next up previous
Up: Time Dependent Fluctuations Previous: Relaxation & Time Correlation

Subsections


Thermodynamic Forces & Kinetic Coefficients. Onsager Law

Kinetic Coefficients

Thermodynamic force:
We introduced before conjugated variables:

\begin{displaymath}
X=\frac{\partial A}{\partial x} = -\frac{\partial S}{\partial x}\end{displaymath}

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:

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

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

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

From these equations  
 \begin{displaymath}
 \frac{d\left\langle x\right\rangle}{dt} = - \frac{\lambda}{...
 ...eft\langle X\right\rangle = -\Gamma \left\langle X\right\rangle\end{displaymath} (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 xi. We have many thermodynamic forces Xi. Equations for relaxation:

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

Onsager Law:
The coefficients are symmetric:

\begin{displaymath}
\Gamma_{ik}=\Gamma_{ki}
 \end{displaymath}

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


next up previous
Up: Time Dependent Fluctuations Previous: Relaxation & Time Correlation

© 1997 Boris Veytsman and Michael Kotelyanskii
Tue Oct 28 22:13:33 EST 1997