Nonequilibrium steady state

A stationary state of driven dynamics with sustained probability/physical currents and positive entropy production; typically violates detailed balance.

Nonequilibrium steady state

A nonequilibrium steady state (NESS) is a time-stationary state of a system that is maintained away from thermal equilibrium by driving (external fields, boundary gradients, multiple reservoirs, etc.). Unlike equilibrium Gibbs states, a NESS typically carries nonzero currents and violates detailed balance.

This knowl is framed in the language of Markov dynamics (see Markov chains and Markov semigroups ).

Definition (Markov dynamics)

Consider a continuous-time Markov jump process on a discrete state space with transition rates $w_{i\to j}$ and time-dependent probabilities $p_i(t)$ . The master equation (see master equation ) is

\dot p_i(t) \;=\; \sum_{j}\Bigl(p_j(t)\,w_{j\to i} - p_i(t)\,w_{i\to j}\Bigr).

A probability vector $\pi$ is a steady state if it is stationary:

0 \;=\; \sum_{j}\Bigl(\pi_j\,w_{j\to i} - \pi_i\,w_{i\to j}\Bigr) \quad \text{for all } i.

A nonequilibrium steady state (NESS) is a steady state that is not an equilibrium state, typically characterized by:

violation of detailed balance, and/or
nonzero steady currents.

Currents and detailed balance

Define the steady probability current along the oriented edge $i\to j$ by

J_{i\to j} \;=\; \pi_i\,w_{i\to j} - \pi_j\,w_{j\to i}.

If detailed balance holds (see detailed balance ), then $J_{i\to j}=0$ for every pair $(i,j)$ .
A NESS typically has $J_{i\to j}\neq 0$ for some edges, producing persistent cycles of probability flow even though $\pi$ is time-independent.

Entropy production rate

A standard quantitative measure of nonequilibrium is the steady-state entropy production rate:

\sigma \;=\; \frac{1}{2}\sum_{i,j} J_{i\to j}\, \log\!\left(\frac{\pi_i\,w_{i\to j}}{\pi_j\,w_{j\to i}}\right), \qquad \sigma \ge 0.

In equilibrium (detailed balance), $\sigma=0$ .
In a genuine NESS with sustained currents, typically $\sigma>0$ .

Connections to information-theoretic quantities are often expressed using relative entropy (KL divergence) .

Example: biased random walk on a ring

Take states $i\in\{1,\dots,N\}$ with rates

$w_{i\to i+1}=p$ (clockwise),
$w_{i\to i-1}=q$ (counterclockwise), with indices modulo $N$ .

Then:

the stationary distribution is uniform, $\pi_i=1/N$ ,
the steady current is constant, $J_{i\to i+1} \;=\; \frac{p-q}{N},$

which is nonzero if $p\neq q$ .

Thus the system is in a NESS whenever $p\neq q$ : stationary but with persistent circulation and positive entropy production.

Linear response and fluctuation relations (context)

Near equilibrium (weak driving), transport coefficients can often be expressed through equilibrium correlation functions via Green–Kubo relations and the fluctuation–dissipation theorem .

Far from equilibrium, work and current fluctuations satisfy exact identities and inequalities, including Crooks fluctuation theorem and Jarzynski equality .

Definition (Markov dynamics)

Currents and detailed balance

Entropy production rate

Example: biased random walk on a ring

Linear response and fluctuation relations (context)

Prerequisites