Non-Equilibrium Steady State Simulator

Non-Equilibrium Lattice

Temperature difference ΔT 1.5

Coupling / relaxation rate κ 0.35

Entropy production Ṡ

–

Heat flux |J|

–

T (hot side)

–

T (cold side)

–

How it works

A 1D chain of cells sits between two reservoirs — hot on the left, cold on the right — and diffuses energy between neighbours at rate κ every step: T becomes T + κ·(T_left − 2·T + T_right)·dt, with the two end cells clamped to the reservoir temperatures. That is Fourier's law applied cell-by-cell. When ΔT > 0 the chain relaxes into a non-equilibrium steady state: a linear temperature profile with a constant, nonzero heat flux J flowing from hot to cold forever — the system never stops exchanging energy, but its macroscopic state stops changing.

Jᵢ = κ·(Tᵢ − Tᵢ₊₁)
σᵢ = Jᵢ·(1/Tᵢ₊₁ − 1/Tᵢ) ≥ 0
Ṡ = Σᵢ σᵢ

ΔT — the drive. Set it to zero and the two reservoirs become identical: the lattice relaxes to a flat profile, flux and entropy production both fall to zero — ordinary equilibrium is just the ΔT→0 limit of the same equations.
κ — how fast neighbouring cells exchange energy; higher κ reaches the steady state faster but doesn't change its final flux, which is set by ΔT and the chain length.
Entropy production Ṡ — the sum of Jᵢ·Δ(1/T) over every link. It is always ≥ 0 (second law), grows with ΔT, and is the quantity that is exactly zero at equilibrium and strictly positive in any steady flow — the signature that distinguishes non-equilibrium steady states from equilibrium.
The drifting dots ride the flux itself — their speed and density along the chain track |J| directly, not just a decorative overlay.

Real-world relevance: this is the same math behind heat leaking through a wall, ionic currents across a cell membrane, or any system Onsager's linear irreversible thermodynamics describes — a constant flux sustained by a constant external gradient, dissipating free energy at a steady rate.

🔥 HOT❄ COLD

Hot cell Cold cell Flux particle

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.