Subsections

• Kinetic Coefficients
• Multi-Variate Case. Onsager Law

Thermodynamic Forces & Kinetic Coefficients. Onsager Law

Kinetic Coefficients

Thermodynamic force:

We introduced before conjugated variables:

$$X=\frac{\partial A}{\partial x} = -\frac{\partial S}{\partial x}$$

Let us set zero at the equilibrium value: $X\to X-X_0$. This is called thermodynamic force.

Examples:

If x is volume, the force is X=-(P-P₀), if x is the number of particles, $X=\mu-\mu_0$, etc.

In equilibrium (x=0) we have:

$$\frac{d\left\langle x\right\rangle}{dt} = 0,\quad X = 0$$

When x>0, $d\left\langle x\right\rangle/dt<0$ and X>0:

$$\frac{d\left\langle x\right\rangle}{dt} = -\lambda\left\lang... ...left\langle X\right\rangle = \gamma \left\langle x\right\rangle$$

From these equations  

$$\frac{d\left\langle x\right\rangle}{dt} = - \frac{\lambda}{... ...eft\langle X\right\rangle = -\Gamma \left\langle X\right\rangle$$ (9)

The new variable $\Gamma$ is called kinetic (or Onsager) coefficient. We have $\Gamma\gt$ (why?).

Two ways to create a fluctuation:

1.

Apply an external force X

2.

Wait long enough, and it will happen

Equation (9) shows that these two ways are equivalent. No matter how we created a fluctuation, it will relax in the same way.

Multi-Variate Case. Onsager Law

Suppose we have many variables x_i. We have many thermodynamic forces X_i. Equations for relaxation:

$$\frac{d\left\langle x_i\right\rangle}{dt} = - \sum_k \Gamma_{ik} \left\langle X_k\right\rangle$$

Onsager Law:

The coefficients are symmetric:

$$\Gamma_{ik}=\Gamma_{ki}$$

This equation can be proved from the symmetry with respect to time inversion: $t\to-t$.