ and $\boldsymbol{\pi}^{\mathrm{eq}}$. Our main result is that $\boldsymbol{\pi}$ almost surely satisfies $\|\boldsymbol{\delta}\|_\infty \to 0$ as $N \to \infty$.

State-wise relative distance tending to zero is a particularly strong way for $\boldsymbol{\pi}$-probabilities to become like those of $\boldsymbol{\pi}^{\mathrm{eq}}$. For example, it implies that the total variation between these distributions tends to zero. This result is useful because $\boldsymbol{\pi}$ is generally a complicated function of all the transition rates.³ It means that $\boldsymbol{\pi}^{\mathrm{eq}}$ can replace $\boldsymbol{\pi}$ in calculations, which has many interesting applications.

Application. One example in the paper is a nonequilibrium fluctuation–dissipation theorem (FDT). The setting is a steady state at inverse temperature $\beta$, an underlying energy landscape $\mathcal{E}_i$ that depends on a control parameter $\eta$, and a function or “observable” $O$ over the states. The standard (static) FDT says that

Here, $\langle O \rangle_{\mathrm{eq}}$ denotes the average of $O$ with respect to $\boldsymbol{\pi}^{\mathrm{eq}}$, $V_i = -\partial_\eta \mathcal{E}_i$ is the “coordinate conjugate to” $\mathcal{E}_i$, and $\mathrm{Cov}_{\mathrm{eq}}(O,V)$ is the covariance between $O$ and $V$ under $\boldsymbol{\pi}^{\mathrm{eq}}$. The first of our main results implies that the same holds when the average of $O$ is taken with respect to $\boldsymbol{\pi}$ instead, so long as we multiply the right-hand side by $1+o(1)$. Doing so predicts that verifying the static FDT in either form or value does not suffice to conclude that a steady state is in thermal equilibrium.

“Low rattling” is asymptotically exact §

The proof of our first main result further shows that the analogous result holds when $\boldsymbol{\pi}^{\mathrm{eq}}$ is replaced by the distribution $\boldsymbol{\nu}$ that is proportional to the expected holding times in each state, i.e., $\nu_i \propto 1/w_i$. Equivalently, the “energy” $E_i$ in the Boltzmann-like distribution $\pi_i^{\mathrm{eq}}$ can be replaced by the quantity $\mathcal{R}_i = \log w_i$ called rattling.⁴ In particular, it implies that the effective potential $-\log \pi_{\boldsymbol{u}}$ of a uniformly random state $\boldsymbol{u}$ has asymptotically perfect linear correlation with $\mathcal{R}_{\boldsymbol{u}}$. This is the precise formulation of the active-matter heuristic of “low rattling.”⁵ Our second main result verifies that the heuristic is asymptotically exact for NESS on dense networks.