A non-equilibrium stationary state is a state in which the macroscopic state variables of a system do not change with time, although the system is still away from thermodynamic equilibrium and irreversible processes continue to occur.
The important distinction is that stationary does not mean equilibrium.
At equilibrium, the thermodynamic driving forces and the corresponding net fluxes vanish, and therefore
\[
\boxed{\sigma=0}
\]
In a stationary non-equilibrium state, the state variables remain constant with time, but one or more fluxes can remain non-zero. Consequently, entropy continues to be produced:
\[
\boxed{\sigma>0}
\]
A simple example is a system maintained between two reservoirs at different temperatures. If the temperature difference is maintained continuously, heat flows through the system at a constant rate. The temperature distribution can become time-independent, but heat transfer and entropy production continue.
Thus,
\[
\boxed{
\frac{\partial(\text{state variables})}{\partial t}=0
}
\]
does not necessarily imply
\[
\boxed{\sigma=0}
\]
It only means that the system has reached a stationary condition.
| Equilibrium state | Stationary non-equilibrium state |
|---|---|
| State variables are independent of time | State variables are independent of time |
| Driving forces vanish | Driving forces may remain finite |
| Net irreversible fluxes are zero | Irreversible fluxes may be non-zero |
| Entropy production is zero | Entropy production is generally non-zero |
| No continuous transport is required | Continuous exchange of energy or matter may maintain the state |
Consider a system in which a temperature gradient and a concentration gradient are maintained. Let the corresponding thermodynamic forces be \(X_{\mathrm{th}}\) and \(X_{\mathrm{m}}\).
In the region close to equilibrium, the heat flux and matter flux can be written as
\[
J_{\mathrm{th}}
=
L_{11}X_{\mathrm{th}}
+
L_{12}X_{\mathrm{m}}
\]
and
\[
J_{\mathrm{m}}
=
L_{21}X_{\mathrm{th}}
+
L_{22}X_{\mathrm{m}}
\]
Here \(L_{ij}\) are phenomenological coefficients. The cross-coefficients \(L_{12}\) and \(L_{21}\) describe the coupling between heat and matter transport.
The entropy production is
\[
\sigma
=
J_{\mathrm{th}}X_{\mathrm{th}}
+
J_{\mathrm{m}}X_{\mathrm{m}}
\]
Substituting the two flux equations gives
\[
\sigma
=
\left(
L_{11}X_{\mathrm{th}}
+
L_{12}X_{\mathrm{m}}
\right)X_{\mathrm{th}}
+
\left(
L_{21}X_{\mathrm{th}}
+
L_{22}X_{\mathrm{m}}
\right)X_{\mathrm{m}}
\]
On applying the reciprocal relation
\[
L_{12}=L_{21}
\]
the expression becomes
\[
\sigma
=
L_{11}X_{\mathrm{th}}^2
+
L_{12}X_{\mathrm{m}}X_{\mathrm{th}}
+
L_{12}X_{\mathrm{th}}X_{\mathrm{m}}
+
L_{22}X_{\mathrm{m}}^2
\]
Therefore,
\[
\boxed{
\sigma
=
L_{11}X_{\mathrm{th}}^2
+
2L_{12}X_{\mathrm{th}}X_{\mathrm{m}}
+
L_{22}X_{\mathrm{m}}^2
}
\]
Consider a stationary state in which the matter flux is zero, while the thermal driving force is maintained externally:
\[
\boxed{J_{\mathrm{m}}=0}
\]
From the phenomenological equation for matter flow,
\[
J_{\mathrm{m}}
=
L_{21}X_{\mathrm{th}}
+
L_{22}X_{\mathrm{m}}
\]
Therefore, at the stationary state,
\[
L_{21}X_{\mathrm{th}}
+
L_{22}X_{\mathrm{m}}
=0
\]
Hence,
\[
\boxed{
X_{\mathrm{m}}
=
-\frac{L_{21}}{L_{22}}X_{\mathrm{th}}
}
\]
Using the reciprocal relation \(L_{21}=L_{12}\),
\[
\boxed{
X_{\mathrm{m}}
=
-\frac{L_{12}}{L_{22}}X_{\mathrm{th}}
}
\]
Thus, although the matter flux is zero, the matter-related thermodynamic force need not be zero. It adjusts itself so that the two coupled contributions to the matter flux cancel each other.
The entropy production is
\[
\sigma
=
L_{11}X_{\mathrm{th}}^2
+
2L_{12}X_{\mathrm{th}}X_{\mathrm{m}}
+
L_{22}X_{\mathrm{m}}^2
\]
Let \(X_{\mathrm{th}}\) remain fixed and vary \(X_{\mathrm{m}}\). The condition for an extremum of entropy production is
\[
\left(
\frac{\partial\sigma}
{\partial X_{\mathrm{m}}}
\right)_{X_{\mathrm{th}}}
=0
\]
Differentiating,
\[
\frac{\partial\sigma}{\partial X_{\mathrm{m}}}
=
2L_{12}X_{\mathrm{th}}
+
2L_{22}X_{\mathrm{m}}
\]
Taking \(2\) common,
\[
\frac{\partial\sigma}{\partial X_{\mathrm{m}}}
=
2
\left(
L_{12}X_{\mathrm{th}}
+
L_{22}X_{\mathrm{m}}
\right)
\]
But
\[
J_{\mathrm{m}}
=
L_{21}X_{\mathrm{th}}
+
L_{22}X_{\mathrm{m}}
\]
and since
\[
L_{21}=L_{12}
\]
we have
\[
\boxed{
\frac{\partial\sigma}{\partial X_{\mathrm{m}}}
=
2J_{\mathrm{m}}
}
\]
Therefore, in the stationary state, where
\[
J_{\mathrm{m}}=0
\]
we obtain
\[
\boxed{
\frac{\partial\sigma}{\partial X_{\mathrm{m}}}=0
}
\]
Hence, the entropy production has an extremum at the stationary state.
To determine whether the extremum is a maximum or a minimum, differentiate once more with respect to \(X_{\mathrm{m}}\):
\[
\frac{\partial^2\sigma}
{\partial X_{\mathrm{m}}^2}
=
2L_{22}
\]
For a physically permissible irreversible process, the diagonal phenomenological coefficient \(L_{22}\) is positive:
\[
L_{22}>0
\]
Therefore,
\[
\boxed{
\frac{\partial^2\sigma}
{\partial X_{\mathrm{m}}^2}>0
}
\]
Hence, the extremum is a minimum.
Therefore, for the stationary state,
\[
\boxed{
\left(
\frac{\partial\sigma}
{\partial X_{\mathrm{m}}}
\right)_{X_{\mathrm{th}}}
=0
}
\]
and
\[
\boxed{
\left(
\frac{\partial^2\sigma}
{\partial X_{\mathrm{m}}^2}
\right)_{X_{\mathrm{th}}}>0
}
\]
so that the entropy production has its minimum value.
The result can be generalized to a system having several independent thermodynamic forces. Suppose there are \(n\) independent forces
\[
X_1,X_2,\ldots,X_n
\]
and the entropy production is
\[
\boxed{
\sigma=\sum_iJ_iX_i
}
\]
If some of the forces are fixed externally and the remaining forces are free to adjust, the stationary state is characterized by the vanishing of the fluxes corresponding to the unrestricted forces:
\[
\boxed{
J_j=0
\qquad
(j=k+1,\ldots,n)
}
\]
These conditions are equivalent to the minimum conditions
\[
\boxed{
\left(
\frac{\partial\sigma}{\partial X_j}
\right)_{\text{fixed forces}}
=0
}
\]
with the second derivative condition giving a minimum under the appropriate linear-regime conditions.
This result is known as Prigogine’s principle of minimum entropy production. It applies to stationary non-equilibrium states in the linear region close to equilibrium, with specified external constraints. 0
The physical meaning is that the system cannot reach ordinary equilibrium because some external conditions continuously maintain the driving forces. It instead settles into a stationary state in which the irreversible processes are balanced by the imposed exchange with the surroundings.
If the external constraints are removed, the driving forces disappear and the system can proceed towards equilibrium:
\[
\boxed{
\text{Non-equilibrium}
\rightarrow
\text{stationary non-equilibrium state}
\rightarrow
\text{equilibrium}
}
\]
when the external constraints maintaining the non-equilibrium condition are no longer present.
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