Linear Response Theory

Non-equilibrium thermodynamics uses a phenomenological approach since enough information is not really available.

G Formalism 1st + 2nd Law 1st + 2nd Law Gibbs equation Gibbs equation 1st + 2nd Law->Gibbs equation Rate of Entropy Production Rate of Entropy Production Gibbs equation->Rate of Entropy Production Level of Energy Dissipation Level of Energy Dissipation Rate of Entropy Production->Level of Energy Dissipation Balance equations Balance equations Balance equations->Rate of Entropy Production

The Gibbs equation combines the first and second laws of thermodynamics. Together with the general balance equations under local thermodynamic equilibrium, the rate of entropy production can be obtained. This leads to the level of energy dissipation during a process, and in describing coupled phenomena.

We will not go far from equilibrium. The transport and rate equations will be expressed in linear forms, and the Onsager reciprocal relations are valid.

Stationary States

Intensive properties that specify the state of a substance are time independent in equilibrium systems and in nonequilibrium stationary states (which only happens in non-isolated systems of course).

Extensive properties specifying the state of a system with boundaries are also independent of time, and the boundaries are stationary in a coordinate system.

The stationary state of a substance at any point is related to the stationary state of the system

The total entropy will not change:

\[ \frac{dS}{dt} = \frac{dS_e}{dt} + \frac{dS_i}{dt} = 0 \]

Also in terms of entropy currents (in and out) a continuity equation can be written:

\[ \frac{dS_i}{dt} + (J_{s,in} - J_{s,out}) = 0 \]

\[ \begin{align} \frac{dS_e}{dt} &= \text{Entropy exchange between system and surroundings} \\ \frac{dS_i}{dt} &= \text{Entropy production inside} \end{align} \]

Entropy production (\(dS_i \geq 0\)) and since \(dS_e/dt\) is greater than 0 (system need not be isolated):

\[ \frac{dS_e}{dt} = -\frac{dS_i}{dt} = (J_{s,in} - J_{s,out}) < 0 \]

Entropy exchange with the environment is negative

The stationary state is maintained through the decrease in entropy exchanged between the system and its surrounding.

The total entropy produced within the system must be discharged across the boundary at stationary state.

D Formalism Stationary state Stationary state Boundary conditions do not change with time Boundary conditions do not change with time Stationary state->Boundary conditions do not change with time Isolated system Isolated system Steady state not possible Steady state not possible Isolated system->Steady state not possible dS_e/dt = 0 dS_e/dt = 0 Steady state not possible->dS_e/dt = 0 Entropy created cannot be discharged to maintain steady state (dS_e/dt = 0) Entropy created cannot be discharged to maintain steady state (dS_e/dt = 0) Steady state not possible->Entropy created cannot be discharged to maintain steady state (dS_e/dt = 0) Adiabatic system Adiabatic system Adiabatic system->Steady state not possible Equilibrium Equilibrium dS/dt = dS_e/dt = dS_i/dt = 0 dS/dt = dS_e/dt = dS_i/dt = 0 Equilibrium->dS/dt = dS_e/dt = dS_i/dt = 0