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Time Evolution and Ergodicity

Gibbs Postulate

Trajectory--a line in phase space. Many-body systems: a complex entangled trajectory. Assumption: after long time the system ``forgets'' the initial conditions. Probability of each state does not depend on where we started!

What can it depend upon? Energy, total momentum, total volume,...

Microcanonical Ensemble:
Closed system $\Rightarrow$ all states have same energy, momentum,...
Gibbs Postulate:
All states in microcanonical ensemble are equivalent. They have the same probability to occur.

Averaging and Ergodicity

Two ways of averaging:
I take one system and observe it for a long time (time averaging) :  
 \left\langle f(p,q)\right\rangle_{\tau} = \lim_{\tau \right...
 ...infty}\frac{1}{\tau}\int_0^{\tau} f\bigl(p(t),q(t)\bigr)\,dt 
 \end{displaymath} (1)
I take many systems from my ensemble (i.e. with same energy) and observe them all in one time (ensemble average):  
 \left\langle f(p,q)\right\rangle_E = \frac{1}{\Omega} 
 \int_{E=E_0} f(p,q)\, dp^N\,dq^N
 \end{displaymath} (2)
 \Omega = \int_{E=E_0} dp^N\,dq^N
 \end{displaymath} (3)

Ergodicity hypothesis: ensemble average is equal to time average.

\left\langle f(p,q)\right\rangle_\tau = \left\langle f(p,q)\right\rangle_E\end{displaymath}

Is ergodic hypothesis alway correct?

Gibbs postulate: yes.

But how large should be $\tau$ in (1) for the limit to work? What happens if it is a year? Twenty years? The mountains flowed down at Thy presence [*], but we are mortals....

Examples: water is liquid at frequencies $\approx1\,\mathrm{Hz}$, but solid at $10^{12}\,\mathrm{Hz}$. Window panes flow in hundred years...

Ergodic systems:
$\tau$ is small, and ergodicity works. Gases, liquids, crystals.
Non-ergodic systems:
$\tau$ is large, and we cannot wait for equilibrium. Glasses.
The same system can be both ergodic and non-ergodic depending on your time scale. Polymers.

Computer simulations: is the simulated system ergodic? Do we study its equilibrium or transient features?


Each system in the ensemble is a point in the phase space. How many points are there?


Geometric interpretation: cell model
 S = k \ln W\end{displaymath} (4)
Boltzmann constant: $k=1.38\times10^{-23}\,\mathrm{J/K}$

Thermodynamic limit

Suppose we increase the volume of our system V and the number of particles N.

Thermodynamic limit ($\text{T-limit}$):
$N\to\infty$, $V\to\infty$,and $N/V=\mathit{const}$. This limit is always taken!
Intensive variables:
A variable A is intensive if

\lim_{\text{T-limit}} A = \mathit{const}

Examples: temperature, pressure, density, concentration,...
Extensive variables:
A variable A is intensive, if

\lim_{\text{T-limit}} A = \infty, \lim_{\text{T-limit}} A/V = \mathit{const}

Examples: number of particles, volume, energy,...

Entropy of two non-interacting subsystems:
The total number of states:

W = W_1\times W_2\end{displaymath}


S = S1 + S2

Thermodynamic limit: many weakly interacting parts!
Entropy is an extensive quantity

Distribution Function

Let $\rho(p,q)$ be the probability to meet the given state in the ensemble. Ensemble averaging:

\left\langle f(p,q)\right\rangle_E = 
 \int \rho(p,q) f(p,q)\, dp^N\,dq^N\end{displaymath}

$\rho$ is the distribution function . It depends on the ensemble. Normalization:  
 \int \rho(p,q) \, dp^N\,dq^N = 1\end{displaymath} (5)
What is the distribution function for the microcanonical ensemble?
It is zero if $E\ne E_0$, and satisfies (5). Therefore

\rho_{\text{microcanonical}} = \delta(E-E_0)

next up previous
Next: Quiz Up: Microscopic and Macroscopic States. Previous: Macroscopic State

© 1997 Boris Veytsman and Michael Kotelyanskii
Wed Sep 3 22:59:36 EDT 1997