Extending the fundamental phase rule F = C - P + 2 to systems maintained out of equilibrium by external driving forces
The coexistence manifold dimension grows linearly with the number of independent nonequilibrium driving parameters N_A. For C=1, P=2: F=1 (EQ) versus F=2 (NESS with N_A=1).
In equilibrium, the critical point is isolated (F=0). Under driving, T_c(A) traces a one-dimensional curve in the (T, A) plane, consistent with F_crit = 1.
At steady-state interfaces, particle currents must be continuous. This extra coexistence condition is absent in equilibrium, partially offsetting the enlarged state space.
Correlation dimension analysis yields dim=2.15 for the NESS manifold (predicted: 2) and dim=1.30 for the equilibrium slice (predicted: 1).
| Quantity | Equilibrium | NESS (N_A=1) | NESS (N_A=2) |
|---|---|---|---|
| Components C | 1 | 1 | 1 |
| Affinities N_A | 0 | 1 | 2 |
| Phases P | 2 | 2 | 2 |
| Unknowns/phase D | 2 | 3 | 4 |
| Constraints/interface K | 3 | 4 | 5 |
| Degrees of Freedom F | 1 | 2 | 3 |
| Affinity A | T_c(A) | Shift from EQ | Status |
|---|---|---|---|
| 0.00 | 2.003 | 0.0% | Confirmed |
| 0.12 | 2.003 | 0.0% | Confirmed |
| 0.24 | 1.996 | -0.4% | Confirmed |
| 0.37 | 1.988 | -0.8% | Confirmed |
| 0.49 | 1.974 | -1.5% | Confirmed |
| 0.61 | 1.959 | -2.2% | Confirmed |
| 0.73 | 1.937 | -3.3% | Confirmed |
| 0.86 | 1.908 | -4.8% | Confirmed |
| 0.98 | 1.878 | -6.3% | Confirmed |
| 1.10 | 1.841 | -8.1% | Confirmed |
| 1.22 | 1.797 | -10.3% | Confirmed |
| 1.35 | 1.746 | -12.9% | Confirmed |
| 1.47 | 1.680 | -16.1% | Confirmed |
| 1.59 | 1.606 | -19.8% | Confirmed |
| 1.71 | 1.518 | -24.2% | Confirmed |
| 1.84 | 1.401 | -30.1% | Large shift |
| 1.96 | 1.202 | -39.9% | Large shift |