4.1 Microcanonical ensemble

**1.**- If the system is divided into two subsystems that may
freely exchange energy, then the equilibrium state is the one in which
the available energy
is distributed such that

(4.1) **temperature**. **2.**- In equilibrium the entropy is additive:

(4.2)

Now consider two systems ( ) and ( ) in

(4.3) |

(4.4) |

(4.5) |

(4.6) |

(4.7) |

where is that partial energy of system which maximizes the product .

In other words, at thermal contact between two systems isolated from the outside world there will be a regular flow of energy until the quantity () is equal in both systems. Since the combined system has then the largest extension in phase space, this will be the most probable distribution of energies upon the two systems. There may be fluctuations around the optimal energy distribution, but due to the extreme sharpness of the maximum of these deviations remain very small.

It should be noted that these conclusions, although of eminent physical
significance, may be derived quite simply from the geometrical properties
of high-dimensional spheres.

Example:Consider two systems with and initial energies , . Now bring the systems in thermal contact. The maximum value of the product occurs at , and the respective phase space volume is

How does another partitioning of the total energy - say, instead of - compare to the optimal one, in terms of phase space volume and thus probability?

(4.9) |

We can see that the energy fluctuations in these small systems are relatively large: . However, for larger particle numbers decreases as : ; thus .

Let the system under consideration be in mechanical or thermal contact with other systems. The macroscopic conditions () may then undergo changes, but we assume that this happens in a

Defining (in addition to ) the pressure by

(4.11) |

(4.12) |

Example:Classical ideal gas with

(4.13) |

Solving this equation for we find for theinternal energythe explicit formula

(4.14) |

From thermodynamics we know that ; therefore

(4.15) |

From this we conclude, in agreement with experiment, that thespecific heatof the ideal gas is

(4.16) |

Thepressuremay be found from :

(4.17) |

We found it a simple matter to derive the entropy of an ideal gas as a function of , and ; and once we had the subsequent derivation of thermodynamics was easy. To keep things so simple we had to do some cheating, in that we assumed no interaction between the particles. Statistical mechanics is powerful enough to yield solutions even if there are such interactions. A discussion of the pertinent methods - virial expansion, integral equation theories etc. - is beyond the scope of this tutorial. However, there is a more pragmatic method of investigating the thermodynamic properties of an arbitrary model system: computer simulation. The classical equations of motion of mass points interacting via a physically plausible pair potential such as the one introduced by Lennard-Jones read

If at some time the microstate is given we can solve these equations of motion for a short time step by numerical approximation; the new positions and velocities at time are used as starting values for the next integration step and so forth. This procedure is known as molecular dynamics simulation.

Since we assume no external forces but only forces between the particles, the total energy of the system remains constant: the trajectory in phase space is confined to the energy surface . If the systems is chaotic it will visit all states on this hypersurface with the same frequency. An average over the trajectory is therefore equivalent to an average over the microcanonical ensemble. For example, the internal energy may be calculated according to , where may be determined at any time from the particle velocities, and from the positions. By the same token the temperature may be computed via , while the pressure is the average of the so-called ``virial''; that is the quantity . In particular we have .

In the case of hard spheres the particle trajectories are computed in a different manner. For given the time span to the next collision between any two particles in the system is determined. Calling these prospective collision partners and we first move all spheres along their specific flight directions by and then simulate the collision (), computing the new directions and speeds of the two partners according to the laws of elastic collisions. Now we have gone full circle and can do the next and .

Further details of the MD method may be found in [VESELY 94] or [ALLEN 90]

2005-01-25