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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,...
Ergodicity hypothesis: ensemble average is equal to time average.
Is ergodic hypothesis alway correct?
Gibbs postulate: yes.
But how large should be 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 , but solid at . Window panes flow in hundred years...
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?
The cannot be obtained in classical statistics. Quantum answer:
Geometric interpretation: cell model
Suppose we increase the volume of our system V and the number of particles N.
Examples: temperature, pressure, density, concentration,...
Examples: number of particles, volume, energy,...
Entropy of two non-interacting subsystems:
The total number of states:
S = S1 + S2Thermodynamic limit: many weakly interacting parts!
Let be the probability to meet the given state in the ensemble. Ensemble averaging:
is the distribution function . It depends on the ensemble. Normalization:
© 1997 Boris Veytsman and Michael Kotelyanskii